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8/10/2019 Lord Kelvin Volume 4
http://slidepdf.com/reader/full/lord-kelvin-volume-4 1/580 P u b l
i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
Mathematical and physical papers, by Sir William Thomson. Collected
from different scientific periodicals from May, 1841, to the present
time.
Kelvin, William Thomson, Baron, 1824-1907.
Cambridge, University Press, 1882-1911.
http://hdl.handle.net/2027/miun.aat1571.0004.001
Public Domain
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use that is made, additional rights may need to be obtained
independently of anything we can address.
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R SITYPR ESS
NE E. C .
ER
R INC ESSTREET
.
C H AU S
N A M S S O N S
a lcutta : MA C MILLA NA NDCO . LTD.
P u b l i c D o m a i n
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GENERALDYNAMICS
R A B L E
N B A R O N K E L V I N
O . L L .D . D . C . L. S C .D . M . D . . . .
R . A S S O C . I NS T IT U T E O F F R A NC E
T H E L E GI O N O F H O N O U R K T P RU S S I A N O R D ER P O U R L E M IA R IT E
H E U N I V E R SI T Y O F G L AS G O W
E TE R S C O L L E GE C A MB R I D G E
EDWITHB R IE A NNOTA TIONSB Y
R D . Sc . L L .D . S E C. R .S .
SOR O F MATHEMA TIC SINTHEUNIV ER SITYO F CA MB R IDGE
S T J O H N S C O L LE GE
R ESS
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L A Y M . A.
R ESS
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esofthisreprintthe paperswere
y f romItoC IV thosewhichhad
hev o lumeof " PapersonElectrostaticsandMagnetism ( Macmillans 1872 reprinted1884 ,
withareferencetotheir placein
inhimselfbeganthepreparationof
dmateria lw hichhadbeenstandingintype
sultimatelyprintedoff asan
9 , to thevo lumeof " B a lt imoreLectures
henumberingofthesepapersi sin
nsafterwardsmadeto V olumeIII.
hev o lumeof " B a lt imoreLectures
itherinthete torasA ppendices aconsiderablenumberof la terpapersconnectedwiththeDynamica l
erattheend ofV olumeIII.ofthe
sicalPapers ( 1890 he inserteda
nly j ustw ritten onthere lations
amicpropagation stimulatedthereto
natingcurrentsin cables andof
esinspace boththenundergo inge ploration. A lsoaconsiderablenumberof the lessabstract
edandreprintedwithoutregardto date
PopularLecturesandA ddresses
C onstitutionofMatter 1889 V o l. II.
sics 1894 V o l. III. Na igationa l
w henhewasre uestedbyLady
geof thecompletionof theco llectededit ion
o r , t h at i n c on s e u e nc e o f th e v a r i e ty o f
eprints anyattemptatcontinuing
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ofthepapersmustbe abandoned.It
thataclearerv iew of these uenceofLord
acti ity couldbeobta inedbyclassif y ingthe
numberofbroadheadings co llecting
nchronologicalorder thematerial
datthesametimema ingtherecord
ludingtitlesof otherpaperswith
heretheyhad alreadybeenrepublished.Thisprocedurehasbeencarriedbac intimefar
nne ionw ithLordK el in sow n
isearlierw or asreprintedinthe
thisco llection. Inordertosecurethe
ntinuityundervariousheadings w here
d ithasbeenthoughtad isable
adyincludedin the" B altimore
re .
edinthis rearrangementofthe
y theuseof tw o importantbibliographiesofLordK el in sw or . F orthepapersupto1884
of theR oyalSociety sC ata logueof
a ilable andfortheremainingperiod
hroughthecourtesyofP ro f . McLeod to
the continuationofthatcatalogue.
presentv o lumeandallthene t
graphyof661titlesappendedto
n sLifeofLordKel inhask indlybeen
amepurpose.Inthat listthecrossreferencestoreprintsor abstractsofthev ariouspapersare
suchisthecomple ityof themateria l
hhasbeenre uiredtoestablishthe
ariousentries.Aconsiderableproportionofthe listconsistsoftitles ofv erbalcommunications
ng madetolearnedSocieties where
yanaccidenta lpressabstract hasbeen
specti tisofcourse morecomplete
publicationswhichaloneisgi en
thetableofContents.
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pp.270-456 onW a esonW ater.
ysicalside asub ectpre-eminently
mathematicalmachinerywasformulated
-perhapsmainlyow ingtothere uirementsof thescientif icengineersw hode eloped f irstthe
andaf terw ardsthedesigningofships
ngthedrainon thepropellingenergy
ductionofw a es. LordKel in s
ac uiredasayachtsman ledhimdirectly
ef fecto fw indandcurrent inw hich
LordR ay leigh and mainlyonthe
Helmholt w hile thesearchforthe
sistanceto ships andthee perimentsofO sborneReynoldsonthedemarcationbetweensmooth
mpteddiff icult in estigationsinv iscous
mplete.Themodein whichthe
arofalimitedregular trainof
t hasfeatureswhichare importanta lso
ead anceofabeamof radiation
um w hiletheregulartra insofstanding
acurrent by f low o erasubmerged
modeofgenesiso fw a ymotionw hichmay
eorologicalatmosphericphenomena.
tionisthegraphicalrepresentations
sareduetotheRoya lSocietyofEdinburgh
of thenumerousdiagrams.
eneralDynamics pp.457-5 1
ariousfragmentarypapers beginning
nofthepartitionofthermalmolecular
ntoapplicationsofthePrinciple ofAction
odstothesub ectofperiodicorbits
micalastronomy andtothegraphical
lems.Asfollowingnaturallyonthis
e iscompletedbyabrie f sectiononElastic
2-560 w hichislitt lemorethanachrono logica ll istof t it lesofpapers mainlyofoptica landelectrical
t ionandrefle ioninordinarye lasticsolid
les w hichha ebeenrepublished
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s. Thespecia lsub ecto fpropagation
sideredinitsmoremodernelectric
e n re s er e d f or t h e ne t v o l um e .
. oftheMathematicalandPhysical
s now arrangedready forpress on
micaland GeologicalPhysics ElectrodynamicsandElectrolysis MolecularandCrystallineTheory
onicTheory withperhapssomeaddressesandothermiscellaneousscientificmatter.
nowledgemuche pertassistance
dhistas .Manyoftheproofsheets
o ug h b y Mr W . J . H A RR I SO N F e l l o w of
ge.Inthecorrectionofthe latterhalf
gilanceofMrGEO R GEGR EEN w how as
tif icsecretary forthe lateryearsofhislif e
ecia lk now ledgetobear hasensured
mallo ersights inaddit iontomore
e plicitlymentionedinfootnotes.
. O R R F . R . S. w hoseassistancew asspecially
onw iththedif f iculttopicstreatedonp. 33 0
nproofa llthesubse uentsheetsof the
a lsosuppliedmostof the listo ferratabelongingtotheearlierpart. F orgenera lad icere latingto
theEditorisunderobligationtoLordR A YLEIGH
, andtoDrJ . T . B O TTOMLEY. Inthe
rerrorsandmisprintsha ebeencorrected
a ll importantchangesha ebeenreferred
he Editor whichareenclosedin
ways totheofficialsofthe Cambridge
hee ce llenceof theirw or , andtheir
E GE C A MB R I D G E.
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PAGE
oms. . .. . . .. 1
Motion. .. . . .. 1
tryV elocityofaC ircularV orte R ing. 67
nofF reeSo lidsthroughaLi uid. 69
indandC apillarityonWa esinw ater
76
es. . .. . . . 86
cese periencedbySolidsimmersedina
9
sandRepulsionsdueto V ibration. 98
tionofR igidSo lidsinaLi uidcirculating
orationsinthemorina
01
atics. . .. . . . 115
essiona lMotionofaLi uid[ Li uid
nets[ i l lustratingV orte -Systems . . 1 5
tationa lOscil la t ionsofR otatingWater. 141
mationofC ore lessV orticesby theMotion
iscidIncompressibleF luid 149
faC o lumnarV orte . . . . 152
ityofSteadyandofPeriodicF luidMotion 166
bngInf inity inLordR ay leigh sSo lution
orte Stratum.. . 186
ragePressuredueto impulseofV orte -R ings
uresofE uil ibriumofaR otatingMassof
89
nofaLi uidw ithinanEll ipso idalHo llow 19
ityandSmallOscil la t ionofaPerfect
ightC ore lessV ortices. 202
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PAGE
ff iciencyofSa ils Windmills ScrewPropellersinWaterandA ir andA eroplanes. 205
sistanceofaF luidtoaP lanek eptmo ing
inedto itata small
nofaHeterogeneousLi uid commencing
Motionof itsB oundary. 211
rneofDiscontinuityofF luidMotion in
anceagainstaSolid
id.. . . . 215
ES.
edErrorinLaplace sTheoryof theT ides. 2 1
O scil lat ionsof theF irstSpecies in
Tides.... 248
ationofLaplace sDif ferentialE uation
R .
ryWa esinF low ingWater. . . 270
esproducedbyaSingle Impulse inWater
spersi eMedium. . 30
rontandRearofaF reeProcessionofWa es
07
a e s .. . .. . .. 3 0 7
PropagationofLaminarMotionthrougha
iscidLi uid... 3 08
otionofV iscousF luidbetw eentwo
1
aterTw o-DimensionalWa esproducedby
urbance. 33 8
rontandRearofaF reeProcessionofWa es
51
rShip-Wa es. .. . . . 368
hip-Wa es. .. . . .. 3 94
Deep-SeaWa esofThreeC lasses: ( 1 f rom
2 f romaGroupofE ua l
s ( 3 ) byaPeriodica lly
re... 419
lanationof theMac ere lS y . . . 457
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PAGE
inematica landDynamicalTheorems. 458
ormofC entrifuga lGo ernor. . . 460
w A stronomica lC loc , andaPendulum
Motion. . .. 46
bationsoftheCompassproducedby the
64
orm ofA stronomica lC loc w ithF ree
ntlyGo ernedU niform
heel.... 470
wedasPossiblyaModeofMotion. 472
saK ineticTheoryofMatter... 474
ticW or ingModeloftheMagneticCompass.......... 475
periments...... 482
tC asesfortheMa w ell-B o lt mann
utionofEnergy.. 484
eTest-casedispro ingtheMa w ell-B o lt mannDoctrineregardingDistributionofKinetic
otionofaF initeC onser ati eSystem. 497
reminP laneKineticTrigonometrysuggested
C ur atura Integra. . 51
tyofPeriodic Motion.... 515
olutionof DynamicalProblems.. 516
e eryProblemofTw oF reedomsinC onser ati eDynamicstotheDraw ingofGeodetic
enSpecif icC ur ature. 521
onof " Mercator s P ro ectionperformedby
ts.... 52
catorChartononeSheetrepresenting
lyContinuousClosed
lProblems...... 5 1
nturyCloudso ertheDynamicalTheory
1
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N.
PAGE
InductionofElectric CurrentsinSubmarineTelegraphW ires..... 5 2
ngthroughSubmarineC ables i l lustratedby
ghaModelSubmarine
MirrorGa l anometerandby
2
ustrationsoftheMagneticandthe HelicoidalRotatoryEffectsofTransparentB odieson
2
ndWa esinaStretchedUniformC hain
.... 5 3
TheoryofLight.. . . .. 5 8
sandGreen sDoctrineofE traneousF orce
F resne l sK inematicsof
8
thesisforElectro-magneticInductionof
thconse uentE uationsof
dHomogeneousSo lidMatter 5 9
erenceofElectricitywithina HomogeneousSolidConductor..... 545
icationsofF ourier sLaw ofDif fusion i l lustratedbyaDiagramofC ur esw ithA bso lute
546
ghtningConductorsattheB ritish
ionandR ef ractionofLight. . . 547
ticity andPonderableMatter. .. 547
anismfortheC onstitutionofEther. . 547
iscousLi uid E uil ibriumorMotionof
l ibriumorMotionofan
re ity " Ether" ; Mechanica lRepresentationofMagneticF orce. . 547
perimentsforComparingtheDischarge
Dif ferentB ranchesofa
o r d K e l i n a nd A l e a n de r
aTheoryofR ef raction Dispersion and
. 551
ndulatoryTheoryofC ondensationa lrarefactionalW a esinGases Li uids andSo lids
nSo lids o fElectricWa es
f Transmittingthem
isibleLight U ltra - io let
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X V
PAGE
tionandRefractionofSolitaryPlane
rfacebetweenTw oIsotropic
So lid orEther. . 551
ellmeier sDynamicalTheorytothe
oducedbySodium- apour. 551
ationofF orcew ithinaLim itedSpace
herica lSo litaryWa es or
es ofbothSpecies E ui oluminalandIrrotational inanElasticSolid. 552
producedin anInfiniteElasticSolidby
ceoccupiedby itofa
yAttractionorRepulsion. 552
of EtherforElectricityandMagnetism. 55
yingMethodforStress andStraininan
56
ctro -etherealTheoryof theV elocityofLight
a n d So l id s .. 5 6 0
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f oo t f o r m om e nt r e ad m o me n tu m , 7 , C f . L a mb s H y dr o dy n am i cs ~ ~ 1 2 9 1 0 , 1 0 2 f i rs t e . f o r r re a d T , 1 6 l i ne 8 f r om f o ot f o r di m in i sh i ng r e ad i n cr e as i ng , 1 6 , , 2 , f o r - re a d + , , 1 4 1 , , 3 , f o r pp . 9 7- 1 09 r e ad p p . 10 9 -1 1 6
, e . ( 8 ) , f o r - - d2 r ea d2 d -2 , 1 4 , l i n e1 0 fr om f oo t o m it i n af te r U s in g , 1 4 4 , , 5 f or 0 r ea d a , 1 6 , , 8 f or i = l r ea d i= 0 , 1 6 0 , , 1 2 f or 4 r ea d 2 , 1 65 f oo tn ot e r e ad [ s up ra p . 1 , , 1 7 5 l i n e7 o mi t e e r y .C f. L am b s H yd ro dy na mi cs ~ 1 64 , 1 86 s ee f oo tn ot es p . 3 3 4 , 2 5 5 e . ( 3 ) a nd ( 4 , d e le te r , 2 5 5 , , ( 6 , f o r r ea d r2 2 56 , , ( 8 , ( 9 , ( 1 1 , ( 1 2 , f o r r re ad r 2 , 3 0 5 , , ( 1 4 , f o r 22 i n la st e p re ss io n re ad 2 , 3 1 7 se . Re fe re nc e ma y be m ad e to W . M .H ic s B r i t .A ss oc . Re po rt 1 88 5
p. 517 a lsop. 9 0: a lsotosameauthor Phil.
8 , p.3 3 , on" SpiralorGyrostaticV orte
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T M S .
ya lSocietyofEdinburgh V o l. v I pp. 94-105
o l . x x x i 1 8 67 p p . 15 - 24 .
ot ' sadmirabledisco eryof the law of
rfectli u id- thatis inaf luidperfectly
orf luidf rict ion - theauthorsa idthatthis
suggeststhe ideathatHelmholt ' sringsare
rtheonlyprete tseemingto j ustif y
nofinfinitelystrongandinfinitelyrigid
ee istenceofw hichisassertedasaprobable
greatestmodernchemistsin their
rystatements isthaturgedbyLucretius
hatitseemsnecessarytoaccountfor
hingq ualitiesofdifferentk indsof
haspro edanabsolute lyuna lterable
anyportionofaperfectli u idinw hich
heca lls" Wirbe lbewegung hasbeen
rtionofa perfectli uidwhichhas
hasonerecommendationofLucretius satoms
cq uality.Togenerateortodestroy
naperfectf luidcanonlybeanactofcreati e
mdoesnote pla inanyof theproperties
1
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ngthemtothe atomitself.Thusthe
a s i t ha s b ee n w el l c al l ed h a s be e n in o e d b y
countforthe elasticityofgases.E ery
assimilarlyre uiredanassumption
gtothe atom.Itisas easy( andas
moreso toassumew hate erspecif icforces
ortionof matterwhichpossessesthe
asinaso lidindi isiblepieceofmatter and
hasnoprimafaciead antageo erthe
nificentdisplayofsmo e- rings w hich
reof witnessinginProfessorTait s
edbyonethenumberofassumptionsre uiredtoe plainthepropertiesofmatteron thehypothesisthat
orte atomsinaperfecthomogeneous
ngsw eref re uently seentoboundobli ue ly
ingv io lently f romtheef fectsof theshoc .
lartothatobser ableintwolargeindiarubberrings stri ingoneanotherinthe air.Theelasticityof
dnofurtherfromperfectionthanmightbe
- rubberringof thesameshape f rom
scosityof india - rubber. O fcoursethis
isperfecte lasticity forv orte ringsina
astasgoodabeginningasthe" clashof
eelasticity ofgases.Probablythe
ofD. B ernoull i Herapath J oule K ronig
ll onthevariousthermodynamicproperties
lltheposit i eassumptionstheyha ebeen
stomutua lforcesbetw eentw oatomsand
edby indi idua latomsormolecules satisf ied
thoutre uiringanyotherproperty inthe
posesthemthaninertiaandincompressibleoccupationofspace.A fullmathematicalin estigation
weentw ovorte ringsofanygi en
espassingoneanotherinany twolines
ercomeneareroneanotherthana
meterofe ither isaperfectly so l able
andtheno eltyof thecircumstances
fficultiesofane citingcharacter.Its
undationoftheproposednew k inetic
bilityoffoundinga theoryofelastic
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M S
dynamicsofmoreclose ly-pac edv orte
anticipated.Itmayberemar edin
icipation thatthemeretit leo fR an ine s
ortices communicatedtotheRoya l
849and1850 w asamostsuggesti e
swereshowntothe Societyto
ittedv orte atoms theendlessvarietyof
ansuff icienttoe pla inthev arieties
nsimplebodiesandtheirmutualaffinities.
ttworingatomslin edtogetherorone
w ithitsendsmeeting constituteasystem
aybealteredinshape canne erde iatef rom
plecontinuity itbeingimpossiblefor
orte motiontogothroughthe lineof
motionoranyotherpartof itsownline.
orte core islitera lly indi isiblebyany
rte motion.
ntoa v eryimportantpropertyof
ithreferencetothenowcelebratedspectrumanalysispracticallyestablishedbythedisco eriesandlaboursof
. Thedynamica ltheoryof thissub ect
adtaughtto theauthorofthepresent
852 andwhichhehastaught inhis
sityofGlasgow f romthattimeforw ard
econstitutionofsimple bodiesshould
amentalperiodsofv ibration ashasa
ormorestrings oranelasticsolid
tuning-for srigidlyconnected.To
ntheLucretiusatom isatoncetogi e
andelasticity forthee planationofw hich
tebodies theatomicconstitutionw as
hen thehypothesiso fatomsandv acuum
hisfollowersto benecessarytoaccount
mpressibilityoftangiblesolidsand fluids
twouldbenecessarythatthe moleculeof
shouldbenotanatom butagroupofatoms
nthem.Suchamoleculecould notbe
dthusitlosestheonerecommendation
egreeofacceptance ithashadamong
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sthee perimentsshow ntotheSociety
atomhasperfectlydefinite fundamenta l
dependingsole lyonthatmotionthee istence
disco eryofthesefundamentalmodes
tingproblemofpuremathematics.
o lt ring theanaly ticaldif f icult ieswhich
formidablecharacter butcerta inly far
sentstateof mathematicalscience.
ommunicationhadnot attempted
ute ceptforaninf inite ly long stra ight
orthiscasehew asw or ingoutso lutions
possibledescriptionofinfinitesimalv ibration andintendedtoincludethemin amathematicalpaperwhich
ocommunicatetotheRoyalSociety.
whichhe couldnowstateisthe following.
enw ithitssectiondif feringf rome act
tesimalharmonicde iationoforderi.
a esroundthea isofthecylinder in
vorte rotation w ithanangularv e locity
heangularve locityof thisrotation. Hence as
w holecircumference ise ua lto i f o ran
rderithereare i-1periodsofv ibration
ionof thevorte . F orthecase i= 1
andthesolutione pressesmerelyaninf initesimallydisplacedv orte w ithitscircularformunchanged.
dstoellipticdeformationofthe circular
eriodofv ibrationis therefore simply
. Theseresultsare o fcourse applicable
henthediameterof theappro imately
comparisonwiththediameterofthe
o e- ringse hibitedtotheSociety . The
esof thetw ok indsof trans ersev ibrationsofaring suchasthev ibrationsthatw ereseeninthe
bemuchgra erthantheell ipticv ibrationof
thev ibrationswhichconstitutethe
apourareanalogoustothosewhichthe
bited andit isthereforeprobablethatthe
rotationof theatomsofsodium- apouris
illionthofthemillionthof asecond
ey theperiodofv ibrationof theye llow
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M S
inasmuchasthislightconsistso f tw o
istentinslightlydif ferentperiods e ua l
mej uststated andofasnearlyascan
tensit ies thesodiumatommustha etw o
bration ha ingthoseforthe irrespecti e
ute ua llye citablebysuchforcesasthe
ncandescentv apour.Thislastcondition
twofundamentalmodesconcerned
ar( andnotmere lydifferentordersof
oconcurv erynearlyintheirperiods of
pro imatelycircularanduniformdis o f
ntalmodesof trans ersev ibration w ith
adrants fulf ilboththecondit ions. Inan
anduniformringofe lasticso lidthesecondit ionsarefulfi l ledforthef le ura lv ibrationsinitsplane anda lso
tionsperpendicularto itsow nplane. B ut
g if createdw ithonepartsomew hatthic er
tremainso butw oulde perience longitudina lv ibrationsrounditsow ncircumference andcouldnot
mentalmodesofv ibrationsimilarin
atelye ua linperiod. Thesameassertion
bepractica llye tendedtoanyatomconsistingofasinglev orte ring how e erin o l ed asil lustrated
wnto theSocietywhichconsistedof
edinvariousw ays. Itseems therefore
tommaynotconsist ofasinglev orte
probablyconsisto f tw oappro imatelye ua l
hroughoneanotherli e tw o lin sofacha in.
uitecerta inthatav apourconsistingofsuch
o lumesandangularve locit iesinthetw o
uldactprecise lyasincandescentsodium apouracts- thatistosay w ouldfulf i lthe" spectrumtest for
geoftemperatureon thefundamentalmodescannotbepronounceduponwithoutmathematical
toe ecuted andthereforew ecannotsay
867. -Theauthorhasseenreasonforbelie ingthatthe
htbereali edbyacertain configurationofasingleline
bedescribedinthemathematica lpaperw hichheintendsto
ty.
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M S
isthroughthecentreof thering for
esidethe lineofmotionof - theringsees as
heposit ionofhiseye acon e 4outline
e inf ronto f thering. Thiscon e
dingsurfacebetweentheq uantityof
forw ardwiththeringinitsmotionand
ieldsto letitpass.It isnotsoeasy
ondingcon e outlinebehindthering
smo eisgenerallyleftin therear.In
gsurface oftheportioncarriedforward
itesymmetricalontheanteriorand
eplane ofthering.Themotionof
beprecisely thesameasitwould beif
e wereoccupiedbyasmoothsolid
itis inastateof rapidmotion
ara isofthering withincreasing
nearerandnearertothering itself.The
lmotionmaybe imaginedthus:-Leta
ber o fcircularsection w ithadiameter
ength bebentintoacircle anditstwo
thersothat itmayk eepthecircular
lettheapertureofthe ringbeclosedby
tanimpulsi epressurebeapplieda ll
ensity sodistributedastoproducethedef inite
cifiedasfollows andinstantlythereafter
ed. Thismotionis inaccordancewith
w s tobea longthosecur esw hichw ouldbe
inplaceof the india -rubbercircle w eresubstitutedaringe lectromagnetS andtheve locit iesatdif ferentpo ints
ntsprecise ly thecon e outlinereferredto andthe lines
luidcarrieda longby thev orte forthecaseofadouble
twoinf inite ly long para lle l stra ightv orticesofe ua lrotations
ecur esha ebeendraw nbyMrD. M F arlane f rom
erformedbymeansof thee uationofthesystemof
x + b
w h e r el o g N = a
a
nthemathematicalpaperwhichthe authorintendsto
eto theRoyalSocietyofEdinburgh.
arconductorwitha currentofelectricitymaintained
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heintensitiesofthe magneticforcesin
fthemagneticfield.The motion as
w illf ulf i lthisdef inition andw illcontinue
ingv e locit iesate erypo into f thef ilm
anebe inproportiontotheintensities
ecorrespondingpointsofthemagnetic
mo edperpendiculartoitsown plane
otionofthefluidthroughthe middle
ocityverysmallincomparisonw iththat
f thering. A largeappro imately
willbecarriedforwardwiththe ring.
ringbeincreased thevo lumeof f luid
minishedine erydiameter butmostin
tdiameter anditsshapew illthusbecome
asingthev elocityofthering forward
blatenessw ill increase until insteadofbe ing
il lbeconca ebeforeandbehind roundthe
theforw ardve locityof theringbe
te ua ltotheve locityof thefluidthrough
hea ia lsectionof theoutlineof the
ardwill becomealemniscate.Ifthe
orward theportionofit carriedwith
be itselfannular and re lati e ly tothe
f luidw illbebac w ardsthroughthecentre.
eportionof fluidcarriedforwardand
symmetrica l bothre lati e ly tothe
thetwosidesof thee uatoria lplane. A ny
nthusdescribedmightofcoursebe
derdescribed orby f irstgi ingav e locity
gthefluidin motionbyaidofan
byapply ingthetw oinit ia ti eactions
amountof the impulsere uired or
f fecti emomentumof themotion or
hemotion isthesumof theintegral
ontheringandonthefilmre uiredto
etwocomponentsofthewholemotion.
sthediameterof theringisv erysmall
ameterof thecirculara is the impulse
ysmallincomparisonwiththe impulse
e locitygi entotheringismuchgreater
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M S
entra lpartso f thef ilm. Hence unless
theringissov erygreatastoreducethe
d forwardwithittosomethingnotincomparablygreaterthanthe v olumeofthesolidring itself the
lconf igurationsofmotionsw eha ebeen
bybutinsensibleq uantitiesthemomentum
Theva lueof thismomentumiseasily
onof Green sformula.Thusthe
ortionoffluid carriedforward( being
of thesamedensitymo ingw iththe
getherw ithane ui a lentforthe inertiao f the
s isappro imately thesameina llthese
oaGreen sintegra le pressingthew hole
Thee ualityo f theef fecti emomentumfordif ferentve locit iesof theringiseasilyv erif iedw ithout
otsogreatastocausesensiblede iations
portionoffluidcarriedforward.Thus
of thea iso f theportionof thef luid
nedbyfindingthe pointinthea isof
ocity ise ua ltotheve locityof thering.
heplaneof theringthatv e locityv aries
of aninfinitesimalmagnetonapoint
lyasthecubeof thedistancef romthecentre .
usoftheappro imatelyglobular
simplein erseproportiontothe
andtherefore itsmomentumisconstantfor
ering. Tothismustbeadded asw as
uantitye ua ltoha lf itsow namount asan
rtiaof thee ternalf luid andthesumis
mentumofthemotion.Henceweseenot
emomentumisindependentofthe
butthatitsamountisthesameasthe
orrespondingringelectromagnet.The
btainedbytheGreen sintegralreferred
uste plainedisnotconfinedtothe
ngspecia lly re ferredto butise ua lly
ngs ofanyform detachedfromone
roughoneanotherinanyw ay ortoasingle
reeandq ualityo f " mult iplecontinuity "
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ysoastoha enoend. Ine erypossible
hef luidate erypo int w hetherof the
luidf i ll inga llspaceroundit isperfectly
' sformulhew hentheshapeof thecore
ticin estigationnow e pla inedpro es
ntumofthewholefluidmotionagreesin
iththemagneticmomentofthe correspondingelectromagnet.Hence stillconsideringforsimplicity
o fcore le tthislinebepro ectedon
htanglestooneanother.Theareas
bta ined(toberec onedaccordingto
autotomic astheywillgenera llybe are
ntumperpendiculartothesethreeplanes.
sultwill beagoode erciseon" multiple
uthorisnotyetsuf f icientlyac ua intedw ith
leresearchesonthisbranchofanaly tical
therornota llthek indsof " multiple
tedareincludedinhis classificationand
in estigationinwhichathin solid
emo inginanydirectionthrougha
e motionpre iouslye citedinit re uires
te erypo inttobe inf inite ly smallin
sofcur atureof itsa isandw iththe
ny otherpartoftheci rcuitfromthat
entheeffecti emomeno f thew hole
ndforav orte w ithinfinitely thincore
mberofsuchvortices how e ernearone
edsimultaneously andthew holeef fecti e
anddirectionwillbethe resultantof
ntcomponentv orticeseachestimated
etheremar ableproposit ionthatthe
anypossiblemotioninan infiniteincompressiblefluidagreesindirectionandmagnitudewiththe magnetic
ingelectromagnetinHelmholt ' stheory.
ethemathematicalformulaee pressing
mentinthemoredeta iledpaper w hichhe
eto laybeforetheR oyalSociety .
rstoanyone eitherobser ingthe
ingsorin estigatingthetheory -What
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M S
si eof theringinanycase Helmholt ' sin estigationpro esthattheangularv orte v e locityof the
tslength orin erse lyasitssectiona larea .
electriccurrentinthe electromagnet
tely thinvorte core remainsconstant
thmaybealteredinthecourseof the
periencesbythemotionofthe fluid.
the largerthediameterof theringfor
trengthofvorte motionsinanordinary
reateristhew holek ineticenergyof the
sthemomentum andw ethereforesee
lmholt ringaredeterminatewhen
hof thev orte motionaregi en and
ineticenergyorthemomentumof thew hole
ceif afteranynumberofcollisions
lt ringescapestoagreatdistancef rom
ornearly f ree f romv ibrations itsdiameter
edordim inishedaccordingasithasta en
nenergy to theothers. A fulltheoryof the
omsbyele ationof temperature istobe
rinciple .
fe hibitingsmo e-ringsisasfo llow s: A largerectangularbo openatoneside hasacircularho leof6
ntheoppositeside.A commonrough
tcube orthereabout w il lansw erthepurpose
eof thebo isclosedbyastouttowelor
eet ofindia-rubberstretchedacrossit.
sidecausesacircularvorte ringtoshoot
therside . Thevorte ringsthus
hebo isf i l ledw ithsmo e. Oneof the
ofdoingthisistousetw oretortsw ith
o lesmadeforthepurpose inoneof the
llquantityofmuriaticacidisputinto
ndofstrongli uidammonia intotheother.
romtime totimetooneor otherof
cloudofsal-ammoniacisreadilymaintained
A curiousandinterestinge periment
esthusarranged andplacedeither
notherorfacingone anothersoasto
meetingf rom oppositedirections-orin
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NA MICS [ 1
tions soastogi esmo e-ringsproceedingin
erat anyangle andpassingone
nces.Aninterestingv ariationofthe
debyusingcleara irw ithoutsmo einone
siblev orte ringspro ectedf romitrender
ysensiblewhentheycomenearanyof
dingf romtheotherbo .
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T IO N .
oya lSocietyofEdinburgh V o l. xx v . 1869
pril 1867.
daugmented28thA ugustto12thNo ember 1868.
ofthepresentpaperhasbeen
hypothesis thatspaceiscontinuously
siblefrictionlessli uidactedonbyno
lphenomenaofe eryk inddependso le lyon
uid. B utIta e inthef irstplace as
on a f initemassof incompressible frict ionless f luidcomplete lyenclosedinarigidf i edboundary .
elmaybeeithersimplyor multiply
uentlyconsidersolidssurrounded
w hicha lsomaybeeithersimplyormultiplycontinuous. Itw il lnotbenecessary toe cludethesupposit ionthat
theouterboundaryo ersomefinite
notsurroundedby the li uid buteach
rroundedby the li uidornot andw hether
mustbeconsideredasaparto f thew hole
egi enatrest andletnoforce
heconta iningv esse l orf romthesurfacesof
eractonanyparto f it . Lettherebe
erfectly incompressible andof thesame
te itherperfectly rigid ormoreorless
efinedasamass continuouslyoccupyingspace whose
ononeanothere erywheree actlyinthedirection
ceseparatingthem.
egralederhydrodynamischenGleichungen Iwelcheden
prechen C re lle ( 1858 translatedbyTa it inPhil. Mag.
dt eausderA nalysissitus & amp c. C re lle (1857 . Seea lso
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ctorimperfecte lasticity. Someof thesemay
oserigidity andbecomeperfectly li u id
dmaybesupposedtoac uirerigidity
s.Let thesolidsacton oneanother
res f rictions ormutua ldistantactions
of " actionandreaction. Letmotions
andinthe li uid e itherby thenatura l
sor bythearbitraryapplicationof
elimited time.Itisof noconse uence
ha ereactionsonmatteroutsidethe
thattheymightbeca lled" natura lforces in
e ( whichadmitsactionandreactionat a
edarbitrarilyby supernaturalactionwithout
mlocution and atthesametime toconformtoacommonusage w esha llca llthemimpressedforces.
usnessastodensityofthe contentsof
sse l itf o llow sthatthecentreof inertiaof
idandsolidsimmersedini tremainsat
the integra lmomentumof themotionis
nandTait sNaturalPhilosophy ~ 297 the
fthecomponentsofpressureonthe
ra lle ltoany f i edline ise ua ltothetimeintegra lo f thesumof thecomponentsof impressedforcespara lle l
ua litye ists o fcourse ateachinstant
mpressedforces andcontinuestoe ist
o f the irt imeintegra ls a f tertheyha e
bse uentmotionof theso lids andof the
othem w hate erpressuremaycometo
ssel w hetherf romthef luidorf romsome
ntactwithit thecomponentsofthis
y f i edline summedfore erye lementof
esse l mustvanishfore ery inter a lo f
ressedforcesact. If f o re ample one
conta iningv esse l therewillbeanimpulsi epressureof thef luido era lltheresto f thef i edcontaining
sumof itscomponentspara lleltoany line
othe correspondingcomponentof the
ntrarytodesignatemerelydirectionalopposition and
dw ordopposite tosignifycontraryandinoneline .
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T I O N
solidonthepartof thissurfacewhich
a n d co n si d er o b li u e i mp u ls e o f an i n ne r m o i n g
lidsphericalboundary . B ut a f terthe impressedforcesceasetoact andaslongasthecontainingvesse lis
so lids the integra lamountof thecomponentof f luidpressureonit paralle ltoany line v anishes.
dtostopthewholemotionof fluid
2 is done if theso lidsarebroughttorestby
esonly thetimeintegra lso f thesums
eforces paralle ltoanystatedlines may
e ualandcontrary tothetimeintegrals
sofcomponentsof the initia tingimpressedforces( ~ 3 ) . B utw esha llsee( ~~ 19 21 thatif the
f inite ly large anda llo f themo ingso lidsbe
ingthe wholemotion theremustbe
nq uestionbetweenthetimeintegra ls
arydirectionsof theinitiatingand
s buttheremustbe( ~ 21 complete ly
betweenthetwosystems.
on henceforthIshallusetheun ualifiedtermimpulsetosignifyasystemof impulsi e forces to
narigid body.Thusthemostgeneral
toanimpulsi e force andcouple inplane
ordingtoPo insot ortotw oimpulsi e
g accordingtohispredecessors.F urther
pulseofthemotionatanyinstant in
hesystemof impulsi e forcesonthemo eable
teitfromrest oranyothersystem
alenttothatone if theso lidswerea llrigid
honeanother as forinstance the
eforceandminimumcouple.Theline
eforcew illbeca lledtheresultanta is
emomentof theminimumcouple( w hose
is line willbecalledthe rotational
sdef inedthetermsIintendtouse I
errorsthatmightbefalleninto remar
wholemotionsofsolids andli uidis
e d e fi n ed a s t he i m pu l se b u t ( ~ 4 i s e u a l
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rce-resultanto f " the impulse andthe
ertedonthe li uidby theconta iningvesse l
emotion:and thatthemomentof
motionroundthecentreof inertiaofthe
snote ua ltotherotationa lmoment asI
se ua ltothemomentof thecoupleconstitutedby" the impulse andthe impulsi epressureof the
e li uid. Itmustbeborne inmindthat
owe erdistanta llroundf romthemo eable
esse lmaybe ite ercisesaf inite inf luence
mentofmomentumofthewholemotion
nte ly large andinfinitelydistanta ll
tdoessoby infinitelyslowmotionthrough
f luid ande ercisesnof inite inf luence
solidsorof theneighbouringfluid.
ood ifforaninstant wesupposethe
tobenotf i ed butq uite f reetomo eas
Themomentumofthe wholemotion
bute actlye ua ltotheforce- resultanto f
andthemomentofmomentumofthe
ntreofinertia willbepreciselye ual
ecouplefoundbytransposingthe constituentimpulsi eforcestothis pointafterthemannerofPoinsot.
the immersedsolids andof thefluidin
hweshallcallthe fieldofmotion will
edifference whetherthecontaining
rle f tf ree pro ideditbe infinitelydistant
therefore essentia lly indif ferent
edorletitbef ree. Theformersuppositionismorecon enientinsomerespects the latterinothers but
ntto lea eanyambiguity andIsha ll
e formerina llthatfo llow s.
impulseofthemotion andits
ceandcouple accordingtothepre ious
shedfromthemomentum andthemoment
w holecontentsof thev esse l le tthevesse l
epressureonthe li uidw illa lw aysbe
nt inalinethrough itscentre which
a landcontrary totheforce- resultanto f " the
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T I O N
erefore w ithitw illconstitute ingenera la
ofthiscoupleandthecouple-resultantof
ua ltothemomentofmomentumof the
ntreofthe sphere( whichisthecentre
esse lbe inf inite ly large andinf inite ly
emo eablesolids themomentof
motionisirre le ant andw hatis
sthe impulseanditsforceandcoupleresultants asdef inedabo e.
a ti n g ( ~ ~ 1 0 1 2 , a n d pr o i n g
5 , a fundamenta lproposit ioninf luidmotionw illbe
of the impulse w hetherof themo eable
consideredorofv ortices.
ntumofe erysphericalportionof
re lati e ly tothecentreof thesphere is
oatanyoneinstantfore eryspherica lportion
sf irsttoberemar ed thatthemoment
tof the li uidw hichatany instant
spherica lspacecane periencenochange
teofchangevanishesatthatinstant ,
eonit( ~ 1 , be ingperpendicularto its
reprecise ly tow ardsitscentre . Hence if the
thematterinthe fi edsphericalspace
y themomentofmomentumof thematter
g e actlythatofthematter which
e la t er ( ~ ~ 2 0 1 7 1 8 t h at t h is b a la n ci n g is
e itheramo ingso lid oro f someof the
ofw hichspherica lportionspossessmoment
efi edspherica lspace butit isperfect
10 asw illbepro edin~ 15.
ethefo llow ingpure lymathematical
narynotationu v , w forthecomponentsof
nt ( x , y z ) .
di t io n ( l a s t c l au s e o f ~ 1 0 r e u i re s t ha t
b e a c om p le t e di f fe r en t ia l , a t w ha t e e r i ns t an t
artofthe fluidtheconditionholds.
I b e li e e f i rs t p ro e d b y St o e s i n hi s p a pe r " O n t h e
Motion andtheE uil ibriumandMotionofElasticSo lids "
Transactions 14thApril 1845.
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v d y + w d b e a c om p le t e di f fe r en t ia l
onofx , y z , throughany finitespaceof
t thecondit ionof~ 10ho ldsthroughthat
e s p r oo f of L e mm a ( 1 : - i r st f o r
w hethersub ecttothecondit ionof~ 10or
nentmomentofmomentumroundOX of
hitscentre at0.Denotingbyfff
pacew eha e
d d yd . .. .. .. .. .. .. .. .. 1 .
( d w /d y o & a m p c . d en o te t h e v a l u es a t 0 o f th e
eha e byMaclaurin stheorem
d j '
ememberingthat( dw / d ) o & amp c. areconstantsforthespacethroughw hichthe integrationisperformed
. y d w
- dy y2 d d y d + - f f y d d yd
d 0
pleintegralsv anish becausee ery
ussphere isaprincipa la is andifA
ntumofthesphericalv olumeroundits
hesecond
inthee pressionforL w ef ind
.. . .. . . .. . 2 .
ccordingtotheconditionof~ 10 and
of theinfinitesimalspherenowconsidered
nt ofspacethroughwhichthiscondition
emustha e throughoutthatspace
. . 3 ) ;
1 .
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T I O N
2 , l et
d ( 4 ;
onentmomentofmomentumround
n y sp h er i ca l s pa c e wi t h 0 in c e nt r e. W e h a e [ ( 1
- v z ) . .. . .. . .. . .. . .. . . ( 5 ,
oughthisspace( notnowinfinitesimal .
d y - d. p. ( 6 ;
d - d = . . .. . .. . .. . .. . . 6 ;
onwithreferenceto- inthesystem
I r s u ch t h at
sin. . . .. . . .. . . .. . . .. . . 7 .
5 to thissystemofco-ordinates w eha e
. . .. . . .. . . .. 8 .
ce isspherical w iththeoriginofcoordinatesinitscentre w emaydi ide it into inf initesimalcircular
s ha ingeachfornormalsectionan
hd anddpforsides.Integrating
erings w eha e
ause( isa single- a luedfunctionof thecoordinates. HenceL= 0 w hichpro esLemma( 2 .
ynamicalproposition statedat
forthepromisedproof le tR denotethe
tyof thefluidacrossanyelement do o f
i tu a te d a t ( x , y z ) ; a n d le t u v , w b e t h e
esultantve locityatthispoint sothat
. . .. . . .. . . .. . . .. . . . 9 .
ingthehollowsphericalspaceacross
edtisR d. dt andthemomentof
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ngmassroundthecentrehas forcomponent
omponentofthemomentofmomentum
e sphericalsurfaceatanyinstant t
R d o . .. . .. . .. . .. . .. . .. 0 .
1 o f ~ 1 2 a n d th e n ot a ti o n of ~ 1 4 w e ha e
ariationperunitlengthperpendicular
thatisdifferentiationwithreferenceto r
beingdirectionalrelati elytothe
rd i na r y po l ar c o -o r di n at e s r 0 * , w e h a e
Od. . . . .. . . .. . . .. . ( 11 .
f continuity foranincompressible li uid
f o r e e r y po i nt w i th i n th e s ph e ri c al s p ac e a n d
dTait A pp. B ]
amp c. . .. . . .. . . .. . . .. . . . 12 ,
w h er e S o de n ot e s a co n st a nt a n d S1 S 2 & a m p c .
rdersindicated.
2 rS - 3 r 2 + & a m p c . .. . .. . .. . .. . 1 ) .
thesisofthe mostgeneralsurface
sect a l andtesseralharmonics[ Thomson
thatdSi/ drisasurfaceharmonicof thesame
w aysunderstandd2/ d 2+ d2/ dy2+ d2/ d 2.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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wasstatedabo e( ~ 5 . Topro ethis
Inow proceed.
surfacestobe describedrounda
simmersedina li uid. Thesurrounding
1 perpendicularly andthereforew henany
eneratedby impulsi e forcesappliedtotheso lids
meterofthemomentumofthe matter
ceatthef irst instant mustbee actly
those impulsi e forcesroundthisline .
sline ofthemomentumofthematter
woconcentric sphericalsurfacesisz ero
nyso lid andpro idedthat if thereare
no impulseactsonthem.
w hatw eha edef inedas" the impulse
6 , w eseethatitsmomentroundany line is
momentumroundthesameline o fa llthe
alsurfaceha ingitscentrein thisline
rto whichanyconstituentofthe
still hold thoughthereareother
rhood andimpulsesareappliedto
mentsofmomentumofthoseonlywhich
ntoaccount andpro idednoneof themis
11 regardingf luidoccupy ingatany
lsurface areapplicablew ithoutchangeto
ingthe spaceboundedbyS becauseof
hatnosolidiscutbyS. Hencee ery
15 asfarase uation( 11 , maybenow
hnS butinsteadof ( 12 w enow ha e
7 6 , i f we d en ot e by T 1 T 2 & a m p c . a no th er
monics
+ & a mp c .
a mp c . ' . . . .. . .
reatestand smallestsphericalsurface
ha ingnoso lidsinit becausethroughall
becausethisw ouldgi e inthe integra lo f f low across
e afiniteamountofflowout oforintothe space
rationordestructionofmatter.
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T I O N
ndthee uationofcontinuitypro ethat
te ad o f ( 1 ) , w e no wh a e
* * * * * ( 1 5 .
mp c .
d s in .. . ( 1 6 .
d r
5 ne itheranymo eableso lids norany
stwithinanyfinitedistanceof Sall
amp c. musteachbeinf inite ly small: andtherefore
0. Thispro esthepropositionassertedin~ 5:
escannotha ez eromomentrounde ery
iteportionofspace w ithoutha ing
-resultanteache ualtoz ero.
o lidshasnotbeenta eninto
hemmaybeli uef ied( ~ 3) w ithoutv io lating
19. Tosa ecircumlocutions Inow def ine
of f luidha inganymotionthatitcouldnot
transmittedthroughitselffrom its
e ly forbre ity Isha llusethee pression
so lidoravorte , oragroupofsolidsor
o edmaybenowstatedin terms
w hichwerenotusedin~ 5 andsobecomessimply this: -The impulseof themotionofaso lidorgroup
dthesurroundingli uidremainsconstantas
sufferedfromtheinfluenceofothersolids
conta iningv esse l.
~ 6 , thatthemagnitudesofthe
ationalmomentoftheimpulseremain
tionof itsa isin ariable .
mof thestaticsofarigidbodyw emay
ceand couplealongandroundthe
a lresultantforcea longtheparalle ll ine
agreatercouplethe resultantofthe
couple andacouple intheplaneof thetw o
momente ua ltotheproductof the irdistance
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we maypassfromtheforce-resultant
heimpulsealongand rounditsa is
antandgreatermomentof impulse by
a ny p oi n t Q , n o t in t h e a i s ( ~ 6 o f t he
entis( ~ 18 e ua ltothemomentof
ntQ , ofthemotionwithinanyspherical
ascentre w hichenclosesa llthev ortices
lidsorv orticesw hicha lwaysk eep
of f initeradius orasinglebody mo ing
anha enopermanenta eragemotionof
obli uetothedirectionof theforceresultantoftheimpulse ifthereisa finiteforce-resultant.F or
sphericalsurfaceenclosingthemo ing
ha emomentofmomentumroundthe
y.
motionof translationw hentheforceresultanto f the impulsev anishes andtherew illbe fore ample
hapedli e thescrew-prope llero fasteamer
omogeneousli uid andsetinmotionby
ndiculartothea iso f thescrew .
sultantofthe impulsedoesnot
enomotionof translation ortheremaye en
tionoppositeto it. Thus fore ample
motion established( ~ 6 ) throughit w il l
a trest. A ndifatany timeurgedbyan
n thelineofthe force-resultantofthe
n itwillcommenceandcontinue
agemotionof translationinthatdirection a
m andthesameasif therewereno
ringis symmetrical.Ifthetranslatory
cyclicimpulse butlessinmagnitude
aryto thewholeforce-resultant
e ualandoppositetothe cyclic
nslationwithz eroforce-resultantimpulse-anothere ampleofwhatisassertedin ~ 24.Inthiscase
mmetrical orofanyothershape such
w hi c h t o f i i d ea s w e h a e s u p p os e d gi e n
P u b l i c D o m a i n
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T I O N
antpressurein allsensiblyq uiescent
disappearsfrompineach ofthe
0 b e ca u se a s o li d i s e u i li b ra t ed b y e u a l
hetimeintegra l( 2 , w eha e
. 6 ;
( LMN denotethechangesintheforceandcouple-componentsof the impulseproducedby theco ll ision
f 2 dt , Y = & a mp c . Z = & a m p c .
( ) + f 2 dt , M l= & a m p c . N = & a mp c . .. .. .. .. 7 .
uiescentintheneighbourhoodofthe
o ingbodyorgroupofbodiesisinfinitely
sthatbeforethecommencementand
onw eha ef= 0 ate erypo into f the
dy. Hence fore eryva lueof trepresenting
nof theco ll ision theprecedinge pressionsbecome
dt Y = & a m p c . Z = & a m p c .
f d t M = & a mp c . N = & a mp c .' " ) ;
ntegra lchangeof impulsee periencedbya
inpassingbesideaf i edbodywithout
egardedasasystemof impulsi eattractions
erywhere inthedirectionof thenormal and
runito farea. B utitmustnotbeforgottenthattheterm binthee pression[ ~ 3 1( 5 ] forpproduces
, aninf luenceduringthecoll ision the integra l
pearsf romthee pression[ ~ 3 2( 7 ] for
sioniscompleted thatis( ~ 29 a f terthe
sedawaysofarasto lea enosensible f luid
odofthefi edbody.
m ~ 2 , w e s ee t h at w h en t h er e i s no
ga instthef i edbody andw henthe
fsolidspassesaltogetherononeside of
irectionof thetranslationw illbedeflected as
hole anattractiontow ardsthef i edbody
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ccordingas( ~ 25 thetranslationisin
seoroppositeto it. F or ineach
redby theintroductionofanimpulse
uponthemo ingbodyorbodiesasthey
) thetranslationbeforeandaf tertheco llisionis
impulse andisalteredin direction
silyunderstoodfromthe diagrams
presentsthef i edbody thedottedline
dsII , thedirectionsof theforce- resultant
i etimes andthefulla rrow-headsTT ,
tion.
A I S
2.
o theclassillustratedbyfig. 1.
withcyclicmotion( ~ 25 established
stotheclassil lustratedby f ig. 2 if the
ghthefluidin thedirectionperpendicular
ntrarytothecyclic motionthroughits
w esubstitutev orticesforthemo ingso lids
thatthetranslationisprobablyalw aysinthe
. Hence asil lustratedby f ig. 1 there is
sifbyattraction w henagroupofv ortices
i edbody. Thisiseasilyobser ed fora
bysendingsmo e-ringsona largesca le
t splan insuchdirectionsastopass
f i edsurface. A nordinary12- inchglobe
dhungbyathincord answersveryw ell
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T I O N
o f ~ 3 0 3 1 3 2 i s c l e a rl y a pp l ic a bl e t o
g b o dy o r t o a gr o up o f v o r t i ce s o r mo i n g
lwaysnearoneanother( ~ 2 ) , passingneara
i edboundary andbeing beforeandaf ter
a tav erygreatdistancef rome erypartof the
Helmholt ring pro ectedsoastopass
eof twow alls showsadef lectionof its
attractiontowardsthecorner.
force- resultanto f the impulse is asw e
37 , determinatew henthef low of the li uid
anysurfacecompletelyenclosingthesolids
b u t no t s o f r om s u ch d a ta e i th e r th e a i s
nalmoment asw eseeatoncebyconsidering
whichmayafterwardsbesupposed
tonbyaforce inany linenotthroughthe
inaplaneperpendicularto it. F orthisline
andthe impulsi ecouplew illbetherotationa l
tonof thesolidandli uid. B utthe
w i ll m o e e a c tl y a s it w o ul d i f th e i mp u ls e
eforceofe ua lamountinaparalle ll ine
sphere withthereforethissecondline
ro forrotationa lmoment. F oril lustrationof
nnglatentina li uid( w ithorw ithout
festbyactions tendingtoa lteritsa is or
uga lforcedueto it see~ 66andothers
pulse inanydirectionise ua ltothe
tmomentumofthemassenclosedwithin
ngalltheplacesof applicationofthe
iththato f the impulsi epressureoutwardson
matterenclosedbyS( w hethera ll l i u id
ly so lid iso funiformdensity its
a lto itsmassmultipliedintotheve locity
ofthespacewithin thesurfaceSsupposed
ea lw aysthesamematter andw illtherefore
malmotionofS thatistosay onthe
elocityinthedirectionof thenormalat
he impulsi e f luidpressure corresponding
ctualmotionfromrest beingthetime
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R T EX M O T IO N I l
dff
d y d
.
theothertw otermsof ( 3 oncesimply
ThomsonandTait A pp. A ( a ] toasurface
( N - c os a d o. .. .. .. .. 4 ;
sition anda lso o fcourse thatif therebe
va lueof thesecondmemberisz ero .
ingthemagneticandhydro inetic
esoleconditionthatat e erypointof
facethemagneticpotentialis e ualto
weconcludethat47rtimesthe magnetic
smwithinanysurface inthemagnetic
otheforce- resultanto f the impulseof theso lids
orrespondingsurface inthehydro inetic
irectionsof themagnetica isandof the
lsearethesame.F orthetheoryof
stingtoremar thatindeterminatedistributionsofmagnetismwithinthesolids orportionsoffluidtowhich
) w ereapplied ordeterminatedistributions
he irsurfaces maybefound w hich
ternaltothemshallproducethe same
-potential andthereforethesamedistributionofforceas thedistributionofv elocitythroughthewhole
henthemagneticforce intheinterior
hemannere pla inedin~ 48( 2 o fmy
agnetism itise pressiblethrougha ll
oef ficientsofapotentia l and onthe
eticsystemud + v dy+ w d isnotacomplete
ughthespacesoccupiedbythesolids
esultantforceandresultantflowholds
teriortothemagnetsandsolids inthe
ystemsrespecti e ly . B utif theother
ewithina magnet[ Math.Theoryof
1 orThomson sElectrica lPapers Macmillan 1869.
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f oot-note and~ 78 , publishedinpreparation
n Electro -magnets. ( st i l l inmyhandsin
completed , andw hicha lonecanbeadopted
-magneticmattertra ersedbyelectric
forcehasnota potentialwithinsuch
ee( ~ 68 thatdeterminatedistributionsof
oughspacescorrespondingto thesolids
emcanbefoundwhichshallgi e fore ery
rtra ersedbyelectriccurrentsornot a
agreeinginmagnitudeanddirection
etherofso lidorf luid atthecorresponding
csystem.Thisthoroughagreementfor
o-magneticanaloguepreferabletothe
ingbegunw iththemagneticana logoussystem
enienceforthedemonstrationof~ 3 8 w e
ethepurelyeloctro-magneticanalogue.
a us e d in a n ti c ip a ti o n i n ~ 3 7 ( 1 w e
4 2 4 ) f i nd t h e mo me n tu m of t h e wh o le m at t er
ore enso lidalone-atany instantw ithin
ermsof thenormalcomponentve locityof
f thissurface or w hichisthesame the
surface itse lf ifw esuppose ittovarysoas
ematter.
meof thespaceboundedbyany
.As yetweneednotsupposeV constant.
o -o r di n at e s of t h e ce n tr e o f gr a i t y. W e h a e
. . .. . .. . .. . .. . .. . .. . .. . .( 5 ,
hatthee pressionw ithinit istobeta en
S. Now asSvariesw iththetime the
ista enw ill ingenera lvary butthe
swhichite periencesatdifferentparts
ea inthe inf inite ly smallt imedt contributeno incrementsordecrementsto f f x 2dyd ] , asw eseemost
tobeasurfacee erywherecon e
Z 2 d t d ] . .. .. .( 6
i i d t6
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O R T EX M O T IO N 3 3
locityw ithw hichthesurfacemo esinthe
r ma l a t ( x , y z ) , w e h a e i n t he p r ec e di n g e p r es s io n
eoutwardnormaltoO X . Hence
] .
im itsindicatedby [ ] areclearly
otinganelementof thesurface suchthat
erthew holesurface. Thusw eha e
. .. . . .. . . .. . 7 .
isconstant thisbecomes
8 .
. . .. . . .. . . .. . 8 .
ce S istheboundaryofaportionof the
form densityunity withwhose
thex -componentmomentumofthis
n d t h e r ef o re e u a ti o n ( 8 i s t he r e u i re d
n.
( 7 a n d ( 8 a r e pr o e d m or e s ho r tl y o f
ay ticalprocessgi enbyPo isson and
ch s u b e c ts t h us i n s ho r t. L e t u v , w
ocity o fanymatter compressibleor
p o in t ( x , y z ) w i th i n S a n d le t c d en o te
of d u /d + d / d y + d w /d , s o t ha t
.. . . .. . . .. ( 9 .
omponentmomentum of thew holematter
yat theinstantconsidered
x d yd - -| d , d dy d . .... . 1 0 .
~ 60.
Magnetism.
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d
d d yd - x - + dd d d yd ,
J v J j J \ dy d j v
yd = ( v d d + wd d y .
anda lteringthee pressiontoasurface
n an d T ai t A p p. A ( a , w e h a e
u dyd + v d d + w d d y - ff fc d d yd
d y d . . .. . .. . .. . .. . .. . .. . .. . . 1 1 ,
7 .
mpressible w eha ec= 0by theformula
on" ofcontinuity ( ThomsonandTait
on ( 8 .
pretationofthedifferentialcoefficientsdu/d , & amp c. andofthee uationofcontinuity when asat
o f f luidandso lids u v , w arediscontinuousfunctionsha ingabruptlyvary ingva lues presentsno
pulseappliedtothecollision
orticesmo ingthroughali uid theforceresultanto f the impulsecorresponds asw eha eseen precise ly
mofasolid intheordinarytheoryof
aybefeltin understandinghowthe
4 o f thew holemassiscomposed therebeing
tumofsolidsandfluidsin thedirectionof
esnear theplaceofits application
C onsider fore ample thesimplecase
truc byasingle impulse inthe lineof its
inthedirectionof the impulse beforeand
thecontrarydirection inthespaceround
esofdi idingthew holemo ingmass
llustrati eof thedistributionofmomentum
ow ingproposit ions( ~ 45 w ithreferenceto
d ( ~ ~ 4 6 4 7 4 8 .
ro f f initeperiphery notnecessarily circular complete ly surroundingthev ortices( ormo ing
rsurroundingnone andconsiderthe in
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T I O N
riouslymo ingmatteratany instant
ylinders.Thecomponentmomentum
hef irst ise ua ltothecomponentof the
edirection andthatofthe secondis
herical surfaces oneenclosing
ingso lids andtheothernone. The
hewholematterenclosedbythefirstis
ulse andise ua lto2of itsva lue.
mofthewholefluidenclosedbythe
a llmo edw iththesamevelocity and
satitscentre .
planesat afinitedistancefrom
eld ofmotion butneithercuttingany
omponentperpendiculartothemof the
occupyingatanyinstantthe space
rthisincludesnone some ora llo f the
l id s i s z e r o .
ositions:Considerineithercaseafinite lengthoftheprisme tendingtoav erygreatdistanceineach directionfromthefieldof
byplaneorcur edends. Then the
aysuppose(~ 61 startedf romrestby
thesolids[ or( ~ 66 ontheportionsof fluid
s ; the impulsi e f luidpressureonthe
eratenomomentumparalleltothe
momentuminthisdirection therewill
mpressedimpulsi e forcesontheso lids andthe
esontheends butincase2therew illbe
pressureontheends. Now the impulsi e
dsdiminish[ ~ 50( 15 ] accordingtothe
distancef romthef ie ldofmotion w henthe
direction andarethereforeinfinitely
nfinitelylong eachway.Whence the
i c e p a ns i on s ~ 1 9 ( 1 4 , ( 1 5 , i n t he
3 7 ( 1 , ( 2 ; a n d t h e fu n da m en t al t h eo r em
0
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T h o ms o n an d T ai t A p p. B ( 1 6 ] a n d
ase andT i= 0fortheother w epro ethe
45 immediately .
~ 45 thew ell- now ntheoryofelectric
tor maybecon eniently referredto .
sthenormalcomponentforceatany
duetoanydistribution pa o fmatterin
eoftheplane adistributionofmatter
gN1/27rforsurfacedensityateachpo int
spthroughallthespaceontheotherside
ereforethatthewholequantityofmatterin
se ua ltothew holequantityofmatter
odenotingintegrationo erthe inf initeplane
. . .. . ( 12 ,
matterinp bez ero . Hence ifNbethe
throughspace onbothsidesofthe
wholequantityofmatteroneachside
.. . .. . . ( 1 ) ;
parts foreachofwhichseparately
ranslatedintohydro inetics showsthatthe
ssany inf initeplane iszeroate ery
lidsorv ortices. Hence andf romthe
ch(~ 3 ) w eassume thecentreofgra ity
twoinfinitefi edparallelplaneshas
perpendiculartothematanytime
o ingso lidiscutbye ither: w hichisP rop. III.
nd D u b. M at h . J o u r n al 1 8 49 L i ou i l le s J o u rn a l 1 8 45
ofElectricalPapers(Macmillan 1869 .
tica llyw ithease bydirectintegrationsshow ing( w hether
neco-ordinates that
w e h a e
d yd _
z 2
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T I O N
tteracross anysurfacewhate er
w holevo lumeof thef inite f i edconta ining
partsisnecessarily zero becauseof the
andthereforethemomentumofallthe
aralle lplanes e tendingtothe inner
esse l andtheportionof thissurface
mhasalwaysz eroforitscomponent
anes w hetherornotmo ingso lidsor
erorboththeseplanes. B utit isremar ablethatw henanymo ingso lidorv orte iscutbyaplane the
ssthis plane( ifthecontainingv essel
desf romthef ie ldofmotion , con ergestoagenerally f initev a lue astheplane ise tendedtov ery
omthe fieldofmotion whicharestill
onwiththedistancesto thecontaining
sf romthatf initev a luetozerobyanother
thedistancestow hichtheplane ise tended
parablewith andultimatelybecome
esof thecur e inw hichitcutstheconta ining
wit isthattheconditionof neither
gso lidorv orte isnecessary toa llow ~ 45
eferencetotheconta iningv esse l and
ua lity toz eroassertedinthisproposit ion
appro imatedtow hentheplanesare
a llround w hich thoughinf inite ly shorto f
a iningvesse l areverygreatincomparison
stancesfromthemostdistant partsof
concernedin~ 45 I. III. maybe
ulartotheresultantimpulsedrawany
esof thef ie ldofmotion w itha llthe
ticesbetweenthem anddi ideaportionof
tofiniteprismaticportions bycylindrical
erpendiculartothem. Supposenow oneof
includeall themo ingsolidsand
ta lteringtheprismaticboundary le tthe
edinoppositedirectionsto distanceseach
a incomparisonw iththedistanceof themost
o lidsorv ortices. B y~ 45 I. themomen
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thisprismaticspace is( appro imately
tant I o f the impulse andthatof the
theothersis( appro imate ly z ero .
a p p ro i m at e ly z e r o v a l ue s m us t o n
bee ualto - I if theportionsof theplanes
prismaticspacesbee tendedto
omparisonwiththe distancebetweenthe
s w eha eonly toremar thatifb
entialatapoint distantDfromthe
dx f romaplanethroughthemiddleperpendiculartothe impulse w eha e( ~ 5 ) appro imately
omparisonwiththeradiusof thesmallest
mo ingso lidsorv ortices. Hence putting
planesunderconsideration denotingbyA the
ftheprismaticportions andcallingD
forthisarea w eha e( ~ 45 forthe
tionwithinthisprismatic space
mo ingso lidsorv ortices
2isan infinitelysmallfraction( asa/Dis
sf inite ifA / D2isf inite pro ideda/Dbe
sintegra lva lue( compare~ 48 footnote con ergesto- w hentheportionofarea includedinthe
tilla/Dis infinitelysmallforallpoints of
mathematicaltheoryofthe con ergenceofdefiniteintegrals andasillustratingthedistributionof
isinterestingtoremar that udenoting
ra l le l t o x , a t a ny p o in t ( x , y z ) , t h e in t eg r al
p re ss in g mo me nt um m ay a s is r ea di ly p ro e d h a e
ooaccordingtotheportionsofspace
n.
thedistributionofmomentum
lbesphericaloffinite radiusa.
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O R T EX M O T IO N 3 9
9
a m p c . 1 4 ,
a mp c ..
pro idedrislessthana andgreaterthan
concentricsphericalsurfaceenclosingall
ow by thecondit ionthattherebeno
nta iningsurface w emustha e
a . .. . .. . .. . .. . .. . .. . .. 1 5 ,
1 = T ~ . 1 6 ;
a2i+1. . . .. . . .. . . .. . . .. . . .. . . . 16 ;
+ 3 I + 2 + & a mp c .. .. .. .. .. 1 7
] i f t he w h ol e a mo u nt o f t he x - c om p on e nt o f i mp u ls i e
fluidwithin thesphericalsurfaceofradius
tbedenotedbyF , w eha e
. .. . . .. . . .. . . .. . 18 ,
o f theradiusthroughdo. Now
cof thef irstorder andthereforea llthe
pansion e ceptthef irst disappearinthe
uentlybecomes
T ic o s 0 r. .. . .. . .. . .. 1 9 .
- . + . . . . .. . . .. . . .. . . .. ( 20 ,
n d Ta i t A p p. B , ~ ~ i j ] t h e mo s t ge n er a l
eharmonicof thefirstorder. Weha e
efore( byspherica lharmonics orby the
mentsofinertiaofauniformspherical
4 7r A
o - 3 . . . . .. . .. 2 1 ;
4 A .. . .. . . 2 2 .
= l + 2 - . . .. . .. .. . .. . .. . . 2 2 .
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ethex -momentumof thef luidatany instant
oncentric sphericalsurfacesofradius
.. . .. . .. . ( 2 ) .
itely smallincomparisonw itha this
asitoughttodo inaccordancewith~ 45 II.
4. 47rA
inginthefluid outsidethesphericalsurface
ua landoppositetothat( ~ 45 II. o f
therf luidorso lid w ithinthatsurface.
n d ~ 5 2 w e se e t ha t i f X , Y Z b e
ftheforce-resultantoftheimpulse the
ce pansion( 14 isasfo llows: T - -2X x + Yy+ Z z . . . .. . . .. . . . (25 ,
andv orticesta enintoaccountarewithin
adiusis v erysmallincomparisonwith
orticesormo ingso lids andw iththe
edboundingsurface.
splendidpaperonV orte Motion has
ntremar thatacerta infundamenta l
w hichhasbeenusedtodemonstratethe
onsinhydro inetics issub ecttoe ceptionw henthefunctionsin o l edha emultiplev a lues. Thisca lls
nde tensionofe lementaryhydro inetic
proceed.
m( 1 o fThomsonandTait A pp. A
d+ d __
dr + d od o d d yd
dy d d C u
d O V 2 0 t= f d f - b ff -d d yd , ' V , 2 .. .. .. .. .( 1 ,
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T I O N
ceptionif4andb denoteany tw osingle a luedfunctionsofx , y z ; f fd dyd integrationthroughthe
niteclosedsurface S f do integration
rface andbrateofvaria tionperunito f
ctionatanypo into f it . ThisisGreen s
Helmholt ' sl im itationadded( inita lics .
forhimself.
isamany - a luedfunction andthe
c/ d , . .. d d / d , . .. e a ch s i ng l e- a l ue d
1 cannotbegenera lly true. Itsf irst
mbiguous buttheprocessofintegration
beror thethirdmemberisfound would
rO ismany- a lued. Inonecasethe
ote ua ltotheambiguoussecond w ould
p r o i d ed b i s n ot a l so m a ny - a l ue d a n d in
mber thoughnote ua ltothethird w ould
pro idedf isnotmany - a lued.
' = tan- Y . . . . . .. . . .. . . .. . . .. . . .. . . .. 2 ,
tionsoftwo planesperpendicularto
tw eentw ocircularcy lindersha ingOZ for
ionsof thesecylindersinterceptedbetw eenthe
dricalboundarye cludesfromthe
elineOZ w hereO ' hasaninfinitenumber
/d , a nd d o / d h a e i nf in it e v a lu es . W e ha e
x Y . .. . .. . .. . .. . .. . .. . . 3 ) ,
2 y2 .( " 3 )
b / ' = 0 . T h en i f 4 b e si n gl e - a l ue d
rocesspro ingthee ua litybetw een
bersof ( 1 , w hichbecomes
d x d yd = o . .. .. .. .. .. .. .. 4 .
t o e nd .
becomes
2 d d y d . .. . .. . .. 5 ,
J x ' v v
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biguousintegrationofthefirstmember
edbyS asw eseebye amining inthis
aningofeachstep oftheordinaryprocess
esforpro ingGreen stheorem. It is
ddto( 5 aterm
alueof tan-ly / isrec onedcontinuously
eplaneZ OX totheother: or
O
onesideofZ O Ytotheother torender
st m e mb e r of ( 1 . T h us t a i n g fo r
rmof theaddedterm w enow ha eforthe
to n ( 1 f o r th e c as e o f b = t a n- l y / , b a n y
andSthesurface composedof thetwo
oparalle lplanesspecif iedabo e:
d
0 27 x d ( d ~ f d ot an -Y -Y ' 1 + y 2 i \ d y/ y= O X
. .. . . .. . . .. . . .. . . . 6 .
nybarrierstoppingcirculationround
allambiguitybecomesimpossible and
1 ho lds. F orinstance if thebarrierbe
O X , interceptedbetw eentheco-a a l
nesconstitutingtheSof~ 55 sothat
integrationo ereachsideof thisrectangular
ssimply thestrictapplicationof ( 1 tothecase
ceptiona linterpretationofGreen s
asese emplif iedin~~ 55and56 depends
ayha edif ferentva luesw henrec oned
entcur es draw nwithinthespace
po intP toapo intQ; dsbe inganinf initesimale lementof thecur e andF therateofv aria tionofGper
tPC Q, PC Q betw ocur esforw hich
a lues andletbothliew hollyw ithinS.
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T I O N
omPtoQ; ma eitf irstco incidewith
y itgradua llyuntil itco incidesw ithPC Q ; it
ediateformscuttheboundingsurface
dy + d
ta inedw ithinS anddc/ d , do / dy do/ d
uousbyhypothesis whichimpliesthat
esfora llgradua lvaria tionsofonecur e
chly ingw hollyw ithinS. Now inasimply
r e j o iningthepo intsPandQ maybe
nycur ePC Q toanyotherPC Q , and
ainedwithinS besimplycontinuous
nthemultiplicityofv a lueofGorO'
ermultiplycontinuous( . 58 thespace
maybee adedifw eanne toSasurface
eryapertureorpassageonthe opennessof
itydepends fortheseanne edsurfaces
nospace donotdisturbthetriple
dw ill therefore nota ltertheva luesof itsf irst
ingthemultiplicityo fcontinuity they
sby parts bywhichitssecondor third
f romallambiguity . Toa o idcircumlocution w esha llca ll/ theaddit ionthusmadetoS andfurther
s( ~ 58 notmere lydoublybuttriply
remultiply continuous w esha lldesignateby
/ ; a nd s oo n t he s e e r a l pa rt s of / r e u ir ed
iplecontinuityofthe space.These
tedetachedf romoneanother asw henthe
ueto detachedrings orseparate
utonepart/ maycutthroughpart
he n t wo r i ng s ( ~ 5 8 d i ag r am l i n e d i nt o o ne
constitutepartoftheboundaryof the
sha lldenoteby J ' ds integrationo er
r a ny o n e of i t s pa r ts / , / 2 & a m p c . L et n o w
lynearapo intB , o f / butonthetwo
c denotetheva lueof f dsa longany
espaceboundedbyS andj o iningPQ
r thisva luebeingthesamefora ll
ra llposit ionsofB tow hichitmaybebrought
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andw ithoutma inge itherPorQ passthrough
say K cisasingleconstantw henthe
ublycontinuous butitdenotesoneor
C 2 . . . Kmc w hichmaybealldifferentf rom
space isn-plycontinuous. Lastly le tK'
t r e la t i e l y to 4 a s K c r e l a ti e l y to b .
psoftheintegrationsbyparts now
biguity theaddit ions
f f dS b f. . .. . .. . .. . .. . . 7 ,
embersof ( 1 : 2denotingsummation
edif ferentconstituents1 2 / 2 . . . o f /
enthespace is( ~ 58 notmorethan
stheoremthuscorrectedbecomes
bd b ' \ d y d
fd sb ff - f 0 2 d d yd
f - j l V 2 d d yd . .. .. . 8 .. .. .. ) .
ogyofR iemann ask now ntome
Isha llca lla f initeposit ionofspacen-ply
ingsurfaceissuch thattherearen
enanytwopointsinit. Topre ent
Iadd( 1 , thatbyaportionofspaceImean
o into f itmaybetra e lledto f romany
tcuttingtheboundingsurface ( 2 , that
nofa ll l iew ithintheportionofspacereferred
hatby irreconcilablepathsbetw eentw opo intsP
such thata linedraw nf irsta longoneof
changedtillit coincideswiththeother
ingthroughPandQ andalwaysw holly
considered.Thus whenallthepaths
ereconcilable thespaceissimply
are j usttw osetsofpaths sothateach
withanyoneofthe otherset the
s whentherearethreesuchsets itis
soon. Toa o idcircumlocutions w esha ll
daryofahollowspacein theinterior
thatnooperationsw hichw esha llconsider
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T I O N
peningtothespaceoutside it. A tunnel
at eachendintotheinterior space
edoublycontinuous andifmore
erynew oneaddsonetothedegreeof
onesuchtunnelhasbeenmade the
ntinuouswiththewholebounding
sidered andinrec oningdegreesof
nse uencewhethertheendsofany f resh
otherofthis wholesurface.Thus if
y side aholeanywhereopeningfrom
raddsonetothe degreeofmultiple
hedfromtheouterboundingsolid
dormo able inthe interiorspace addsto
olatedportion butdoesnotinterfere
ult iplecontinuity. Thus ifw ebegin
paceboundedoutsideby theinner
ternalso lid andinterna llyby the
solidin itsinterior andifwedrill a
cedoublecontinuity.Twoholes or
achwithonehole( suchastwoordinary
tutetriplecontinuity andsoon. A spongeli eso lidw hoseporescommunicatew ithoneanother i l lustratesa
ontinuity andit iso fnoconse uence
thee ternalboundingso lidorisan
. Anothertypeofmultiplecontinuity
gslin edinoneanother w asreferred
edintooneanotherinv arious
ecomplicatedmutualintersectionsofthe
/ , 2 . . . re uiredtostopa llmult iple
a inganyportionof thebounding
nthatcase inw hichoneatleasto f thetw o
ev arietiesofmultiplecontinuitycuriously
edbya singleordinarystraightor
sufficientlybythesimplesttypes which
unnelalongalineagreeingin formwith
reonw hichasimplek notist ied andby
w irew ithak notonitto thebounding
ceof thew ireshallbecomeparto f the
paceconsidered thek notnotbe ing
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ebeingarrangednot totouchitselfin
gak nottedw ire w ithitsendsunited in
Noamountofk nottingork nitt ing
nthecordw hosea isindicatesthe lineof
nywaythecontinuityofthe space
esimplicityof thebarriersurfacere uired
utit isotherwisew henak nottedor
fthe boundingsolid.Asinglesimple
gonlydoublecontinuity re uiresacuriously
ppingbarrier:which initsformof
utifully show nby the li uidf i lmadhering
ethef irstf igure dippedinasoapso lution
omplicationof thesetypes oro fcombinationsof themw ithoneanother e ludesthestatementsand
N o . - D e c. 1 8 69 [ ~ 5 9 -~ 6 4 ( / ] .
namica llemma forthe immediate
applyGreen scorrectedtheorem( ~ 57 to
roughamultiplycontinuousspace.
dby ittov erysimpledemonstrationsof
entaltheoremsofvorte motion andshall
asubstituteforthecommone uations
sf initetube o f inf initesimalnormal
fullo f l i u id( w hethercirculatinground
withitsendsdone awaybyunitingthemtogether.
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T I O N
isa lteredinshape length andnormal
andw ithanyspeed. Thea erageva lueof
o f thef luida longthetube rec oneda ll
pecti e lyof thenormalsection , v aries
of thecircuit.
s considerf irstasingleparticleofunit
force andmo inga longasmoothguiding
edandbentaboutquitearbitrarily . Letp
ure andA rthecomponentve locit iesof
w ardsthecentreofcur ature andperpendiculartotheplaneofcur ature atthepo intP throughw hich
assingatany instant. Let4bethe
eparticleitself alongtheinstantaneous
hroughP . Thus: V , 4arethree
fthev elocityoftheparticleitself.Let
hedirectionof4 o f thew holeforceon
m en t ar y k i n e t ic s
+ 7. .. . . .. . . .. . . ..
h ithertopublished ) w il lbegi eninthesecondv o lumeof
uralPhilosophy.Itmaybe pro edanalyticallyfromthe
emotionofaparticlea longav ary ingguide-cur e( Walton
ournal 1842 F ebruary ; ormoresynthetica lly thusLet1 m nbethedirectioncosinesofPT thetangenttotheguideatthepo int
e ispassingatany instant ( x , y z ) theco-ordinatesof
y z ) i t s co m po n en t v e l o c it i es p a ra l le l t o fi e d r ec t an g ul a r a e s .
a nd Z = 1 + ni m+ n ,
y+ n + i + 1 i = + i = + i +
d( ThomsonandTait sNaturalPhilosophy ~ 9 tobemade
intinasecondedit ion thattheangularve locityw ith
n i s e u a l to / i 2 + i Z 2 + - i 2 a n d i f t hi s b e de n ot e d
f the lineP perpendiculartoPT intheplane inw hich
ndontheside towardswhichitturns.Hence
ntv e locityofPa longP . Now if thecur ew eref i edw e
by thek inematicdef init ionofcur ature(ThomsonandTa it
e inw hichPTchangesdirectionwouldbetheplaneofcur ature.
supposed thereisalsoin thisplaneanadditionalangular
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YNA MIC S [ 2
usofcur ature andd: /ds dr/ dsrates
f rompointtopointa longthecur eat
adofasingleparticleofunitmass le tan
, o fa li uid f i l l ingthesupposedendless
twbe theareaofthenormalsection of
re / a is andSsthe lengtha longthetube
t atany instant sothat( asthedensity
,
denotetherateofvaria tionof thef luidpressure
t
1
. .. . ( 2 .
thetwoendsof thearcasmo ew iththe
t he k i n e m at i cs o f a v a r y in g c ur e
.. . . .. . . .. ( 3 ) ;
= s s d d . . .. . .. . .. . .. . .. ( 4 .
/ d t it s v a l u e b y ( 2 w e ha e
p d \ )
s
= ( I q 2 _ p . . .. . .. . .. . .. . .. . .. . . 5 ,
8 l 2 . ( 5 ,
s andacomponentangularve locity intheplaneofPTandV ,
othenormalmotionof thev ary ingcur e . Hencethew hole
resultantoftwocomponents
planeofr7.
) d= K
f t he t e t i s p ro e d .
f t he t e t i s p ro e d .
P u b l i c D o m a i n
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T I O N
tf luidve locity and8thedifferencesfor
s.Integratingthisthroughthelength
hefluid itsendsP1 P2mo ingw ith
-p 2 - Q -p \ ... ... .. ( 6 ,
heva luesof thebrac etedfunction atthe
ecti e ly andE2denotingintegrationa long
now P2bemo edforward orP ibac w ard t i l lthesepo intsco incide andthearcP1P2becomesthe
tE denoteintegrationroundthewhole
comes
sremainsconstant how e erthetube
ropositiontobepro ed asthe" a erage
is f o un d b y di i d in g ( 8 s b y t he l e ng t h of
maginedinthepreceding hashadnoother
by itsinnersurface normalpressureonthe
cethe proposition atthebeginning
f romw hich asw eha eseen thatproposit ionfo llow simmediate ly
aterease andnotmere ly foranincompressible f luid butfor
sityisafunctionof thepressure bythemethodof
ordinatesfromtheordinaryhydro inetice uations.
D iD d
d d '
ria tionperunito f t ime o fany functiondependingon
gw iththef luid andw = J dp/ p pdenotingdensity . In
natesweha e
w .
x
t + & a m p c .
m p 4 - 5 , an d D= 0 w.
uationsreducetheprecedingto
d dy d - . .
t ion e uation( 6 genera lisedtoapply tocompressible f luids.
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y closedringoffluid formingpartofan
e tendinginalldirections throughany
andmo inginanypossibleway andthe
6 areapplicabletoany infinitesimalorinf inite
tmet. Thusinw ordsPR OP. ( 1 . The line- integra lo f thetangentia lcomponentv e locity
ofamo ingf luidremainsconstantthrough
T h e ra t e of a u gm e nt a ti o n p e r un i t of t i me o f
elocityalonganyterminatedarcofthe
e s s o f th e v a l u e of q 2 - p a t t he e n d
velocity isrec onedasposit i e abo e
nd.
t ha t u d + v d y + w d i s a c om p le t e
bo e( ~ 1 ) tobethecriterionof irrotationa l
60 ( a ] isthesamein alldifferent
sfromonetoanotherof anytwopoints
histhesamething
60 ( a ] isz erorounde eryclosedcur e
dtoa pointwithoutpassingoutofa
which thecriterionholds.
j u s t pr o e d w e se e t ha t t hi s c on d it i on
ranyportionofamo ingf luidforw hich
andthusw eha eanotherproofof
t h eo r em ( 1 6 g i i n g us a n e w v i e w of i t s
w hich[ seefore ample~ 60( g ] w esha ll
nthetheoryofv orte motion.
naclosedcur e capableofbe ingcontractedtoapo intw ithoutpassingoutofspaceoccupiedby
uid thatthecirculationisnecessarily
otion. In~ 57w esaw thatacontinuous
doublyormultiplycontinuousspace may
tona lly yetsoastoha ef initecirculation
Q ' P p r o i d ed P P Q a n d PQ ' Q a r e " i r r ec o nc i l ab l e pa t hs b e tw e en P a nd Q . T h a t t he c i rc u la t io n mu s t be t h e
cilableclosedcur es( compare~ 57 , is
ncefromthenow pro ed[ ~ 59( P rop. 2 ]
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T I O N
60( a ] ina llmutua lly reconcilableconterminousarcs. F orby lea ingoneparto faclosedcur eunchanged andvary ingtheremainingarccontinuously nochange
inthispart and by repetit ionsof the
emaybechangedtoanyotherreconcilable
ntarypropositions.( a Thelineintegralofthetangentialcomponentv elocityalonganyfinite
d inamo ingf luid isca lledthef low inthat
thatis if it f o rmsaclosedcur eor
ow isca lledcirculation. Theuseof theseterms
mentsofP roposit ions(2 and( 1 o f~ 59to
Prop. ( 2 ] . Therateofaugmentation perunito f t ime
tedlinew hichmo esw iththef luid is
f thev a lueof~ q 2-pattheendf romw hich
eendtow ardsw hich posit i e f low isrec oned.
] . T h e ci r cu l at i on i n a ny c l os e d li n e mo i n g
constantthroughalltime.
esurface ly ingaltogetherw ithinaf luid
raw nacrossit thecirculationinthe
e ua ltothesumof thecirculationsin
rs. Thisisob ious asthe lattersum
sit i eandnegati e f low ineachportionof
woparts addedtothesumof theflows
hesingle boundaryofthewhole
ulationroundtheboundariesof inf initesimalareas inf inite lynearoneanotherinoneplane aresimply
s.
anyparto f thef luidrotateasaso lid
ngingshape ; orconsidersimply therotation
ion intheboundaryofanyplanef igure
a ltotw icetheareaenclosed multipliedby
e locity inthatplane( orroundana is
e . F or ta ingr 0todenotepo larcoordinatesofanypointintheboundary A theenclosedarea and
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elocity intheplane andcontinuing
w e ha e . r d
o O r 2 3 = o x 2 A .
o r a f l ui d m o i n g in a n y ma n ne r t h e
daryofan infinitesimalplanearea
area isca lledthecomponentrotationin
a isperpendiculartothatplane o f the
inglew ord" rotation isusedfor
ton: andthedef init ionisj ustifiedby ( c
1 ( 2 a bo e a pp li ed t o ( p b el ow .I t ag re es
w iththedef init ionof rotationinf luidmotion
le e b y S to e s a n d us e d by H e lm h ol t i n
Motion a lso inThomsonandTa it sNatural
a n d 19 0 ( j ) .
7 ' b e t he c o mp o ne n ts o f r ot a ti o n at
id roundthreea esatrightanglestoone
mponentroundana is ma ingw iththem
1 m n
neperpendicularto thelast-mentioned
einA B , C . Thecirculationinthe
B C i s b y ( b e u a l to t h e su m of t h e
riesPB C PC A andPA B . Hence
ry theareasof thesefourtriangles w eha e
.
e th e p ro e c ti o ns o f A o n th e p la n es o f t he p a ir s o f
andsotheproposit ionispro ed.
thatthecomposit ionof rotationsina
ompositionsofangularv elocitiesofa
ti e s o f f or c es & a m p c .
nf initesimalparto f thef luid the
peripheryofe eryplaneareapassing
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8/10/2019 Lord Kelvin Volume 4
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T I O N
posit ion andthereforeu v , w maybe
ofx , y z . Inapure lyana ly tica ll ight
ntbearingonthetheoryofthe integrationofcompleteorincompletedifferentials.Itwasfirst gi en
oreanalyticalproofthanthepreceding
N a tu r al P h il o so p hy ~ 1 9 0 ( j ) .
h ( j ) ( n ( o o f th e pr es en t se ct io n ( ~ 6 0
andw ithhisintegrationforassociated
ionalmotioninan unboundedfluid to
stitutehisgenera ltheoryofv orte motion.
p ur el y k i ne ma ti ca l ( h a nd ( j ) a re d yn am ic al .
ca llacircuitanyclosedcur enot
point inamultiplycontinuousspace.
s any twosuchclosedcur esif
58 butdif ferentmutually reconcilable
calleddifferentcircuits.
p l y co n ti n uo u s sp a ce i s a s pa c e fo r w hi c h
n dif ferentcircuits. Thisismerely the
bre iatedby thedef initeuseof theword
pose.Thegeneralterminologyregarding
nuousspacesis asIha efoundsince
ogetherduetoHelmholt ; R iemann ssuggestion tow hichherefers ha ingbeenconf inedtotw o-dimensional
edsomew hatf romtheformofdefinit ion
mh o lt , i n o l i n g a s i t do e s t h e di f fi c ul t
arrier andsubstitutedforitthe
ndirreconcilablepaths.Itis noteasy
gbarriero fanyoneof thef irstthree
tounderstanditssingleness butit iseasy
ethreecases any twoclosedcur es
erepresentedinthediagramsarereconcilable accordingtothedef init ionof thistermgi enin~ 58 and
conceptionw ecanma enouseof thetheoryofmultiple
tics( see~~ 61-6 ) , andHelmholt ' sdef init ionis therefore
ltothat whichIha esubstitutedforit.Mr Cler
. B . List inghasmorerecently treatedthesub ectofmultiple
pletemannerinanarticle entitled" DerCensusriumlicher
. Ges. Gottingen 1861. Seea lsoProf . C ay ley " O nthePartit ion
1861.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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enceofanysuch solidaddsonlyoneto the
space inwhichitis placed.
rt it ion asurfacew hichseparatesa
rs and ashitherto abarrier anysurface
f thespace Helmholt ' sdef init ionof
estatedshortly thus: A space is( n+ l plycontinuousifnbarrierscanbedraw n
hisapartit ion.
aspo intedoutthe importance inhydro- . ineticsofmany- a luedfunctions suchastan- ly / , w hichha e
fgra itation e lectricity ormagnetism
presse lectro-magneticpotentia ls andthe
hepartof thef luidwhichmo esirrotationally invorte motion. It is therefore con enient before
shouldf i uponatermino logy w ith
hatk ind w hichmaysa euscircumlocutionshereaf ter.
, y z ) w i ll b e c al l ed c y cl i c if i t e p e ri e nc e s
e ery timeapo intP o fw hichx , y z are
ordinates iscarriedfromanyposition
hesamepositionagain withoutpassing
hiche itherdb/d , d4/ dy ordo/ d
alueofthisaugmentationwillbecalled
particularcircuit.Thecyclicconstant
amevaluefora llcircuitsmutua lly reconcilable( 58 inspacethroughoutw hichthethreedif ferential
e.
ctioniscyclicw ithreferencetose era l
cilablecircuits itiscalledpolycyclic.
onesetofcircuits it isca lledmonocyclic.
tareaofacircleas seenfromapoint
y w he r e in s p ac e i s a m on o cy c li c f un c ti o n of x , y z , o f
47r.
e cur eofthe( 2 n thdegree
osed(thatisf initeendless branches
eenclosedw ithinothers isann-cyclic
n-cyclicconstantsareessentia llye ua l
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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T I O N
m on g t hr e e v a r i ab l es ( x y z m a ye a si l y
uouscur es constitutingoneormore
essbranches( w hichmaybek notted asshown
of~ 58 orlin edintooneanother as
Theintegra le pressingw hat forbre ity
areaofsuchacur e isacyclicfunction
hasessentia llye ua lv a luesfora ll itscyclic
rentareaofaf initeendlesscur e( tortuous
thesumof theapparentareasofa llbarriers
andraw withoutma ingapartit ion.
erypolycyclicfunctionmaybe
ocyclicfunctions.
nisca lledcyclicunlessthecirculationis
paththroughthef luid w henit iscalled
s( e essentia lly cyclic.
t io n ma y [ ~ 5 9 ( f ] b e e it h er a c yc l ic o r
yclicifthereis onlyonedistinct
therearese era ldist inctcircuits inw hich
relycyclicif theboundaryofthe
nallymo ingfluidisat rest.Ifthe
hemotionof thef luidiscyclic it isacyclic
preparedto in estigatethemostgenera l
nofasinglecontinuousfluid mass
rmultiplycontinuousspace w ith for
dary anormalcomponentve locitygi en
nly totheconditionthatthew holevo lume
licmotion. Commencing asin~ 3 , w ith
hout letallmultiplicityofthe continuityofthespaceoccupiedbyit bedoneawaywithby
es 8i 3 2. . . stoppingthecircuits as
boundingsurfaceof thef luid w hich
ner surfaceofthecontainingv essel
tendedtoincludeeachside ofeachof
asin~ 3 , anypossiblemotionbe
boundingsurface. The li uidisconse uently setinmotion pure ly throughf luidpressure andthe
5 or60 59 throughoutirro tationa l. Hence
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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gtheprescribedsurfaceconditionsis
lmotionis o fcourse( astheso lutionof
unambiguous. B utf romthisbarephysica l
ensuspect w hatthefo llowingsimple
uationpro es thatthesurfacenormal
terminestheinterior motionirrespecti ely
of themotionf romrest.
f irro tationa lmotioninsimplycontinuousspace. In~ 57( 1 , w hichisimmediate lyapplicable as
l y co n ti n uo u s m a e 0 = p a n d pu t
maybethev e locitypotentia lo fanincompressible f luid. Thatdoublee uationbecomesthefo llow ing
y d = b
ionffdomustnowincludeeach sideof
c s i , 2 . .. . H en c e i f b o = 0 f o r e e r y
face w emustha e
d yd = O ,
snomotionof theboundarysurface in
a therecanbenomotionof the
terior whenceitfollowsthatthere
ernalirrotationalmotionswiththe
ponentv elocities.Thus asaparticular
luidatrest le titsboundarybesetin
gaintorestatany instant a f terha ing
anye tent throughanyseriesof
dcomestorest atthatinstant.
portanttheorem whichdiffers
ding andincludeswhatthepreceding
lyanalyticalproofofthe possibilityof
houtthefluid fulfillingthearbitrary
dabo e aswasfirstpublishedin Thomson
so p hy ~ 3 1 7 ( 3 ) , a n d is t o b e gi e n
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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T I O N
riationande tension. Inthemeantime
oursel esastothepossibilityo f irrotationa l
rioussurface-conditionswithwhichwe
ethesurfacemotionsarepossibleandre uire
d [ ~ 1 0 -1 5 o r ~ 5 9 b e ca u se t h e fl u id c a nn o t
nthroughfluidpressurefromthemotion
goon bya idofGreen se tendedformula
t o p ro e t h e de t er m in a te n es s o f th e i nt e ri o r mo t io n
specifiedfor multiplycontinuous
o ne b y h is u n al t er e d fo r mu l a [ ~ 5 7 ( 1 ] f o r
onalmotion.In thecaseof
61 theva lueof thenormalcomponent
entlyarbitraryo erthewholeboundary
a l u es p o si t i e a n d ne g at i e o n t he t w o
ers/ , 32 & amp c. Wemustnow introduce
rthat w henthebarriersare li uef ied
ybeirrotationalthroughoutthespace
lecontinuity.F oralthoughweha e
omponentv e locity ise ua le erywhere
arrier w eha ehitherto le f tthetangentia lv e locityunheeded. If theyarenote ua lonthetwo
direction therewillbeaf initeslippingof
rfaceleftbythedissolutionof the
mbrane constituting[ ~ 60(m abo e ,
a" v orte sheet. Theanaly tica l
t ionofe ua litybetw eenthetangentia l
aria tionof theve locitypotentialin
bee ualonthetwosides ofeach
ration weseethatthedifferencebetween
ocitypotentia lonthetw osidesmustbethe
eachbarrier. Thiscondit ionre uiresthat
e ua lo erthew holemembrane. F or
titutingof themotion le tpl P2bethe
P2of thef luid andmo ingw iththe
anotheronthe twosidesofoneof the
hepressurew w hichmustbeappliedtothe
differenceoffluidpressure onthetwo
-p2inthedirectionopposedtop . A ndlet
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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itypotentialsatP iandP2 sothatif fdsdenote
a longanypathP IPP2w hate erf rom
oughthef luid( andthereforecuttingnone
and~ thecomponentof f luidve locitya long
f thiscur e w eha e
.. . . .. 1 .
59
. .. . .. . .. . .. . .. 2 ,
notetheresultantf luidv e locit iesatP IandPi.
nentv elocitiesatPiandP2 arenecessarily
if thecomponentspara lle lto thetangent
gmembranearea lsoe ua l w eha e
s
mponentve locit iesatP iandP2arenot
e sa me d i re c ti o n 0 2 - 0 m us t a s w e ha e
erthemembrane andthereforew musta lso
ssurehasbeenapplied foranytime
ofuniformvaluea llo erthemembrane
pliedno longer andthemembrane( ha ing
isdoneawaywith.Thefluid massis
tateofmotion w hichisirro tationa l
Th e " c i r cu l at i on [ ~ 6 0 ( a ] , o r t he c y cl i c
2-01 fore erycircuitreconcilablew ith
e uation
. . (4 ,
ale tendedthroughthewholeperiod
te v alue.
onmaybeperformed oneachofthe
roducedin~ 61toreducethe( n+ 1 fo ld
upiedby thefluid tosimplecontinuity.
anypointof thefluidwillthenbe a
6 0 ( x ) e u a l to t h e su m o f th e s ep a ra t e
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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so lutionb= tan- ly / consideredin~ 56
ti o n V 2 0 = 0 a n d ob i o us l y sa t is f ie s t he
merelyfortheannularspacewithrectangular
onsidered butforthehollowspace
olutionobtainedbycarryingaclosed
n d an y a i s ( O Z ) n o t cu t ti n g th e c ur e
w esha ll infuturecallaho llow circularring.
onpossiblewithinafi edhollow
ev elocitypotentialisproportionalto
ridianplanethroughanypoint anda
so lidangle a subtendedatanypoint
y a n in f in i te s im a l pl a ne a r ea A i n a ny f i e d p os i ti o n
tionV 2a=0. Thisw ell- now nproposit ion
ingA attheorigin andperpendicularto
e
A - . . .. .. . .. 5 ,
3 d ( x 2 + 2 y+ 2 ( 5
erif ied.
at( x , y z ) byanysingleclosed
subtendedatthe samepointbyall
di ideany limitedsurfaceha ingthis
edge. [ C onsiderparticularly cur essuch
efirst threediagramsof~ 58. Hence
lesubtendedat( x , y z ) by thissurface
20isfulf il led. Hence j representsthev e locity
motionpossiblefora li uidcontained
edv esse l w ithinw hichisf i ed atan
uterboundingsurface aninfinitely
mof theclosedcur e inq uestion.
e ampleforwhichthecur eisa
hesimplestspecimenof cyclicirrotational
sthato fE ample(1 is toaseto fpara lle l
entialbeingtheapparentareaofa
eaofaspherica le ll ipse isreadily found
siblereadilyintermsofa completeelliptic
andthereforeintermsofincomplete
andsecondclasses.Thee ui-potential
eablebyaidof Legendre stables.B ut
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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T I O N
w eowetheremar ableandusefuldisco ery
estreamlines( orlinesperpendicularto
aces aree pressible intermsofcomplete
econdclasses.Theyarethereforeeasily
re stables. Theanne eddiagram of
uchuse later showsthesecur esasca lculatedanddraw nbyMrMacfarlanef romHelmholt ' sformula
ctangularco-ordinates. A nimpro ed
escribed inanoteby MrCler
ehask indlya llowedmetoappendtothis
c it y a nd M a gn e ti s m v o l . i â € ”
^ f e Z - Z - y 0
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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h e mo t io n d es c ri b ed i n E a mp l e ( 2 w i ll
eanysolidringformedby solidifyingand
ofthefluidboundedby streamlines
hinwire.Thusweha easolidthic
garing oranendlessk notasil lustrated
of~ 59 o fpeculiarsectiona lfigure
nesroundthe arbitrarycur eof
dthecyclicirro tationa lmotionw hich ifplaced
rmits isthatw hoseve locitypotentia lis
gledefinedgeometricallyin thegeneral
a m pl e ( 2 .
mpoundedacyclicandpolycyclicirrotationalmotion- inetico-statics.Thewor doneintheoperation
lateddirectlyby summingtheproducts
nitesimalareaofthesurface intothe
uidcontiguouswith thisareamo es
mal fora llpartso f thesurface w hether
r wherethegeneticpressureis applied
isionsofthewholetimefromthe
tion.
w or done andfdttime- integration
onupto anyinstant.Atanypre ious
ure q theve locity andbtheve locity
ntiguousto anyelementdoofthe
thedifferenceoffluidpressuresonthetwo
9 o foneof the interna lbarriers andNthe
uidv elocitycontiguoustoeitherdoor
statemente pressedinsymbolsis
o + I f k N d s . .. . .. . .. . .. . .. . .. ( 6 ,
these eralbarriersifthereare more
generalhydro inetictheoremfor
5 9 ( 6 c o mp a re w i th ~ 3 1 ( 5 ] , w i th b e p r es s ed
tsofapo intmo ingw iththef luid w e
7
epressuretobe impulsi e sothatthere is
apeeither oftheboundingsurfaceorof
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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T I O N
fdt.Thiswillalsoimply thatdo/dt
arisonw ithqq 2 sothat
ationof~ 57w eha e
9 .
reachbarriersurface.
f f Ad . .. .. .. .. 1 0 .
ngmotionof theboundingsurfaceand
rs maybev ariedq uitearbitrarilyfrom
ftheimpulse sothatthehistory
c uisitionoftheprescribedfinal
therdif ferent andnote ensimultaneous
oundingsurface. Thusk , andk 2 may
ionsof t pro idedonly f ldtandf 2dt
a lues w hichwesha lldenoteby f ltandt2
e a m pl e w e m ay s u pp o se f t o h a e a t e a c h
reoneandthesameproportionofits final
at t er b e d en o te d b y D a n d if w e p ut . = W . . ( 1 1 ,
natesofposition butmayofcoursebe
hetime. Hence obser ingthat
s 1 ( 1 0 b e co m es
f f bd s . .. . .. . .. . .. 1 2 .
mberof thise uationdoubledagreesw ith
m em be r s of ( 7 ~ 5 7 w it h f a nd l e a ch
ndthef irstmemberof thate uationbecomes
yof thew holemotion. Hence w hen
0 ( 7 o f ~ 5 7 e p re ss es t he e u at io n of e ne rg y
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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ration o f thef luidmotioncorrespondingto
bypressuresvary ingthroughoutaccording
etime thefirstmemberbeingtwice
emotiongenerated andthesecondtwice
ocess.
ample le tussupposethe initia ting
asfirsttogeneratea motioncorrespondingtov elocitypotentialb andafterthattochange the
btob+ b , denotingbybandO ' any tw o
0 = D andeachfulf i ll ingLaplace s
gmentationf romzeroto( , andagain
uniformthroughthew holef luid. Thew or
f o un d a s ab o e ( 1 2 ,
b ds ] . .. .. .. .. .. .. .. 1 ) ,
c . d en ot e th e cy cl ic c on st an ts r el at i e t o f a s fR k 2 & a m p c .
andtheaddit ionalw or doneinthesecondprocess
bp ) d o+ K ' f f( 2 ib + b e ) d A . .. .. .( 1 4 .
a e s e en ( ~ 6 ) t h at t h e ac t ua l f lu i d mo t io n
hollyonthe normalv elocityateach
ceand thev aluesofthecyclic constants
doneingeneratingitoughttobe independentof theorderandlaw of theac uisit ionofv e locityatthe
of theatta inmentof theva luesof these era l
t h e su m o f ( 1 ) a n d ( 1 4 o u gh t t o be
t i ff or 4 i n ( 1 2 w e su bs ti tu te b + c , t he
a lueandthatof thesumof ( 1 ) and(14
b do + , ( S c J B f ' d s- ~ ' f fb ds ] . .. 15 ;
iff erencebetweenthetwoe ualsecond
57forthecaseof
0
ce thee ua lityo f thesecondmembersof
utestheana lytica lreconcilia t ionof thee uations
esofgenerationofthe samefluid
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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Y V E L O C I T Y O F A C I RC U L A R V O R TE X
t stranslationofHelmholt ' s2emoironV orte
x III. 1867 511-512.
a rl y a s ma y b e He l mh o lt ' s n o ta t io n l e t g be
a iso fauniformv orte - ring andathe
score( w hichwillbeappro imate ly
comparisonw ithg , thevorte motion
re isnomolecularrotationin anypart
hiscore andthatinthecoretheangular
arrotationisappro imatelyw orrigorously
stanceX f romthestra ighta is.
f translationisappro imatelye ua lto
meorderasthismultipliedbya/ gbeing
uidatthesurface ofthecoreis appro imatelyconstantande ualtocoa.At thecentreofthering itis
andW respecti e ly andifTbethe
w ethereforeha e
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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NA MIC S [ 3
nslationisv erylargeincomparison
longthea isthroughthecentreof the
sso smallthatlog8g/ais largeincormparisonwith27r.B utthev elocityoftranslationisalwayssmall
elocityofthefluidat thesurfaceofthe
thesmalleristhediameterof thesectionin
eterofthering.
mpletelythedifficultywhichhas
rencetothe translationofinfinitely
Iha eonlysucceededinobta iningthem
fmymathematicalpaper( April29
ietyofEdinburgh buthopetobea llowed
hatpapershould itbeacceptedfor the
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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C SO L U T I O N S A ND
.
ca l M ag a i n e V o l . X L I I. N o . 1 8 71 p p . 3 6 2 - 7 7
Lectures 1904 pp. 584-601.
F F R E E SO L I D S T HR O U G H A L IQ U I D .
producedbyaf ree lymo ingSo lid.
ththefollowinge tractfromthe
u r n al o f d at e J a n ua r y 6 1 8 5 8: L e t X , | 9 Z , X , t 1 j 2 b e r ec t an g ul a r co mp o ne n ts o f a n
mpulsi ecoupleappliedtoaso lidof in ariableshape w ithorw ithoutinertiao f itsow n inaperfect
, w a p a r b e th e c om p on e nt s o f li n ea r a nd
ated. Then if thev isv i at( tw icethe
thew holemotionbe asitcannotbutbe
io n
2 + [ v , ] v 2 + ... + 2[ v , u v + 2 [ w u wu + ...
. ..
[ v , v ] , & amp c. denote21constantcoef f icientsdeterminableby transcendenta lana ly sisf romtheformof thesurface
n o l ingonlyell iptictranscendenta lsw hen
n o l ing o fcourse themomentsof
emustha e
, t v + [ W , n w + [ u u + [ p u p + [ p ] a - = , & amp c .
[ v , W ] v + [ w U ] w + [ W , W ] W + [ p = ] p + [ a- i a- = t & amp c .
eProceedingsoftheRoyalSocietyof Edinburghfor
.fromletterstoProfessorTait ofAugust1871.PartV .
cationSeptember1871.
ad o f I Q , i s u se d t o de n ot e t he " m ec h an i ca l v a l u e " o r a s
k ineticenergy" o f themotion.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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C SO L U T I O N S A ND O B S E RV A T IO N S [ 4
, Y Z , andacontinuouscouple
t o a e s f i e d i n th e b od y i s a pp l ie d a n d if
d en o te t h e im p ul s i e f o rc e a nd c o up l e ca p ab l e of g e ne r at i ng f r om r e st t h e mo t io n u v , w w r p - w hi c h e i s ts i n
merelymathematica lly ifX & amp c. denote
glinearfunctionsof thecomponentsof
ns o f m ot i on a r e as f o ll o ws : dX _ 3 + p = X , d = & a m p c .
d t
mA O - + J a p= L I
p + . Mw = M
hen
Z = 0 L= O , M = O , N = O ... 2 ,
ob iouslyare
c o ns t .. .. . .. . .. . .. . .. . .. . .. 3 ) ,
stant
- = c o ns t .. .. . .. . .. . .. . . 4 ,
mentumconstant and
+ I + p tM + o - = Q . .. .. .. .. 5 .
ommunicatedinaletterto Professor
robably J anuary 1858 andtheyw erereferred
inhisf irstpaperonStream- lines
a lSocietyofLondon , J uly186 .
tedtotheRoyalSocietyof Edinburgh andthefollowingproofisadded: Thesee uationswillbev erycon enientlycalledtheEuleriane uationsof
ondprecise ly toEuler se uationsfortherotationofa
dethemasaparticularcase. A sEulerseemstoha ebeen
tionsofmotionintermsofco-ordinatecomponentsofve locity
f i edre lati e ly tothemo ingbody itw il lbenotonly
st todesignateas" Euleriane uations" anye uationsofmotion
ence w hetherforposit ion orv e locity ormomentof
orcouple mo ew iththebodyorthebodiesw hosemotion
noteboo o fdate1858 containingthisearly statementof the
asbeenpreser ed. F orde e lopmentsseeLamb s
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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SO L U T I O N S A ND O B S E RV A T IO N S
hisin estigationwasto il lustratedynamica le f fectsofhe ligo idalproperty ( thatis rightorlef t-handed
ofcompleteisotropy withheli9oidal
chthecoef f icientsinthequadratice pression
nditions.
v ] = [ w w ( l et m be th ei rc om mo nv a l ue
= [ o- P ] , , n , , ,
p = [ w , ] , , h , ,
, U ] = [ , v ] = 0 [ p a = [ ] , P = [ o p = 0o
- = [ v , - = [ , C = [ W , = ] = [ w p = 0
2+ w2 + n( W2 + p2 + a2 + 2h( ur+ v p+ wo } ( 11 .
ore theEuleriane uations( 1 become
- wp = X , & a mp c .
= L & a mp c .. 1 1 .
f referencef i edre lati e ly
remainsunchangedw henthe linesof
yother threelinesatrightangles to
i t i s ea s il y s ho wn d i re c tl y f ro m ( 6 ( 7 ,
te r in g t he n o ta t io n w e t a e u v , w t o de n ot e
locityofPparalle lto threef i edrectangularlines andA p athecomponentsof thebody sangular
nes w eha e
amp c .
= L & a mp c .
f referencef i edinspace
ientthantheEuleriane uations. . .. 12 ,
uations whenneitherforcenor
X = 0 & a m p c . L = 0 & a m p c . , p r e se n ts n o
isreadily seenf rom~ 21( " V orte
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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N O F F R E E S O L I DS T H RO U G H A L IQ U I D 7
w henthe impulse isbothtranslatoryand
roundwhichthebody isisotropic
rcleorspira lsoastok eepataconstant
iso f the impulse " andthatthecomponentsofangularv e locity roundthethreef i edrectangulara es
e madebyattachingpro ecting
aglobe inproperposit ions forinstance
atthemiddlesof thetwel eq uadrantsof
idingtheglobe intoe ightquadranta l
theglobeandthevanesof lightpaper a
ghand lightenoughtoillustrateby
motionsofanisotropicheli oid
e li uid. B utcuriousphenomena not
ntin estigation w ill nodoubt onaccount
r e d .
roughaperforatedSo lid. -7/
mo eablerigidbody inf inite ly
ofotherrigidbodies f i edormo eable
raperturesthroughit andlettherebe
circulations( . 60 " V orte Motion )
rbethecomponentsof the" impulse
r ee f i e d a e s a n d X , / t v i t s mo m en t s
abo e w itha llnotationthesame w estil l
o r te M ot io n" )
t d .
a mp c .
aquadraticfunctionof thecomponentsof
e n ow h a e
] U + ...+ 2 u v ] v +... ...... 1 ) ,
ergyofthefluid motionwhenthesolid
u u U 2 + . . . i s t h e s am e q u a d ra t ic a s b ef o re .
, [ u v ] , & amp c.aredeterminablebyatranscendentalanalysis ofwhichthecharacteris notatallinfluenced
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/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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[ 5
I N D A ND C A PI L LA R IT Y O N W A V E S
DF R IC T IONLESS.
indonW a esinw atersupposed
rofessorTait o fdateA ugust16 1871.
allydownwardsand0Yhori ontal let
at . . . .. . . .. . . .. . . .. . . . 1
ectionof thew aterbyaplaneperpendiculartothew a e- ridges andleth( theha lfw a e-height be
sonw ith2rr/ n( thew a e- length . The
elocityof thew ateratthesurface isthen
. . . .. . . .. . . .. . . .. . 2 ;
nf initesimal mustbethev a lueofdl/ d
if4 denotetheve locity -potentia la tanypoint
te r . No w b ec a us e
nofy andafunctionofx w hich
= oo itmustbeof theform
ndentofx andy . Hence ta ingdo/d ,
nd e u at in g it t o ( 2 , w e ha e
n ah c os ( n y -n at ;
a n d e = n a t s o t ha t w eh a e
a t . .. . .. . .. . .. . .. . .. 3 ) .
ed resultssimplyfromtheassumptions
ss thatithasbeenatrest andthat
themannerspecif iedby ( 1 .
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SO L U T I O N S A ND O B S E RV A T IO N S
nowind w henthew a e- lengthis27r/n.
.. . . .. . . .. . . .. . . .. 14 ,
2 ( 1 4 ,
eve locityof thesamew a esw henthere
, inthedirectionofpropagationof thew a es.
w e ha e
& g t } @ . 15 .
owingconclusions:
& lt w V / 1 + ) / /-
t i e a n d ne g at i e t h at i s t o sa y w a e s
stthew ind. Thepositi ev a lue isa lways
w a estra e lfasterw iththanaga instthe
a estra e ll ingaga instthew indisa lw ays
ityw ithoutw ind.
l t 2 w t he v e l oc it y of w a e s t ra e ll in g wi th t he
henV = 2w thev e locityof thew a es
bedby thew ind aresultob iousw ithout
isincorrect andiscorrectedinthereprintinB a lt imore
u lt s ( 1 , ( 2 , ( 3 ) , ( 4 a r e re p la c ed b y t he f o ll o wi n g:
n gt h 2 7 r/ n t h e g r ea t es t w a e - e l oc i ty i s w ^ / 1 + ) , w hi c h is
elocityofthewind.It isinterestingtoseethat with
hanthatof thewa es andinthedirectionof thew a es
nstance thew a e-speedw ithnowind w hichisw isless
ofw t ( orabout1/ 1650 ) thanthespeedwhenthew indisw ith
peed. Thee planationclearly isthatw hentheairismotionlessrelati e ly tothew a ecrestsandho llowsitsinertia isnotca lledintoplay.
- 28 7x
sz ero thatistosay staticcorrugationsofw a e- length
ibratedbywindofve locity
ouldbeunstable.
= 1 + .2 8- 7 1 + 8
e ual.
- & gt
aginary andthereforethewindwouldblowintospin-drift
or s h or t er . T h en f o ll o ws ( 1 6 a n d ( 1 6 ) .
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O F W I N D A N D SU R A C E- T EN S IO N O N W A V E S 7 9
t 2 w t h e v e l oc i ty o f wa e s t ra e l li n g wi t h th e
ocityof thesamew a es w ithoutw ind.
g t w . 1 + - / a- w a e s o f su ch l en gt h th at w wo ul d
utwind cannottra e laga instthew ind.
t w ( 1 + o / / o t he re c an no t be w a e s o f so s ma ll
heundisturbedvelocity isw andthe
risessentia llyunstable. A nd( 1 ) shows
eofw is
.......
ane le e lsurface isunstable if the
ceeds
1 6 ) .
fessorTa it o fdateA ugust2 , 1871.
a eonw aterwhose length
t wh er e
.. .. . .. . .. 1 ,
ua lwaysseeane q uisitepatternof ripplesin
thesurfaceofw aterandmo inghori onta llyatanyspeed fastorslow . Theripple -lengthisthe
tion
18 ,
o f theso lid. The lattermaybeasa il ing esse lorarow -boat apo lehe ldv ertica llyandcarriedhori onta lly ani orypencil- case apen nife -bladee itheredgeor
best a f ishing- linek eptappro imate ly
thangingdownbelowwater whilecarried
perhourbya becalmedv essel.The
otsadmirably ripplesinf ront and
ity ( X thegreaterrooto f samee uation
tobebeca lmedagain Isha lltry
olepattern showingthetransition
ow a es. Whenthespeedw ithw hich
tablee enif thea irw eref rictionless. B . L. reprint.
r= l7centim. ( seePartV . .
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I N D O N W A V E S
gharea lf luidsuchasa irorw ater.
esandhollow sofaf i edso lid(such
must becauseof thev iscosityof the
erforceontheslopesfacingitthanonthe
aregularseriesofw a esatseaconsistedofaso lidbodymo ingw iththeactua lve locityof thew a es
uponit o ritw oulddow or uponthe
ve locityof thew indw eregreaterorlessthan
es. Thiscasedoesnotaf fordane act
w indonw a es becausethesurface
mo eforw ardw iththev e locityof the
urrow edsoliddo. Stil l itmaybee pected
of thew inde ceedsthev e locityofpropagationof thew a es therewillbeagreaterpressureonthe
heanteriorslopesof thewa es and
at w h en t h e v e l o ci t y of t h e wa e s e c e ed s t he
orisinthedirectionoppositetothato f the
eaterpressureontheanteriorthanon
wa es.Inthefirst casethetendency
e inthesecondcasetodiminishit.
seriesofwa esofacertain height
certainforceof windorgradually
otbeing strongenoughtosustain
dof fhand. Tow ardsansw eringitSto es s
r aga instv iscosityofw aterre uiredto
gi esamostimportantandsuggesti e insta lment. B utnotheoretica lso lution andvery litt leo fe perimental
re ferredtow ithrespecttotheeddyingsof
etopsof thew a es tow hich by its
essureontheposteriorthanonthe
uenceofthewindin sustainingandmaintainingwa esischieflyif notaltogetherdue.
encalledthreedaysago byMrF roude
r t on W a e s ( B r i ti s h As s oc i at i on Y o r ,
emar able il lustrationorindicationof the
f theinfluenceofwindon wa es
w indmuste ceedthatof thew a es in
Lethim[ anobser erstudyingthe
C ambridgePhilosophica lSociety 1851( Ef fecto f Interna l
theMotionofPendulums " SectionV . .
6
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SO L U T I O N S A ND O B S E RV A T IO N S
e duringthesuccessi estagesofan
aca lmtoastorm beginhisobser ations
nthesurfaceofthewateris smoothand
imagesofsurroundingob ects. This
ctedbye enaslightmotionof the
lessthanha lfamileanhour( 81in. per
isturbthe smoothnessofthereflecting
rflittingalongthesurfacefrompoint
edtodestroy theperfectionof the
ndondeparting thesurfaceremains
ea irha eav elocityofaboutamilean
waterbecomesless capableofdistinct
ser ingit insuchacondit ion it istobe
nofthis reflectingpowerisowing
minutecorrugationsofthesuperficial
fthethird order.Thesecorrugations
thewateraneffectv erysimilarto
glasswhichweseecorrugatedfor
heirtransparency andthesecorrugationsatoncepre enttheeyefromdistinguishingformsat a
ddiminishtheperfectionofformsreflected
his appearanceiswellk nownas
hwhichthefish seetheircaptors.
cehas thisdistinguishingcircumstance
esurfaceceasealmost simultaneously
disturbing cause sothataspot
edirectactionof thewindremains
f thethirdorderbeingincapableof tra e ll ing
siderabledistance e ceptwhenunder
original disturbingforce. This
f presentforce notofthatwhich
itgi esthatdeepblac nesstothe
ccustomedtoregardasaninde ofthe
oftenastheforerunnerofmore.
o fw a emotionistobeobser edw hen
cting onthesmoothwaterhasincreased
llwa esthenbegintorise uniformly
of thew ater thesearew a esof the
erthew aterw ithconsiderableregularity .
earf romtheridgesof thesew a es but
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I N D O N W A V E S
thehollowsbetweenthem andon
sew a es. Theregularityof thedistributionof thesesecondarywa eso erthesurface isremar able
ch ofamplitude andacoupleof
geasthev e locityordurationof thewa e
ontermina lw a esunite theridgesincrease
hewa esbecomecusped andare
ondorder. Theycontinueenlargingtheir
pthtowhich theyproducetheagitation
ywiththeirmagnitude thesurface
co eredw ithw a esofnearlyuniform
or" wa esofthethird order referred
inignoranceofhisobser ationsonthis
hadca lled" ripples. Theve locityof8- inches
ersecondisprecise ly theve locityhehad
yhisobser ations fortheve locityof
t-ridgedwa esstreamingobli uely
pathofasmallbodymo ingatspeeds
persecond anditagreesremar ably
perimentaldeterminationofthe
e- e locity ( 2 centimetrespersecond
asnote plicitlypo intedoutthathis
nchespersecondwasanabsoluteminimum
. B utthe ideaofaminimumvelocityof
ebeenfarf romhismindw henhef i ed
astheminimumofwindthat can
toappearinNatureon the26th
g i e n e t r ac t s fr o m Ru s se l l s R e po r t
uotationw hichhegi esf romPonce letand
f theF renchInstitutefor1829 showing
sonrippleshadbeen anticipated.Ineed
theseanticipationsdonotinclude
amicaltheorywhichIha egi en and
ew tomew henPartsIII IV andV o f
nwerewritten.
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SO L U T I O N S A ND O B S E RV A T IO N S
e s u nd e r mo t i e p o we r o f Gr a i t y an d C oh e si o n
.
of w i nd c o ns i de r ( 1 ) a n d in t ro d uc e
1 7 i n i t. I t b ec o me s
. .. . . .. 19 .
ue
tothe caseofairandwater we
ebetw eengandg astheva lueofa is
nTandT , a lthoughit istoberemar ed
Tthatisordinarily ca lculatedf rom
ryattraction. F rome perimentsofGayLussac sitappearsthattheva lueofT isabout' 07 o fagramme
thatistosay intermsof thek inetic
grammeasunitofmass
aterunity ( asthatofthe lowerli uid
w emustta eonecentimetreasunito f length.
dasunito f t ime w eha e
n
ncentimetrespersecond corresponding
h e n l/ n = / 0 7 = 2 7 ( t h at i s w h en t h e
ntimetre , thev e locityhasaminimumvalue
cond.
theorywhichrelatestotheeffectof
uidsoccurredtomeinconse uenceof
edasetofveryshortw a esad ancing
ontofabodymo ingslowly throughw ater
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E S .
l . v . 1 87 1 p p. 1 - .
eredcohesionofwater( capillary
w hichw ouldseriouslydisturbsuche perimentsasyouw erema ing if ontoosmallasca le. Parto f its
hewa esgeneratedbytowingyour
. Iha eof tenhadinmymindthe
af fectedbygra ityandcohesionj o intly but
ringittoan issuebyacuriousphenomenonwhichwenoticed atthesurfaceofthe waterrounda
ngoutofOban( beca lmed atabout
hthewater.Thespeedwasso small
inea lmostv erticallydownw ards sothat
gementwasmerelyathinstraightrod
andmo edthroughsmoothw ateratspeeds
three- uartersofamileperhour. I
rs andotherformsofmo ingso lids butthey
eof them sogoodaresultasthef ishing- line.
shing-lineseemedto fa ourtheresult
tsroughnessinterferedmuchwithit. Isha ll
theropportunityof try ingasmoothroundrod
ertica lbya leadw eighthangingdow nunder
w hile it isheldupby theotherend. The
w ithoutanyotherappliancepro edamply
ygoodresults.
asane tremely f ineandnumerous
cedingthesolid muchlongerw a esfo llow ing
obli uew a esstreamingoff intheusual
oneachside intow hichthew a esin
ertoMrW. F roude bySirW . Thomson.
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herearmergedsoasto formabeautiful
thetacticsofw hichIha enotbeen
therto.Thediameterofthe " solid"
g- line be ingonly tw oorthreemill imetres andthe longesto f theobser edw a esf i eorsi centimetres it isclearthatthew a esatdistancesinanydirections
gf if teenortw entycentimetres w ere
stosaymo ingeachasif itw erepart
formparallelwa esundisturbedbyany
esseenrightinf rontandrightinrear
mmediatelyanob iousresultoftheory
thswiththesamev elocityofpropagation.
fallingoff thewa esinrearof thefishinglinebecameshorterandthosein ad ancelonger showinganother
. Thespeedfurtherdim inishing oneset
heotherlengthen untiltheybecome as
h o f thesamelengths andtheobli ue
ter eningpatternopenouttoanobtuse
angles. F orav eryshortt imeasetof
eforeandsomebehindthef ishing- line and
hthesamev elocity w ereseen. Thespeed
ternofwa esdisappearedaltogether.
fishing-line( producedfore ampleby
er causedcircularringsofw a estodi erge
nf rontad ancingatagreaterspeed
an thatofthefishing-line.All these
eryremar ablyageometryofripples
nyyearsagotothe Philosophical
w hi c h h o we e r s o f ar a s I c an r e co l le c t
ectwerenotdiscussed.Thespeedof
stheuniformsystemofpara lle lwa esbefore
rlyanabso luteminimumw a e- e locity
tytowhichthecommonv elocityofthe
shorterw a esinf rontwasreducedby
engtheningthelatterto ane uality
meweightper centimetreofbreadth
ofaw atersurface(ca lculatedf rom
ssac containedinPoisson stheoryof
rpurewateratatemperature so faras
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C ent. andonegrammeasthemassofa
fortheminimumvelocityofpropagation
centimetrespersecond . Theminimum
aw atermaybee pectedtobenotv ery
ouldofcoursebe thesameifthe
waterweregreater thanthatofpure
eratio asthedensity.
r be ingbeca lmedintheSoundof
entopportunity w iththeassistanceof
mybrotherf romB elfast o fdeterminingby
mw a ev elocityw ithsomeapproachto
washungata distanceoftwoorthree
sside soastocutthewateratapo intnot
motionofthe v essel.Thespeedwas
totheseapieces ofpaperpre iously
ingthe irt imesof transitacrosspara lle lplanes
metresasunder f i edre lati e ly tothe
edec andgunwale . B yw atchingcarefully
dwa es w hichconnectedtheripplesin
rear Ihadseenthatit includedasetof
o f fobli ue lyoneachside andpresenting
edthemtobew a esof thecrit ica llength
umspeedofpropagation.Hencethe
efishing-lineperpendiculartothe fronts
etrueminimumvelocity . Tomeasure it
ecessarywasto measuretheanglebetween
esofridgesandhollows slopingaway
e andatthesametimetomeasure
hefishing-linewasdraggedthroughthe
suredbyholdingaj ointed' two-foot
hes asnearlyascouldbe j udged by the
sof linesofw a e- ridges. Theangleto
openedinthisad ustmentw asthe
itdow nonpaper draw ingtw ostra ight
andcompletingasimplegeometrical
properlyintroducedto representthe
hemo ingso lid there uiredminimum
prhour theonlyothermeasurementofve locity e ceptthe
ning w hichoughttobeusedinanypracticalmeasurement
second.
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adilyobta ined. Si obser ationsof thisk ind
w ow erere ectedasnotsatisfactory . The
the otherfour:V elocityof DeducedMinimum
a e - V e l oc i ty .
nd. 2 ' 0centimetrespersecond.
' 8 .
2 .
22 9.
ofthisresultto thetheoreticalestimate
econd w as o fcourse merelyaco incidence
hesi e forceofsea-w ateratthetemperature( notnoted o f theobser ationcannotbeverydif ferentf rom
f romGayLussac sobser ationsfor
hthetheoreticalformulaej ustnow
aperw hichIha ecommunicatedtothe
h andwhichwillprobablyappearsoon
ine. If2 centimetrespersecondbe
speed-theygi e1 7centimetresforthe
gth.
e toca llripples w a esof lengthsless
andgenera lly torestrictthenamew a es
ceedingit. If thisdist inctionisadopted
suchthat theshorterthelengthfrom
thevelocityofpropagation w hile for
engththegreaterthev e locityofpropagation. Themoti e forceof ripplesischie f ly cohesion thato fwa es
of lengthslessthanhalfa centimetre
isscarce ly sensible cohesionisnearly
eof ripplesisthesameasthatof the
d ofthesphericaltendencyofa drop
. Ina llw a esof lengthse ceedingf i e
theef fecto fcohesionispractica lly insensible
beregardedasw hollygra ity . This
echoiceyou ha emadeofdimensions
sconcernsescaping disturbancesdueto
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onintothe theoryofwa ese plains
been feltinconsideringthepatterns
the surfaceofwaterina finger-glass
a moistfingeronitslip. Ifnoother
ityw ereconcerned the lengthf romcrest
56 v ibrationspersecondwouldbea
erippleswould beq uiteundistinguishablewithouttheaidofa microscope andthedisturbanceofthe
i edasadimmingof thereflections
ngcohesionintoaccount If indthe length
pondingtotheperiodofI- ofasecond
a lengthwhichquitecorrespondstoordinary
ect.
ctedtheformulafortheperiod( P in
th( 1 thecohesi etensionof thesurface
o f th e f lu i d ( p , i s
edink ineticunits. F orw aterw eha e
ngtotheestimate Iha eta enf romPoisson
8 2 x - 0 74 = 7 . H e nc e f or w a te r
hanhalfacentimetretheerror from
islessthan5percent. o fP . When1
theerrorfromneglectingcohesionis less
eiod. It istoberemar edthat w hile
ngthtobe insensibletocohesion the
es uarerooto f the length forripples
sibletogra ity theperiodv ariesinthe
e length.
ledmyattentiontoMrScottR usse ll s
r i ti s h As s oc i at i on Y o r , 1 8 44 a s c on t ai n in g
of thephenomenawhichformedthesub ect
him If indinit undertheheading
f onegrammeink ineticunitsofforcecentimetresper
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O r d er o r " C ap il la ry W a e s a m os t
" ripples" ( asIha eca lledthem , seen
o inguniformly throughwater a lsoa
sellf romapaperofdate No . 16 1829
w here itseemsthisclassofw a esw as
afterpremisingthatthephenomenonis
yofa f inerodorbarislightlydippedina
adescriptionof thecur edseriesof ripples
yattentionin themannerdescribedinthe
ell sq uotationconcludesw ithastatement
efo llowing: -. . . ontrou eq uelesrides
ndlav itesseestmoyennementaudessous
to il lustratethislaw . SofarasIcan
ly longw a esfo llow inginrearof themo ing
cribedeitherby PonceletandLesbrosor
yshownintheplanconta inedinR ussell s
eshow nabo etheplan( ob iously intended
thewater-surfacebyav erticalplane
therearaswellastheripplesinf ront and
otescapedtheattentionof thatv eryacute
respecttothecur esof therippleridges R usse lldescribesthemasha ingtheappearanceofa
as whichseemsamorecorrect descriptionthanthatofPonceletandLesbros accordingtowhichthey
riesofparabo liccur es. It isclear
ydynamica ltheory thattheycannotbeaccurate
f ar a s I a m ye t a bl e t o j u d g e R u ss e ll s
misaverygoodrepresentationof the ir
hegeometricaldeterminationofa
bser ingtheanglebetweentheobli ue
esstreamingoutonthetwo sides
inches( 21~ centimetres persecond.
estimateof25centimetresper second
ofso lidrelati e ly to f luidw hichgi es
sse ll stermina lv e locityof211centimetres
mar ableharmonyw ithmytheoryand
chInstitute 1829.
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EX PER IENC EDB YSO LIDSIMMER SED
I D .
sof theR oya lSocietyofEdinburghfor1869-70
nsin[ inPapersonElectrostaticsand
p. 567-571.
[ ~ 60( z ) ] onceestablishedthrough
inamo ablesolidimmersedina li uid
withcirculationor circulationsunchanged
h ow e e r t h es ol id b e mo e d o r be nt a nd w ha te e r
f romotherbodies. Theso lid if rigidand
earlycontinueatrestre lati e ly tothef luid
edistance pro idedtherebenoother
ncefrom it.B utiftherebe any
withinanyfinite distancefromthe
a lforcesbetweenthem w hich ifnot
ationof force w illcausethemtomo e.
briumofrigidbodiesin thesecircumstancesmightbecalledK inetico-statics butitis inrealitya
simply . F orw ek now ofnocaseof true
tallofthe forcesarenotdueto motion
hehydrostaticsofgases than sto
w eperfectlyunderstandthecharactero f
thestaticsof l i u idsandelasticsolids w e
indofmolecularmotionisessentiallyconcerned.ThetheoremswhichI nowproposetobringbeforethe
eforcese periencedbybodiesmutually
hroughthemediationofamo ingli uid
remsofabstracthydro inetics areof
illustratingthegreatq uestionofthe
w ithoutfarthertit learetotheauthor spaperonV orte
shedintheTransactions( 1869 w hichconta insdef init ions
nthepresentarticle. Proofsofsuchofthe propositions
reproofareto befoundina continuationofthatpaper.
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sactionat adistanceareality oris
a ined asw enow belie emagneticand
byactionof inter eningmatter
siderf irstasingle f i edbodywithone
hit asaparticulare ample apieceof
end. Lettherebeirrotationalcirculationof thefluidthroughoneor moresuchapertures.Itis
~ 6 E a m .( 2 ] * t h a t th e v e l o ci t y of t h e
ghbourhoodagreesinmagnitudeand
telectro-magneticforce atthecorrespondingpoint intheneighbourhoodofanelectro-magnet
structedaccordingtothefollow ingspecif ication. The" core " onw hichthe" w ire isw ound istobeof
initediamagneticinducti ecapacity -
i eandshapeastheso lidimmersedin
maninfinitelythin layerorlayers
d eachaperture.Thewholestrength
rec onedinabso lutee lectro -magnetic
a lto thecirculationof thef luidthrough
/4-7r.Theresultantelectro-magnetic
umericallye ualtotheresultantfluid
ondingpointinthehydro ineticsystem
re ample theparticularcaseofa straight
letthediameterbeinfinitelysmall in
h. The" circulation w ille ceedby
uantitytheproductofthe v elocitywithin
ntheneighbourhoodofeachend at
comparisonwiththediameterofthe
sonwiththelength thestreamlines
tingfromtheend.Thev elocity
ndinw ardstow ardstheother w il lthereforebe in erse lyasthes uareof thedistancef romtheend.
bledistancesf romtheends thedis O rf romHelmholt ' sorigina lintegrationof thehydro inetice uations.
ncesare accordingtoF araday sv erye pressi e
to linesofmagneticforce w orseconductorsthana ir.
itediamagneticinducti ecapacityisa substance
flinesofmagneticforce orwhichisperfectlyimper ious
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PER IENC EDB YIMMERSEDSOLIDS
ywillbethe sameasthatofthe magnetic
dofan infinitelythinbarlongitudinally
mendtoend.
parisonbetweenfluidv elocityand
Euler sfancifultheoryofmagnetismis
Thiscomparison whichhasbeenlong
rrelationbetweenthemathematicaltheories
sm conductionofheat andhydro inetics
notdynamical. Whenw epass asw e
astrict lydynamica lcomparisonre lati e ly to
twohardsteelmagnets weshallfind
ctionbetweentw otubes w ithli uid
utw iththisremar abledifference that
nthetw ocases unli epo lesattracting
ginthemagneticsystem w hile inthe
tractionbetw eenli eendsandrepulsion
onsidertw oormoref i edbodies such
op.I.Themutualactionsoftwo of
butinoppositedirections tothose
gelectro-magnets. Theparticular
eshow sustheremar ableresult that
canha easystemofmutualaction in
ew ithforcev ary ingin erse lyasthes uare
f thee itendsof tubes openateachend
hem beplacedintheneighbourhood
eenteringendsbeatinfinitedistances the
besimply attractionsaccordingto
tubesonthissuppositionare
erefore asiseasilypro edf romtheconser ationofenergy thequantit iesf low ingoutperunito f t ime
ctedbythemutualinfluence.[ When
e relati epositionsoftwotubesby
diminutionofk ineticenergyof thef luidis
andatthesametimeanaugmentation
hee ternalspace. Theformerise ua l
e the latterise ua ltothew or done
icenergy fromthew hole li uidissimply
ne .
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e enifoneof thebodiesconsideredbemere lyaso lid w ithorw ithoutapertures ifw ith
circulationthroughthem. Insuchacaseas
gneticsystemconsistsofa magnetor
merelydiamagneticbody notitselfa
gthedistributionofmagneticforcearound
nce. Thus fore ample aspherica l
motionsurroundingaf i edbody
thereis cyclicirrotationalmotion
dpressurearesultantforce throughits
itetothate periencedbyasphereof
city similarlysituatedintheneighbourhoodofthec orrespondingelectro-magnet.Therefore according
the latter andthecomparisonassertedin
erienceaforcef romplacesof lesstow ards
locity irrespecti e lyof thedirectionof
ghbourhood aresulteasilydeduced
ulaforfluidpressurein hydro inetics.
atan elongateddiamagneticbody
tends astendsanelongatedferromagneticbody toplaceitslength alongthelinesofforce.Hence
onaf i eda isthroughitsmiddle inauniform
ndstoplace itslengthperpendicularlyacross
ak now nresult( ThomsonandTait s
3 5 . A ga in tw oglobesheldinauniform
iningthe ircentres re uire forcetopre ent
achingoneanother.Inthemagnetic
ofdiamagneticorferromagneticinducti e
rwhenheldin alineat rightangles
dro ineticresultsimilartothisfor
obes istobefoundinThomsonand
y ~ 3 3 2.
thebodyconsideredin~ III. [bean
and beactedonby forceappliedsoas
tantofthe fluidpressure calculated
II. f o rwhate erpositionitmay
ndif itbe influencedbesidesbyany
sorigina llypublished w ithoutlim itation isob iously fa lse
eonlyperce i edto-day . â € ” Sept. 1872.
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C EDB YIMMERSEDSOLIDS
cessuperimposedontheformer it
ouldmo eunderthe inf luenceof the
ne w erethef luidatrest e ceptinso
eby thebody sow nmotionthroughit.
opositionwasfirstpublishedmany
amesThomson onaccountofw hichhe
orte o f f reemobility to thecyclicirro tationalmotionsymmetrica lroundastraighta is. [ A ddit iona l
mepropositionholdsforaglobeof any
fluidmotionconsistingofcirculationor
f inerigidendlesscur eorcur esfor
dbody inthe li uid. Demonstrationto
ofthe RoyalSocietyofEdinburghfor
oya lSocietyofEdinburgh March4 1872[ inf ra p. 108 .
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N D RE P U L S IO N S D U E T O V I B R A T IO N .
etterstoProf . F . Guthrie f romthePhilosophica l
871 reprintedinPapersonElectrostaticsand
p . 57 1 -4 ~ ~ 7 4 1- .
1 4 th 1 8 70 .
dtheProceedingsof theRoya lSociety
n A pproachcausedbyV ibration "
reatinterest. Thee perimentsyou
beautifulillustrationsofthek nown
einabstracthydro inetics w ithw hich
piedinmathematica lin estigations
-motion.
eorem thea eragepressureatany
frictionlessfluidoriginallyat rest
ptinmotionbyso lidsmo ingtoand
anymanner thoughaf initespaceof
antdiminishedby theproductof the
areof thev e locity . Thisimmediate ly
demonstratedinyoure periments
rages uareofve locity isgreater
earestthetuning- for thanonthe
ouslythecardmustbe attractedby
oundittobe butit isnotsoeasyat
atthes uareof thea eragevelocity
rfacesof thetuning- for ne ttothe
ortionsofthev ibratingsurface.
ation howe er thattheattractionmust
oubtva lid asw emaycon inceourse l es
chbearsthe tuning-for andthe
omo ethroughthef luid. If the
dsthetuning-for andtherew ere
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N S A N D R EP U L S I O N S D U E T O V I B R A T IO N S 9 9
teforceon theremainderofthewhole
andsupport thew holesystemw ould
ndcontinuemo ingw ithanacce lerated
f theforceactingonthe card-an
t indeed bearguedthatthisresult
m ightbesa idthatthek ineticenergy
raduallytransformitselfintok inetic
mo ingthroughthef luid andof the
osingup behindthesolid.B ut
mostsuf f icestoputdow nsuchanargument
aticaltheory especiallythetheoryof
et i cs e p l ai n ed i n m ya r ti c le o n " V o r te m ot i on " n e ga t i e s i t.
tionwhichyouobser edagrees
agneticattractionin acertainideal
ecifiedby theapplicationofaprinciple
ic l e [ ~ 7 3 - 7 4 0 c o mm u ni c at e d to t h e
hinF ebruary last[ 1870 asanabstract
onofmypaperon" V orte -motion.
deal tuning-for twoglobesordis s
fo inthe line j o iningthe ircentres the
lbeabar withpolesofthesamename
leoppositepolein itsmiddle.Again
rdis isane ua landsim ilardiamagnetico fe tremediamagneticinducti ecapacity [ ~ 7 4 .
themagneticandthe diamagnetic
itetothecorrespondinghydro inetic
pplytheanalogy wemustsuppose
ary f romma imummagneti ationto
ane ua landoppositemagneti ationbac
miti emagneti ation andsoonperiodica lly . Theresultanto f f luidpressureonthedis isnotat
ppositetothe magneticforceatthe
butthea erageresultanto f thef luid
ea erageresultanto f themagneticforce .
thediamagneticisgenerallyrepulsion
e erthemagnetbeheld andisunaltered
sa lo f themagneti ation itfo llow sthat
oya lSocietyofEdinburgh read29thA pril 1867.
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thefluidpressure isanattractiononthe
- for intow hate erpositionthetuningfor beturnedre lati e ly to it. .. .
oubt curiouslycloseana logiesbetw een
sofmotionin contiguousfluidsof
thedistributionofmagneticforcein a
esofdifferentinducti ecapacities.
eoccupiedbyfrictionlessincompressible
ortionsthaninothers aso lidbesuddenly
ofthefluid motionfirstgeneratedagree
751-76 be low w iththepermanentlines
espondinglyheterogeneousmedium
ar-magnet tobesubstitutedforthe
cedw ithitsmagnetica isinthe lineof
oamounts thef luidv e locitymultiplied
e ualtotheresultantmagneticforce
articulardef init ion[ the" e lectromagnetic
Postscript ] o f theresultantmagneticforce
neousinducti ecapacity gi eninthe
bo e ~ 48ofmypaperonthe" Mathematica l
* beadopted. B utheretheanalogy
rtueofw hichasolidmo eable inaf luid
magneticinducti ecapacityk eeps
st[ contrast~ 751below inthehydro inetic
t ions J une21 1849. PublishedinPartI. fo r1851.
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. . . w e ha e ( H a mi l to n ia n f or m o f La g ra n ge s
. ( 1
' = K ' . ..
lek ineticenergyof thesystem andb
othesiso f I 7 . . . K ' . . . constant.
n g of X , K , K , X ' , . . . l e t B b e o ne
toberegardedgenera llyasmo able .
rsurfacef2across theapertureto
andconsiderthisbarrierasf i edre lati e ly
ormalcomponentve locity re lati e ly to
anypointo fQ; andletf fdadenote
oleareaof f2: then
. .. . . .. 2 ;
tffNdo-......................... 3 ) ,
pressionof thedef init ionofX . Tothe
withfl atanyinstant letpressurebe
eK perunito farea o erthew holearea
orce( orforceandcouple beappliedto
sitetotheresultanto f thispressuresupposed
rigidmateria lsurfacef2rigidlyconnectedw ithB . Themoti e( thatistosay systemof forces
K onthef luidsurface andforceand
ed constitutesthegenera lisedcomponent
[ T h om s on a n d Ta i t ~ 3 1 ( b ] ; f o r it
motionofB orotherbodiesof thesystem
andifX v arieswor isdoneattherate
orforcesthere maybeinthesystem.
sityof thef luidunity le tK denote
. M . ~ 6 0 ( a ] t o f t he f l ui d i n an y c ir c ui t c ro s si n g
otethetangentia lcomponentof theabsoluteve locityof the
cuit andfdslineintegrationonceround thecircuit.
hedbytheinitialsV .M.aretothe partalreadypublished
nV orte Motion. ( Transactionsof theR oya lSocietyof
1868-9.
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R ING-SHA PEDSOLIDS
nlyonce: it isthisw hichconstitutesthegenera lised
lati e ly toX [ ThomsonandTait ~ 3 1
M .~ 7 2 w eh a e
. .. . . .. . . .. 4 ,
est( orinanystateofmotionforw hich
b y t he m o ti e K d u ri n g ti m e t .
T is o fcourse necessarilyaq uadratic
dmomentum-components , I . . . K, K i . .
y functionsofEl b . . . butnecessarily
' , . . . Inconse uenceof thispeculiarity it is
-& amp c. -8:-/ / ~' -& amp c. ... + ( c K " ' , ... ......... 5 ,
tw oquadraticfunctions. Thisw emayclearly
th e n um b er o f t he v a r i a bl e s: 7 . .. a n d j t h e
thew holenumberofcoeff icientsinthesingle
p re ss in g r is ~ ( i + j ) ( i + + + 1 , w hi ch i s
mb e r of t h e co e ff i ci e nt s 2 i ( i + 1 + - ( j + 1
ctions togetherw iththe ij a a ilable
.. a , / , . .. .. .
uantit iesa a , . . . a . o . thusintroduced
memberthat
d.
' X d i= ' X % , ( 6 ^
5 , a n d us i ng t h es e w e fi n d
. .. . . .. . . ,
c. x = d - ' a - E- c . . . .. .. .. .. .. 8 .
w t h at - a + , - 3 4 - a + , & a m p c . a r e th e c on t ri b ut i on s t o th e f lu a c ro s s Q I 2 , & a m p c . g i e n b y th e s ep a ra t e v e l oc i ty T h e ge n er a l li m it a ti o n f o r im p ul s i e a c ti o n t h at t h e di s pl a ce m en t s ef f ec t ed
ll isnotnecessary inthiscase. Compare~ 5( 11 ,
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A nd( 7 show thattopre entthesolids
w henimpulsesK, K ' , . . . a reappliedtothe
races w emustapply tothemimpulses
ations
amp c r. 7 = / K + / + & a mp c . ... ... ( 9 .
so fmotion w eha e inthef irst
,
' , . . .. . .. . .. . .. . .. . .. ( 1 1 ;
lerationof ic underthe inf luenceofK,
celerationofa massundertheinfluence
hemotionsof theso lids , le t
- & amp c . q 7 0 = 7- / K - / -& a mp c . . ... 12 ;
a m p c . d e no t e v a r i at i on s o f Q o n t he h y po t he s is o f
nt.
, r e me m be r in g t ha t b IT / dr & a m p c . d en o te
he h y po t he s is o f I a 7 . . . f K c , . .. c o ns t an t
a d
d # s + c .
& a mp + - & amp
a m p c . C c. . + d ~ *
- & a mp & a mp c . + . ( 1 ) .
r
a .
+ & a mp C .
) +
, { d / , d / 9 b & gt
. - & a mp C .+ - . .. .. . ( 1 4 .
a c co r di n g to t h e no t at i on o f ( 1 2 , ~ 0 o . .. a r e
etsofthesolids duetotheirownmotion
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R ING-SHA PEDSOLIDS
otionof the li uid andthereforee lim inate: 7 . - . by ( 12 f rom( 14 . Thusw ef ind
& a mp
- + & a mp c .
a d \ ) + & amp c
- & a mp C .+ & a mp C . t- ( 1 5 ,
O d # r ~ ) dc y
po n di n g e u a ti o n fo r o & a m p c . a n d wi t h ( 1 1
m p c . a r e th e d es i re d e u a ti o ns o f m ot i on .
e of a p pl i ca t io n o f K , K ' , . . .( ~ 1 i s
yother( suchasthe inf luenceofgra ityona
mperaturesindif ferentparts isimpossible
hatistosay ahomogeneousincompressible
a e K = 0 K ' = 0 a n d fr o m ( 1 1
. . . a reconstants. [ Theyaresometimescalled
( V . M . ~ ~ 6 2 - 64 . T h e e u a ti o ns o f
omesimply
, c dd d ' 1
d oI d 1+ " ) * '
d y \ } + & a mp c _
ationsfor0 , 0 andw iththefo llow ing
b e tw ee n 0 o 7 0. . . an d d 4 . . .
( Q
0= , 0& amp c......... 17 .
-d + ) + & a m p c . b e d en ot ed b y{ ' , } . 1 8
. .. . .. . .. . .. . .. . .. . . 1 9 .
} , { 0 r , & amp c. linearfunctionsofthecyclic
entsdependingontheconfigurationofthe
enera lly regardedsimplyasgi enfunctionsof
, 0 . . . : andthee uationsofmotionare
+ & a mp 0 * 1 c & amp = b -t
d # ( 2
b . ..Q ( 2 0 .
[ I 0 } 1 O + o .& a mp c .=
0
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amiltonianform Q isregardedasa
q 0 4o . .. w ithitscoef f icientsfunctionsof
mp c. andDiappliedto it indicatesvaria tionsof thesecoeff icients. Ifnow weelim inate: , q 0 o . . . f romQ by the linear
1 7 i s a n ab b re i a te d e p r es s io n a n d so
s a q u a d ra t ic f u nc t io n o f r w 0 . .. w i th
f r q b 0 & a m p c . a n d if w e de n ot e b y
a m p c . v a r ia t io n s of Q d e pe n di n g on v a r ia t io n s of
by d Q / d ~ r d Q / l d & a m p c . v a r i a ti o ns o f Q
s of j , q t & a m p c . w e ha e [ c o mp a re T h om s on
) a n d ( 1 5 ]
21 ;
; Q d Q
dl
otionbecome
b
+ + d
0 B } ~ + . .. .- - -
d
of Lagrange sform withtheremar ableadditionofthetermsin ol ingthev elocitiessimply( in
icconstants dependingonthecyclic
sof thesecondmemberscontaintraces
ninthesymbolsb/ d b/ d+ , . . . .
thesee uations le t-= 0 = 0
s 0 = 0 o = 0 . . . a n d th e re f or e a ls o Q = 0
ot i on ( 1 6 ( n o w e u a t i on s o f e u i l i b ri u m
luenceofappliedforcesP ) , & amp c.
reduetothepo lycyclicmotionK, / C , . . . ,
p c = . .. .. .. .. . .. .. 2 ) ;
licationoftheprincipleof energy
augmentationofthewholeenergyproduced
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R ING-SHA PEDSOLIDS
ce m en t 8 r i s b / d . $ 8 f a n d T8 i s
ppliedforces . It ispro edin~~ 724-7 0ofav o lumeofco llectedpapersonelectricityandmagnetism
h at t Q / d - 4 b Q t / d f & a m p c . a r e th e c om p on e nt s
edbybodiesofperfectdiamagneticinducti e
neticfield analogous tothesupposed
Hencethemoti einfluenceofthe
idupontheso lidsine uil ibriumise ua l
agnetisminthemagneticanalogue.
epaperO ntheF orcese perienced
o ingLi uid w hichre latestothe
pthemo ableso lidsatrest. Thepresent
p.II.ofthatarticle tobefalse.Compare
orthecaseofasingleperforatedmo able
rs agreesubstantia llyw ithe uations( 6
mmunicationStotheR oya lSocietyofEdinburgh
0 0o . . . o f thepresentarticlecorrespond
& a m p c . o f t h e f or m er t h e , A . .. m e an t h e
tionsnowdemonstratedconstitutean
notreadilydisco eredorpro edby that
eprincipleofmomentum andmoment
cha lonewasfoundedthe in estigationof
a ly tica ldefinit ionofQ in~ 3 ( 5 ,
themo ablesolidsisperforated this
e ua ltothew holek ineticenergy ( E ,
onw ouldha ew eretherenomo able
eenergy ( W) w hichw ouldbegi enup
chonthissupposit ionf low sthroughthespace
so lids suddenly rigidif iedandbroughtto
. .. . . . (24 ,
independentoftheco-ordinatesofthe
mayput-W inplaceofQt inthee uations
cleon" TheF orcese periencedbySolidsimmersed
( P r oc e ed i ng s R . S. E . F e b ru a ry 1 8 70 r e pr i nt e d in V o l um e o f
e rs ~ ~ 7 3 - 7 40 .
Session1870-71 orreprintinPhilosophica lMlaga ine No ember1871.
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rthisslightmodif ication neednotbew ritten
ctlydefinedas thewholeq uantityof
o ethemo ablesolids eachtoaninf inite
idha ingaperforationwithcirculation
hthisdef init ion -W maybeputforQ lin
w ithoute clusionofcasesinw hichthere
uresin mo ablesolids.
ry simplecase thesub ectofmy
alSocietyoflastDecember inwhich
thoutproof . Lettherebeonlyonemo ing
lettheperforatedso lidorsolidsbereduced
ablerigidcur eorgroupofcur es
t h at i s e i th e r fi n it e a nd c l os e d o r i nf i ni t e , a n d
edso lid. Therigidcur eorcur esw ill
s asthe irpartissimply thato fcoresfor
otion. Inthiscase it iscon enientto
t he r ec ta ng ul ar c o- or di na te s ( x , y z ) o f th e
globe. Then becausethecores be ing
oobstructiontothemotionof the li uid
be m o i n g th r ou g h it w e ha e
) . . .. . .. . .. .. . .. . .. ( 2 5 ,
softhe globe togetherwithhalfthatof
nce
2 6 .
M
, 0 = y
onoccurs becauseinthepresentcase
a sw en ow ha e it a d + , d y+ y d , i sa
Topro ethis le tV betheve locitypotentia la tanypoint( a b c duetothemotionof theglobe
clicmotionof the li uid. Weha e
usof theglobe and
( y -b 2 + ( Z - c } 2.
theglobe a f teranymotionw hate er greatorsmall.
ninwhichithasbeenbefore the integra lquantityof
hascausedtocrossany f i edarea iszero .
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R ING-SHA PEDSOLIDS
mponentve locityof the li uidat( a b c
M u v ) w eh a e
F ( x , , z , , b + b c . ..... 27 ,
z , a b c = -r ( X -d + /d+ v ) D
b e a ny p o in t o f th e b ar r ie r s ur f ac e Q 2 ( ~ 2 , a n d
dr ec t io n c os i ne s o f th e n or m al . B y ( 2 o f ~ 2 w e s e e
themotionof theglobe isffNdr or
, y z , a b c do...... ( 28 .
a b c d o- = U ,
4 t ha t
... 29 .
on o f ~ 7 ( 1 8 f o r x , y . . . in s te a d of 4 & g t , . . .
, x = 0 { x , y = 0.
e e u a ti o ns o f m ot i on ( 2 2 , w i t h ( 2 4 , b e c om e
W d 2 W . 3 0 .
d y d z
essthattheglobemo esasamaterialparticle
r ce s ( X , Y Z ) e p r es s ly a p pl i ed t o i t w o ul d
ha ingW forpotential.
f courseeasily foundbya idofspherical
e locitypotentia l P o f thepo lycyclicmotion
thegloberemo ed andw hichw emust
w or ingitout( be low it isreadily
hypothetica lundisturbedmotion q denote
ointreally occupiedbythecentre
a e
............... 3 1 ,
a halftimesthev olumeoftheglobe
ticenergyofwhatwemaycall theinternal
pyingforaninstant intheundisturbed
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gidglobeinthe realsystem.Todefine
harmonicana ly sispro estheve locityof the
tationa llymo ingli uidglobetobe
e l oc i ty o f t he l i u i d at i t s ce n tr e a n d co n si d er
of the li uidsphere re lati e ly toarigid
e locityq . Thek ineticenergyof this
isdenotedbyw. R emar a lsothatif
itsparts the li uidglobew eresuddenly
yof thew holew ouldbee ua ltoq; and
r gi enupby therigidif iedglobeand
theglobeis suddenlybroughttorest
r re uiredtostarttheglobew ith
amotionlessli uid.
oc i ty p o te n ti a l at ( x , y z ) i n t he a c tu a l
entherigidglobe isf i ed. Letabethe
st a nc e o f ( x , y z ) f r om i t s ce n tr e a n d ff d o
ce.At anypointofthesurface ofthe
obe thecomponentv e locityperpendicular
ntheundisturbedmotionis(dP / dr . = , ;
epressureonthespherical surface
ve locitypotentia lo f thee ternall i u id
- - u n do e s an a m ou n t of w or e u a l to
mponentf romthatv a luetozero . O n
rnalv elocity-potentialisreducedfromP
undoneinthisprocessis
f ( P + ) d P. .. .. ( 3 2 .
P + Y d ' . . .. . .. . .. . .. . .. . . 3 2 .
elocity-potentialP+ s thereisnoflow
ricalsurface gi es
.. . 3 3 ) .
romtheproposition( ThomsonandTait s
496 thatany functionV , satisfy ingLaplace se uation
y2+ d2V / d 2throughoutasphericalspacehasforitsmeanv a lue
a lueatthecentre. F ordP / d satisf iesLaplace se uation.
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R ING-SHA PEDSOLIDS
p + & a mp )
P i - i - & a m p c .
.. .. .. 3 4 ,
+ & a mp C .
de e lopmentsofPandrre lati e ly to
beasorigin theformernecessarily
thelargestsphericalspacewhichcan be
s centrewithoutenclosinganypartof
cessarily con ergentthroughoutspace
B y ( 3 3 ) w e ha e
. 3 5 .
s
i P
Pi P i = 0
( 2 i + 1 f f d P. .. . .. . ( 3 6 .
taso lidspherica lharmonicof thef irstdegree
n of x , y z , p u t
C . .. .. .. .. .. .. .. .. ( 3 7 ,
A 2 + B 2 + C
2
B 2 + C G 2 . f f. d = q 2 x v o l u me o fg lo be = y 2 .
2 . 3 8 ;
om p ar i so n wi t h ( 3 1 ,
. 4 7 p+ . . .. .. .. .. 3 9
otheglobe isinf inite ly small
. .. . . .. . . .. . . .. . . ( 40 ,
halftimesthev olumeoftheglobule
elocityofthefluidin itsneighbourhood.
mulawhichI ga etwenty-fi eyears
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encedbyasmallsphere( whetherofferromagneticordiamagneticnon-crystallinesubstance inv irtue
cew hichite periencesinamagnetic
testra ightlineforthecoreasimple
ample isa f forded. Inthiscase theundisturbedmotionof thef luidisincirclesha ingtheircentresin
sw emaynow call it , andthe irplanesperpendicularto it. A sisw ellk now n thev e locityof irrotationa l
ghta isisin erse lyproportiona ltodistance
hepotentia lfunction W fortheforce
itesimalso lidsphere inthef luidisin erse ly
tanceof itscentref romthea is and
erse lyasthecubeof thedistance andis
o f thea is. Hence w hentheglobule
ndiculartothea is itdescribesoneor
esianspira lst. If itbepro ectedobli ue ly
onentv e locityparalle lto thea isw illremain
componentwillbe unaffectedbythatone
ftheglobuleonthe planeperpendicular
scribethesameCotesianspiral aswould
nomotionpara lle lto thea is. If the
nypositionitwill commencemo ing
ttractedbyaforcev ary ingin erse lyasthe
remar ablethatittra ersesatright
uidcurrentwithoutanyappliedforceto
asw emighterroneouslyatf irstsighte pect
yswiththeaugmentedstream.Aproperly
encewouldatoncepercei ethatthe
momentumroundthea isre uiresthe
y tow ardsit.
letobe ofthesamedensityas
nf inite ly small it ispro ectedinthe
sE periencedbySmallSpheresunderMagneticInf luence and
resentedbyDiamagneticSubstances ( Cambridgeand
urna l May1847 ; and" R emar sontheF orcese perienced
edF erromagneticorDiamagneticNon-crystallineSubstances ( Phil.lMag.O ctober1850 .ReprintofPapersonElectrostaticsand
4 - 66 8 M a cm i ll a n 1 8 72 .
amicsofaParticle ~ 149( 15 .
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R ING-SHA PEDSOLIDS
ocityof the li uid smotion itw il lmo e
mecirclew iththe li uid butthismotion
dtheneglectedtermw ( 3 9 addstothe
a itandStee le sDynamicsofaParticle
ec i es I V . c a se A = 0 a n d A B f i n i te a l so l i mi t in g
sI.andSpeciesV .Theglobulewill
theoppositedirectionifpro ectedwith
sitetothatofthe fluid.Iftheglobule
thedirectionof the li uid smotionor
e locity lessthanthatof the li uid it w il l
nspira l( SpeciesI. o fTa itandStee le ,
nitetime withaninfinitenumberof
dineitherof thosedirectionswitha
thanthatof the li uid itw il lmo ea long
eciesV . o fTa itandStee le , f romapseto
longtheasymptote ataninfinitedistance
ymptotef romthea isw illbe
+ )
anceof theapsef romthea is and/ c/27rct
uidatthatdistancef romthea is. If the
manypointin thedirectionofany
stdistancefromthea isisp itw il lbe
orescapef romit accordingasthecomponent
rpendiculartothea isislessorgreater
emar edthatine erycase inwhichthe
a is( e ceptthee tremeoneinw hich
itt le lessthanthatof thef luid andits
perpendiculartothe radiusv ector ,
oaches althoughithasalwaysan
o lutions iso f . f inite length andtherefore
entoreachthea isisf inite. C onsidering
aplaneperpendiculartothea is atany
mthea is le ttheglobulebepro ected
ngalinepassingatdistanceponeitherside
8
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C S.
gsof theR oyalSocietyofEdinburgh Session1875-76
Aug.1880.
r issteadymotionofv ortices.
o f " steadymotion. Themotionof
f luid orso lidandf luidmatterissa idtobe
tionremainse ua landsimilar andthe
sparticlese ua l how e ertheconf iguration
andhowe erdistantindi idua lmaterial
e fromthepointshomologoustotheir
ndnotsteadymotion: 1 A rigidbodysymmetricalroundana is settorotate
tscentreofgra ity andle ftf ree performs
odyha ingthreeune ualprincipa l
nyshape inaninf initehomogeneous
rmly roundany a lw aysthesame f i edline
ara lle lto thisline isacaseofsteady
igidbody inaninf inite li uidmo ingin
e(2 , andha ingcyclicirro tationalmotion
sperforations isacaseofsteadymotion.
rotationalmotionofli uidinthe
tionallymo ingportionoffluid ofthe
pro idedthedistributionof the
atthe shapeoftheportionendowed
.Theob ectofthepresentpaper
lconditionsforthefulfilmentofthis
estigate further thecondit ionsofstabil ity
motionsatisfyingtheconditionof
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ConditionforSteadinessofV orte
fluid smolecularrotationatany
stbethesameasif fortherotationa lly
luidw eresubstitutedaso lid w iththe
a iso f thef luid sactua lmolecular
edate erypo into f it andthew hole
scriptionswithit w erecompelledtomo e
gtothedescriptionofe ample( 2 . If
onof anymolecularrotation through
gdistributionoffluid- elocityaresuch
itwillbefulfilled throughalltime.
ditionforSteadinessofV orte
2 4 b e lo w v o r t i c it y a nd " i m pu l se g i e n
a imumoraminimum it isob ious
ysteady butstable . If w ithsame
sama imum-minimum themotionis
ybe eitherunstableorstable.
mholt ringisacaseofstable
nergyma imum-minimumforgi env orticity
rcularv orte ring w ithaninnerirrotationa lannularcore surroundedbyarotationa llymo ingannular
w ithirrotationa lcirculationoutsidea ll
ssteady if theouterandinner
therotationalshellareproperlyshaped
f theshe llbetoothin t. Inthiscase
mum-minimumforcirculargi env orticity
steadymotion the" resultantimpulse ( V . M. + ~ 8 isasimple impulsi e force w ithoutcouple :
dyofe ample( 3 ) isatoro id and
ionaland paralleltothea isofthe
snow w ell- now nfundamenta ltheoremsshowsthat f romn
e erypo into faninf inite f luid theve locityate erypo int
pressedsyntheticallybythesameformulaasthosefor
esultantforce o fapuree lectromagnet. ( Thomson s
ostaticsandMagnetism.
isde letedinacopyannotatedbyLordK el in.
personV orte MotionintheTransactionsof theRoya l
ethus referredtohenceforth.
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dingly interestingcasesofsteadymotion
chthat if appliedtoarigidbody it
cordingtoPo insot smethod toanimpulsi e
andacouplew iththislinefora is.
taindistributionsof v orticitygi ing
w iththic eningsandthinningsof the
esinonedirectionortheotherrounda
hwillbe in estigatedinafuturecommunication
suchcasesthecorrespondingrigid
2 hasbothrotationa landtranslationa l
suppose f irst thevorticity ( def ined
heforceresultanto f the impulsetobe
tionse pla inedbelow ~ 29 suchthatthe
mparisonwiththeaperture.Ta e
apieceofbloc t inpipe w ithitsends
rsw ell , bendit intoano a lform and
dedtwistroundthe longa iso f theo a l
stobenotinoneplane( f ig. 1 . A
llipseofthis
ectly determinate
heforceresultanto f
ationalmoment
~ 6 , a r e a ll g i e n
whatwe
modeofsingle
e motionw ith ig. .
ustratethesecondsteadymode commencew ithacircularringof f le iblew ire andpull itout at
neanother soastoma eit intoasit
nglew ithroundedcorners. Gi enow a
undtheradiustoeachcorner to theplane
athecorner and k eepingthecharacter
othew ire bendit intoacerta indeternlinateshapeproperforthedataof thevorte motion. Thisis
pp e i s s ub s ti t ut e d he r e fo r " v e r y st o ut l e ad w i re . ]
dsecond andthird andhighermodesofsteadymotion
oustothef irst second third andhigherfundamenta l
rator oro fastretchedcord oro f steadyundulatorymotion
al orinanendlesscha inofmutually repulsi e lin s.
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- core inthesecondsteadymodeofsingle
e motionwithrotationalmoment.The
edat by tw istingthecornersofa
dcorners thefourth by tw istingthecorners
ingroundedcorners thef if th by tw isting
on andsoon.
diagramsof toro idalhe licesacircle
j udgmentastotherelie fabo eand
eofthediagramwhich thecur erepresentedineachcasemust beimaginedtoha e.Thecirclemay
tobethecirculara iso fatoro ida lcore
besupposedtobew ound.
Iha esaid" gi earight-handed
eresult ineachcase asinf ig. 1 i l lustratesav orte motionforw hichthecorrespondingrigidbody
ices bya ll itsparticles roundthecentral
ow insteadof right-handedtw iststothe
hecornersof thetriangle s uare pentagon
e l e ft - ha n de d t wi s ts a s i n fi g s. 2 3 , 4 t h e re s ul t i n
motionforw hichthecorresponding
F i g . 4.
handedhelices.Itdepends ofcourse
edirectionsof theforceresultantand
ulse withnoambiguityinanycase
orms andinthelines ofmotionofthe
willberight-handedorleft-handed.
o fmotiontheenergy isama imumminimumforgi enforceresultantandgi encoupleresultanto f
essi e lydescribedabo earesuccessi e
m-minimumproblemof~ 4-adeterminate
so lutionsindicatedabo e butnoother
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rt icity isgi eninasinglesimpleringof the
motion forthecaseofav orte linew ithinf inite ly thincore bearsacloseanalogy tothefo llowing
em: indthecur ew hose lengthsha llbeaminimumw ithgi en
rea andgi enresultantarea lmoment
w ouldbe identica lw iththevorte problem
ey thinvorte - ringofgi env o lume
twerea functionsimplyofitsapertural
etricalproblemclearlyhasmultiple
selytothesolutionsofthe v orte
sofso lutionareclearlyverynearly
ms( infinitelyhighmodesidentical ,
e theso lutionderi edinthemannere pla inedabo e
ofN sides w henN isav erygreat
hate itherproblemmustleadtoaform
longregularspira lspringof theordinary
twoendsmeet andthenha ingitsends
dsoastogi eacontinuousendlessheli
teadof theordinarystra ightline-a is , and
ditscirculara is. Thiscur e Ica lla
se it l iesonatoroid , j ustasthecommon
culartoro idasimpleringgeneratedby there o lutionofanysinglycircumferentia lclosedplanecur eroundanya isinitsplanenotcuttingit. A
renchusage isaringgeneratedby there o lutionofacircle
notcuttingit. Anysimplering oranysolidwith
maybecalledatoroid buttodeser ethisappella tionit
nli eatore .
ofatoroid isaline throughitssubstancepassingsomewhatappro imatelythroughthecentresofgra ityofallits crosssections.An
fatoroidis anyclosedlinein itssurfaceonceroundits
tionofa toroidisanysectionby aplaneorcur edsurface
intotwoseparatetoroids.Itmustcut thesurfaceof
impleclosedcur es oneof themcomplete ly surroundingthe
ce:of courseitisthe spacebetweenthesecur eswhich
toroidalsubstance andtheareaofthe inneroneof
perture.
cuttinge eryaperturalcircumference eachonceand
ss sectionofthetoroid.It consistsessentiallyofasimple
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rcularcy linder. Letabetheradiusof
thea iso f theclosedheli ; le trdenote
tionoftheideal toroidonthesurface
supposedsmallincomparisonw itha and
noftheheli tothenormalsectionof
w ere thestepof thescrew and2 rris
lindricalcore onwhichanyshortpart
telysupposedtobewound.
stant Ithegi enforceresultanto f the
g e n r ot a ti o na l m om e nt . W e h a e ( ~ 2 8
I N 7V r T 2 ( a .
r=
eadofasingle threadwoundspirally
w eha etw oseparatethreadsforming asit
edscrew " andleteachthreadma eaw hole
oroidal core.Thetwothreads each
ically tortuousringslin edtogether and
w hatw illnow beadoublev orte - ring.
btainedfora singlethreadwouldbe
ifK cdenotedthecyclicconstantforthe
ds ortwicethecyclic constantfor
rof turnsofeitheraloneroundthe toroidal
enienttota eNforthenumberof
othatthenumberof turnsofonethread
thecyclicconstantfore itherthreada lone
steadymodesof thedoublev orte - ring
c N 7r r2 a
odeswillcorrespondto smallerand
butinthiscase asinthecaseof thesingle
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adsruntogetherintoone asil lustratedforthe
ne e d d ia g ra m ( f i g 7 .
F i g . 8. " T r ef o il K n o t.
oftheli uidroundtherotational
suchthatthef luid- e locityatany
dinthesamedirectionas theresultant
spondingpointin theneighbourhood
cuit o rga l aniccircuits o f thesameshape
etting-forthofthisanalogytopeople
tura listsare w iththedistribution o f
bourhoodofan electriccircuit does
nderstandingofthestillsomewhat
whichweareatpresent occupied.
onofthe li uidinther otational
eceof Indian- rubbergas-pipestif fened
usualmanner andwithitconstruct
chw eha ebeen
thesymmetricaltre fo ilk not( f ig. 8 ~ 1 ) , unit ingthetwo
y by ty ingthem
twoofstra ightcy lindrica lplug thenturnthetuberound
nuousa is. The
uidvorte -core
titmustbe
d K n o t .
rform ofthe
cularto theplaneofthediagram
isthroughthecentreof thediagram
ane ineachofthecasesrepresented
s.Thewholemotionofthe fluid
issorelatedin itsdifferentpartsto
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andsmallin diameterperpendicularto
e itmaybe. Icannot how e er sayat
atthispossiblesteadymotionis stable
f re o lution de iatinginf inite ly litt le
hthesamevorticity there isthesame
rgy andconsiderationofthegeneral
notreassuringonthepoint ofstability
tioniswanting .
deedsucceededin rigorously
yoftheHelmholt ringinanycase.
imaginetheringtobethic erinoneplace
hegi env orticity insteadofbe ing
alcircularring tobedistributedin a
is butthinnerinonepartthaninthe
ththesamev orticityandthesame
hsucha distributionisgreaterthanwhen
ut nowletthefigureof thecross
teadofbe ingappro imatelycircular be
.Thiswilldiminishthe energywiththe
meimpulse.Thusfrom thefigureof
ntinuouslytootherswithsamev orticity
meenergy. Thus w eseethatthef igure
tedabo e a f igureofma imum-minimum
imum norofabso luteminimumenergy .
mum-minimumproblemw ecannotderi e
enaofsteam-ringsandsmo e-rings
tw ere thenatura lhistoryof thesub ect
andthatthesteadyconf iguration w ith
metersofcoreto diameterofaperture
ationsconnectedwithwhatisrigorously
ostabil ityofv orte co lumns( tobegi en
the RoyalSociety mayleadtoa
stabilityforasimpleHelmholt ring
roportiontodiameterofaperture.B ut
istorynor mathematicsgi esusperfect
nthecrosssection isconsiderablein
perture.
W . T . M ay 1 0 1 88 7.
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statementofgeneralpropositions
plesusedintheprecedingabstract o fw hich
sof papersonv orte motioncommunicatedtotheRoyalSocietyofEdinburghin 1867 -68and-69
actionsfor 1869.Therestwillform
continuationof thatpaper w hichIhope
alSocietybeforethe endofthepresent
ha i n g v o r t e m ot i on i s c al l ed v o r te c o re o r f o r br e i t y s i mp l y " c o r e. A n y fi n it e p or t io n o f li u i d
re andhascontiguousw ithito eritsw ho le
ingli uid isca lledav orte . A v orte
aring ofmatter.Thatitmustbe so
dpublishedbyHelmholt . Sometimesthe
ndedto include irrotationallymo ingli uid
ingintheneighbourhoodofv orte -core
of li u idmaysuccessi e lycomeintothe
e andpassaw ayagain w hile thecore
yofthesamesubstance itismoreproper
termavorte asinthedef init ionIha e
ationofav orte isthecirculation[ V . M. ~ 60( a ] inanyendlesscircuitoncerounditscore .
n fi g ur a ti o ns a v o r te m a yt a e w h et h er o n
diness(~ 1abo e , oronaccountof
ort ices orbyso lidsimmersedinthe
d bo u nd a ry o f t he l i u i d ( i f t h e l i u i d is n o t
ua ti o n r e ma i ns u n ch a ng e d [ V . M . ~ 5 9 P r op . ( 1 ] .
e issometimescalleditscyclic constant.
nethroughaf luidmo ingrotationally
r ed w hosedirectionate erypo int
fmolecularrotationthroughthatpoint
] .
o r t e i s e ss e nt i al l y a cl o se d c ur e [ b e in g
v orte ] * .
edsectionofavorte isany
normally thea ia ll inesthroughe ery
d e le t ed b y L or d K e l i n .
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losedsectionofavorte intosmaller
sthroughthebordersof theseareasform
-tubes. Isha llca ll( a f terHelmholt ) a
ortionofav orte boundedbyav orte tube( notnecessarily inf initesimal . O fcourseacompletev orte
orte - f i lament butit isgenera lly
stermonly toaparto favorte asj ust
yofacompletev orte satisfiesthe
tube.
be isessentia llyendless. Inav orte f i lamentinf inite ly smallina lldiametersofcrosssections" rotation
0 ( e f r om p oi n t to p o in t o f th e l en g th o f t he
etotime in erselyastheareaof thecross
area ofthecrosssectioninto the
circulationorcyclicconstantof the
to designateinageneralwaythe
otationinthemattero fav orte .
orte di idedintoanumberof infinitely
thev orticityw illbecomplete lygi enw hen
mentanditscirculation orcyclicconstant
hapesandposit ionsof thef ilamentsmust
erthatnotonly thev orticity butitsdistribution canberegardedasgi en.
yatanypo into favorte isthecirculationofaninf initesimalf i lamentthroughthispo int di ided
ompletefi lament. Thevorte -density
dforthesameportionoffluid.B y
lla longanyonev orte - f i lament.
into infinitesimalf i lamentsin erse lyas
the ircirculationsaree ua l andletthe
ofunity . Ta ethepro ectionofa ll
.1/nof thesumofthe areasofthese
M . ~ 6 6 2 e u a l to t h e co m po n en t i mp u ls e
iculartothatplane. Ta ethepro ections
anes atrightanglestoone another
ityof theareasof thesethreesetsof
accordingtoPoinsot smethod theresultant
oupleof thethreeforcese ua lrespecti e ly to
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as andactinginlinesthroughthethree
ndiculartothethree planes.Thiswillbe
eforceresultanto f the impulse andthe
orte .
stosay thecouple isalsocalledthe
v o r te ( V . M . ~ 6 .
mentofaplane arearoundany
eareamultipliedintothedistancef rom
dicularto itsplanethroughitscentreof
f thepro ectionofaclosedcur eon
eaofpro ectionisama imumwillbe
ctionof thecur e orsimply theareaof
hepro ectiononanyplaneperpendicular
ntarea iso fcoursez ero .
tanta iso faclosedcur e isa line
ity andperpendiculartotheplaneof
tantarealmomentof aclosedcur e
resultanta iso f theareasof itspro ectionsontw oplanesatrightanglestooneanother andpara lle l
tood o fcourse thattheareasof the
oplanesarenote anescentgenera lly
aplanecur e andthatthe irz ero- a lues
fe ua lposit i eandnegati eportions.
tin generalz ero.
edefinitions theresultantimpulse
of inf inite ly smallcrosssectionandofunit
heresultantareaof itscur e . The
rte isthesameastheresultanta iso f the
ona lmomentise ua ltotheresultantareal
ntav orte - f i lamentinaninf inite
ginfluenceofothervortices oro f so lids
Wenow see f romtheconstancyof the
era lly inV . M. ~ 19 , thattheresultantarea
mentofthe cur eformedbythe
tnthowe eritscur emaybecomecontorted anditsresultanta isremainsthesameline inspace.
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YNA MIC S [ 10
tionsandcontortionsthevorte - f i lamentmay
anymotionof translationthroughspacethis
eragealongtheresultanta is.
ua lv orte madeupofaninf inite
orte - f i laments. If thesebeofv o lumes
othe irvorte -densit ies(~ 25 , sothatthe ir
w enow seef romtheconstancyof the
eresultantareasofall thev orte filamentsremainsconstant andsodoesthe sumoftheirrotational
ntareala isofthemall regardedasone
nspace. Hence asinthecaseofav orte f i lament thetranslation if any throughspace isonthea erage
A llthis o fcourse isonthesupposit ion
rte , andnoso lidimmersedinthe li uid
of the li uid nearenoughtoproduceany
gi env orte .
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N A L M O T I O N O F A L I Q U I D
AT S .
o l . x v . 1 8 77 p p . 29 7 -8 C om m un i ca t ed t o S ec t io n A
onatGlasgow September7 1876.
gthismotionwerelaidbeforethe
la ined buttheanaly tica ltreatmentof
more mathematicalpapertobecommunicatedtotheSectionon Saturday.Thechiefob ectofthe
astoillustratee perimentallyaconclusionfromthistheorywhichhad beenannouncedbytheauthor
theSection , to theef fectthat if the
an oblatespheroidalrigidshellfullof
multipleoftherotationalperiodof the
ofthe spheroidisofthe difference
eastdiameters theprecessionaleffect
ontheshe ll isappro imately thesame
drotatingwiththe samerotational
mentconsistedinshow inga li uidgy rostat
of thinsheetcopperfilledwithwater
dfly-wheelofthe ordinarygyrostat.
e hibited thee uatoria ldiameterof
dedthepo lara isbyaboutone- tenthof
peedtobethirty turnspersecond
w hich if actingonarotatingso lidof the
s w ouldproduceaprecessionha ingits
t ipleof Io fasecond must accordingto
eryappro imately thesameprecessionin
uidasinthe rotatingsolid.Accordinglythemainprecessionalphenomenaoftheli uidgyrostat
entfromthoseofordinarysolid gyrostats
A ddresses( Macmillan , v o l. II. pp. 2 8-272.
9
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onforthesa eofcomparison. It is
r ationwithoutmeasurementmight
rencesbetweentheperformancesofthe
statinthewayofnutationaltremors
caseof theinstrumentwiththefist.
nteitherofspeedsor forceswas
tion andtheauthormerelyshowedthe
ghgenerali l lustration w hichhehoped
nterestingil lustration o f thatvery
ematicalhydro- inetics thequasi- rigidity
i uidby rotation.
tionofthispapertothe Association
peningaddresswhichprecededit onthe
e i edf romProf . HenryNo. 240of theSmithsonianC ontributionstoKnow ledge o fdateOctober 1871 entitled
MotionpresentedbytheGyroscope the
o e s a nd t h e Pe n du l um b y B r e e t M a o r G e n. J . G . B a r n ar d C o l. o f E ng i ne e rs U . S . A . i n w hi c h I fi n d a
nofmypre iously-publishedstatements
ccasionofmyaddresstocorrect e pressed
donotconcur withSirW illiamThomsonintheopinions
f romThomsonandTa it ande pressedinhis
ope( Nature F eb. 1 1872 . Sofaras
erfectrigidity withinaninfinitelyrigid
in therateofprecessionw ouldbeaf fected.
perGen. B arnardspea sof " the
byrotation. Thushehasanticipated
mentscontainedin mypaperonthe
farasregards theeffectofinteriorfluidity
nofa perfectlyrigidellipsoidalshell
tererroro f thatpaper w hichI
dress hadnotbeencorrectedby
attheplausiblereasoningw hichhadledme
mcon incing. F ormyself Icanonly
yearliestopportunity tocorrecttheerrors
erors andthatIdeeply regretany
doneinthe meantime.
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S
uidGyrostats.Thesolidgyrostat
ormanyyearsin theNaturalPhilosophy
tofGlasgow asamechanicali l lustrationof
o lids andithasa lsobeene hibitedin
on ersa ionesof theR oya lSocieties
graphEngineers butnoaccountofit
efollowingbriefdescriptionand
ennowbeacceptableto readersof
essentiallyofa massi efly-wheel
to f inertia pi o tedonthetw oendsof its
dtoanoutercasew hichcomplete ly inclosesit. F ig. 1representsasectionbyaplanethroughthea is
ig. 2asectionbyaplaneatrightanglesto
roughthecase j ustabo ethef ly -w heel.
d withathinpro ectingedgeinthe
whichiscalledthe bearingedge.Its
cur il inearpo lygonofsi teensidesw ith
hefly-wheel.Eachsideof thepolygon
radiusgreaterthanthedistanceof the
efriction ofthefly-wheelwould if
cular causethecasetoroll alongonit
opre entthisef fectthatthecur edpo lygonal
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ndrepresentedinthedrawingisgi ento
apieceofstoutc ordaboutfortyfeet
clearrunofaboutsi ty feetcanbeobta inedarecon enient. Thegyrostatha ingbeenplacedwiththe
ertica l thecordispassedinthroughan
o-and-a-halftimesroundthebobbin-shaped
utagainatan apertureontheopposite
rethattheslac cordisplacedclearofa ll
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S
eef romk in s theoperatorholdsthe
case ispre entedf romturning w hile
throughbyrunning atagradually
y f romtheinstrument w hileho ldingtheend
ufficienttensionisappliedtothe
titf romslippingroundontheshaf t. In aa s
- - -- - --
ngularv e locity iscommunicatedtothef lyw heel suf ficient indeed tok eepitspinningforupw ardsof tw enty
en spunitbeset onitsbearing
a itye actlyo erthebearingpo int on
nesuchasa pieceofplate-glasslyingona
nueapparently stationaryandinstablee uil ibrium. Ifwhile it isinthisposit ionacoupleroundahori onta l
f ly -w heelbeappliedtothecase no
mthevertica lisproduced butitro tates
a is. If ahea yblow w iththef istbe
case it ismetbyw hatseemstothesenses
stiff e lasticbody and fora few seconds
statisinastateofv io lenttremor w hich
rapidly . A stherotationa lve locitygradua lly
ofthetremorsproducedby ablowalso
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ioustonoticethe totteringcondition
asied tremulousnessof thegy rostat w hen
asedtospin.
fly-wheelisreplacedby anoblate
sheetcopper andf il ledw ithwater. The
heinstrumente hibitedisI- thatisto
iametere ceedsthepo larby thatf ractionof
thetw oendsof itspo lara isinbearings
brasssurroundingthespheroid.This
nnectedwiththecur edpolygonalbearingedgewhichliesinthe e uatorialplaneoftheinstrument
forthesupporto f thea iso f the
asectionisrepresentedthroughthe
ty andF ig. 4gi esav iew of thegyrostat
hepro longationof thea is. Topre ent
nthegyrostatfallsdownatthe endofits
tdroundit insuchaw ay thatnoplanecan
eli uidgyrostatissimilarto that
ostat dif feringonly intheuseofavery
largewheelforthe purposeofpulling
onabobbin f reetorotaterounda
sthenpassedtwo-and-a-halftimes
nin theanne edsectionaldrawing
ecircumferenceofthe largewheelto
stantthenturns thewheelwithgradually
whiletheframeofthegyrostatis firmlyheld
onappliedtotheenteringcordtopre ent
pulley.
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5 ]
ETS[ ILLUSTR A TINGV O R TEX -SYSTEMS .
l . x v I II . 18 78 p p. 1 , 1 4.
ericanJ ournalofScience describing
ngmagnetsbyMrA lf redM. Mayer to
mofmutually-repellentmoleculeseach
owardsafi edcentre whichappearedin
. p. 487 mustha e interestedmanyreaders.
ularlybecausethemodeofe perimentingtheredescribed w ithaslightmodification gi esaperfect
easilyreali edwithsatisfactoryenough
thek inetice uil ibriumofgroupsofco lumnar
ncirclesroundthe ircommoncentreofgra ity
ctofacommunicationI hadmadetothe
honthepre iousMonday.InMr
ori onta lresultantrepulsionbetw eenany
saccordingtoacomplicatedfunctionof
lycalculableifthe distributionof
wereaccuratelyk nown.Supposethe
ysimilarin allthebarsand ineachto
nglaw:-Letthe intensityofmagnetisationberigorouslyuniformthroughoutav erylargeportion
of thebar( F ig. 1 , andletitvaryuniformly
ndsA andB . Thebarw illactasif
bstitutedidealmagneticmatter , or
a lled uniformlydistributedthroughthe
thew holequantity inDB tobee ua l
nk indtothatofC A . F ore ample
arityinAB andtruesoutherninB D.
neednotbee ual.LetnowA C D B '
lectrostaticsandMagnetism ~ 469( W. Thomson .
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actlysimilardistributionofmagnetism
dletthetw obeheldpara lle ltooneanother.
aryin erselyasthedistance ifthe
ll incomparisonw ithDB orC A andif
mall incomparisonwithCD.Ifthe
werfulbar-magnetbeheldina linemidwaybetweenB A andB ' A , a tadistancef romtheendsB andB '
sonw ithB B ' andcomparablew iththe
thehori onta lcomponentof itseffecton
cev aryingdirectlyasitsdistance from
rthesecondit ionsMrMayer se periments
fe uil ibriumof two orthree orfour
ints inaplane repellingoneanother
themutua ldistances andeachindependentlyattractedtowardsaf i edcentrew ithaforcev ary ingdirectly
sIshowedin mycommunicationtothe
h istheconfigurationofthegroup of
eofstraightcolumnarv orticeswith
byaplaneperpendiculartothe
ertiaof agroupofidealparticles of
esepointsbe ingthef i edcentre inthe
tyreferredtoby MrMayerhas
numerica lproblem andit isremar able
y orinstabilityisidenticalin the
ms.Inthestatic problemitisofcourse
fthe mutualforcesbetweentheparticles
attractiontowardsaf i edcentre isless
blee uilibriumthanforanyconfigurationdifferinginfinitelylittlefrom it.Thepotentialenergy
a functionofdistancefromthecentral
tance increases andthestatementof
enientlymodif iedtothefo llow ing: oragi env a lueof thisfunctionthemutua lpotentialenergy
minimumforstablee uil ibrium. When
heattracti e forcevariesdirectlyasthe
nergyis
onstants andZ r2thesumof thes uaresof
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MAGNETSILLUSTR A TINGV O R TEX -SYSTEMS 1 7
rticlesfromtheattracti ecentre.And
eenthe particlesisthein ersedistance
ergy ise ua lto
...
ot e co ns ta nt s a nd D D , D , & a m p c . d en ot e th e mu tu al
ticles.Thus
uilibrium becomesthattheproductofthe mutual
ticlesmustbe
i e n v a l u e of t h e
e irdistancesf rom
f irstconclusion
ethat the
particlesmust
eNow thecondit ionofk inetice uil ibriumofagroup B a
thatistosay the
o l e incircles n
eof inertia is D
unicationtothe
h thatthe
ancesmust
mumorama imum-minimumforagi env a lueof the
e irdistances
ofgra ity ; and
netice uil ibriummaybestable isthattheproduct
env a lueof
so f the irdistances
Ta ingfor
tices( oro f the little C
Mayer sproblem , s '
uil ibrium is -
ented by
hecaserepresented g. 1.
headsinF igs. 2and3 representthemotions
dthatw hate erbethecomplicationofmotionsduetomutual
rt ices the ircentreofgra itymustremainatrest. [ O f .
s up ra .
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sroundthe ircentreofgra ity . Itmustbe
feachco lumnre o l esa lsoroundits
hesamedirectionasthegrouproundthe
tyofa ll w ithenormouslygreaterangular
dtheproblemofoscillationsin the
^ â € ” '
g. 3 .
rationofstablee uilibrium.Thegeneral
tsformathematicalanalysishasa v ery
orthecaseofatriadofe ua lvorte
oodoftheanglesof ane uilateral
git k inematicallyisrepresentedin
circulardiscsofcardboardpi otedonpins
anglesofane uilateraltriangle
ne.Theplanecarryingthesethree
entlymadeofacirculardisc ofstiffcard
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N A L O S C I L LA T IO N S O R O T A T IN G W A T ER .
gsof theR oyalSocietyofEdinburgh March17 1879:
o l . x , 1 8 80 p p . A0 ~ l / ]
sub ectinhisDynamicalTheoryof
dea ltw ithinitsutmostgenera litye cept
themotionofeachparticleto be
tal andtheve locity tobea lw ayse ua l
ev ertical.Thisimpliesthatthe
allin comparisonwiththedistance
dtof indthede iationf romle e lnessof the
sensiblefractionofits ma imum
ortcommunicationI adoptthis
ther insteadofsupposingthew atertoco er
fthesurfaceof asolidspheroidasdoes
mplerproblemofanareaofw atersosmall
ureof itssurface isnotsensiblycur ed.
anyshape andofdepthnotnecessarily
atest smallincomparisonw iththe least
dthe waterinitrotateround a
ularve locityw sosmallthatthegreatest
itmaybe asmallfractionofthe radian:
gularv e locitymustbesmallincomparison
rgdenotesgra ity andA thegreatestdiameter
ionsofmotionare
,
componentve locit iesofanypo into f the
mnthroughthepo int( x y , re la ti e ly
, O y r e o l i n g wi t h th e b as i n p t h e p r es s ur e
o f thisco lumn andptheuniformdensity
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onsw emustha ey= a/ V -1 w herea isrea l.
ations.
thetessera ltypewemustha e
n V / - 1 w he r e m an d n a re r e al a n d by p u tt i ng
ginaryconstituentswefind
.. . .. . . .. . . .. . 10 ,
onnectedby thee uation
. . .. . . 11 .
ngva luesofuandv , w eseew hatthe
tbetoallowthesetesseraloscillations
shape. Nobounding- linecanbedrawn
hehori ontalcomponentve locityperpendicularto it iszero . Thereforetoproduceorpermitoscil la t ions
ein respecttoform watermustbe
ternatelyallroundtheboundary or
a ll f o rw hichthehori onta lcomponent
ero.Hencetheoscillationsofwater
ughare notofthesimpleharmonic
andtheproblemoffindingthemremains
thew ell- now nsolutionforw a esina
hichareofthe simpleharmonictype.
t ionsinanendlessC analw ithstraight
para lleltox , thesolutionis
a t . . .. . .. . .. . .. . .. . . ( 1 2 ;
t h e e u a t i on
1 ) ,
1 1 .
w ef indthatv v anishesthroughoutifwe
14 ;
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A T IO N A L O S C I L LA T IO N S O R O T A T IN G W A T ER 1 4 5
1 i n ( 1 2 w e fi n d b y ( 7 ,
- a t . . .. . .. . .. . .. . . 1 5 ;
1 ) w e fi nd
theve locityofpropagationofwa esis
riodasinaf i edcanal. Thusthe
inedto theeffectofthefactor
tingresultsfollowfromtheinterpretation
particular suppositionsastothe
odof theoscil la t ion(27r/ & lt r , theperiod
) a n d th e t im e r e u i re d t o tr a e l a t th e
anal.Themore appro imatelynodal
henorthcoastofthe EnglishChannel
nchcoast andof thetidesonthewest
nnelthanon theeastorEnglish
accountedforontheprinciplerepresented
intoaccounta longw ithf rict iona lresistance
es oftheEnglishChannelmaybe
orepow erfulw a estra e ll ingfromw est
esspowerfulwa estra e ll ingf romeastto
outhernpart oftheIrishChannelby more
ll ingf romsouthtonorthcombinedw ithless
ll ingf romnorthtosouth. Theproblemof
ndlessrotatingcanal issol edbythe
H e - ly c o s ( m - a t - e y ^ ( c o s m + a t }
- at + E l yc os ( m + a - t } . .. 1 7 .
cana l w efa llupontheunso l edproblem
eraloscillations.Ifinstead ofbeing
posethecanalto becircularandendless pro idedthebreadthofthe canalbesmallincomparisonwith
theso lution( 17 sti l lholds. Inthis
cumferenceof thecana l w emustha e
aninteger.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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W A V E S I N CI R CU L A R B A S I N ( P O L A R
( i - a t . . . .. . . .. . . .. . . .. . . .. 18
w hereP isafunctionof r. B y ( 8 ) P
on
. .. .. .. . 1 9 ;
d
r/ P
P. .. .. . 2 0 .
2cr- r
erinacircular basin withor
and.Let abetheradius ofthe
centralislandleta be itsradius. The
fulf i lledare = 0w henr= aand
onetotheotherof thetwoconstants
andthespeed* aof theoscilla t ion are
ntit iestobefoundby thesetwoe uations.
isimmediatelyeliminated andthe
e uationfora.Thereis nodifficulty
nthusf indingasmanyasw epleaseof the
ndw or ingoutthew holemotionof the
tsof thise uation w hicharefoundto
er-Sturm-Liou il le theory arethespeeds
mentalmodes correspondingtothedifferent
onsof the idiametra ldi isionsimpliedby
Thus bygi ingto ithesuccessi ev a lues
a m p c . a n d so l i n g th e t ra n sc e nd e nt a l e u a ti o n so f o un d
efundamenta lmodesofv ibrationof the
posedcircumstances.
rthreetida lreportso f theB rit ishA ssociationthew ord
ncetoasimpleharmonicfunction hasbeenusedtodesignate
abodymo inginacircle inthesameperiod. Thus if T
thespeed v iceversa if abethespeed 27r/ ristheperiod.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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A T IO N A L O S C I L LA T IO N S O R O T A T IN G W A T ER 1 4 7
d theso lutionof ( 19 w hichmust
chPandits differentialcoefficientsare
nceP isw hatisca lledaB esse l sfunction
orderi and accordingtotheestablished
. ( 2 1 .
.. . .. . 21
eforanendlesscircular canalisfallen
ygreatv a luetor. Thus ifw eput27rr/ i= X
wa e- length w eha ei/ r= 27/ X , w hichw ill
otation.We mustnowneglecttheterm
ndthusthedif ferentia le uationbecomes
0
.............................. 22
or2-4w 2 / gD. A so lutionof thise uationis
a - r a n d us i ng t h is i n ( 2 0 a b o e w e fi n d
i n ( m - a t ( a l - 2o m e - l y w h er e m = i O .
a t e a c h bo u nd a ry w e h a e a l = 2 w m w h ic h
lyattheboundaries butthroughoutthespace
atee uation( 22 issuf f icientlynearly true.
v a l u e ab o e w e ha e
g- - o) , )
a b o e .
nicationtotheRoyalSocietytogo
ses andtogi edeta ilso f theso lutionsat
eofwhichpresentgreatinterest inrelation
sinre lationtotheabstracttheoryofvorte
differencesbetweencasesinwhichar
than2oareremar ably interesting and
ectto thetheoryofdiurnaltidesin
d e r B e s se l s c he n F u n c ti o ne n ( L e i p i g 1 8 67 , ~ 5 a n d
d ie B e s s e l s c he n F u n c t io n en ( L e ip i g 1 8 68 , ~ 2 9 .
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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YNA MIC S [ 1
thermoreorless nearlyclosedseasin
thelunarfortnightlytide ofthewhole
edthattheprecedingtheoryis applicable
sinanynarrow la eorportionof thesea
afew degreesof theearth ssurface if f o r
entof theearth sangularv e locity rounda
ality -thatistosay w = ysinI w herey
gularv e locity andIthe latitude.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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itherinouridea lin iscidincompressible
uchasw aterora ir to formavorte -sheet
aceof f initeslip byanynatura laction.
at presentunderconsideration andin
blecaseof twoportionsof li u idmeeting
instance adropof ra infa ll ingdirectlyor
ntalsurfaceofsti l lw ater , isthatcontinuity
uid motionbecomeestablishedatthe
eentwopoints orbetweentwolinesin
mmetrytowhichourpresent sub ect
theseparationof the li uidf romthe
osedglobeorany otherfigureperfectly
is andmo inge actly inthe lineof the
esof thef reedli uidsurfacecomeintocontact
heenclosureoftworings ofv acuum
ich how e er maybeenormously farf rom
on .
ne-integraloftangentialcomponentv elocityroundanyendlesscur eencirclingthering asaring ona
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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C O R E L E SS V O R T IC E S
slin edtogether isdeterminateforeach
andremainsconstantfore eraf ter: unless
ormore orthetw of irstformedunite into
dentsthereis nosecurity.
e thatacore lessring- orte , w ith
nditshollow shallbeleftoscillating in
e u a to r o f th e g lo b e p r o i d ed ( V 2 - P / P
terialoftheglobebe v iscouslyelastic
steadyposit ionroundthee uator ina
calonthetwosidesof thee uatorial
motiongoesonsteadilyhenceforthfore er.
c e ed a c e rt a in l i mi t I s u pp o se c o re l es s
si e ly formedandshedof fbehindtheglobe
uid.
possibil itybe impossible foraglobe it iscerta inly
prolate figuresofre olution.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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A CO L U M N AR V O R T EX .
gsof theR oyalSocietyofEdinburgh March1 1880
pp. 155-168.
ion inwhichthestream-linesare
w iththe ircentresinoneline( thea iso f
eve locit iesappro imate lyconstant andappro imatelye ua late ua ldistancesfromthea is. A sapre lim inary
enienttoe pressthee uationsofmotion
pressiblein iscidfluid( thedescriptionof
t in estigationisconf ined intermsof
s " r 0 z - t ha t i s c o or d in a te s s uc h t ha t
y .
y andifw edenotebyx , y z the
fthefluidparticlewhichattime tispassing
y z ) , a n d by d / dt d / d , d / dy d / d d i f f er e nt i at i on s r es p ec t i e l y on t h e su p po s it i on o f x , y z c o ns t an t t y z
c o ns t an t a n d t x , y c o ns t an t t h e or d in a ry e u a ti o ns
~ 2
I
2 .
y..................... 2
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A C O L U M NA R V O R T EX
arcoordinates weha e
3 )
onsare
d r d r
d ( r O ) d r O )
= . . . . . .. . . .. . . . (5 .
pro imate ly incirclesroundO z , w ith
appro imatelye ua ltoT a functionof r
s assume
t - i0 , r 0 = T + r c os m c os ( n t -i
t - i , p = P + w co sm c os ( n t - iO )
dw arefunctionsof r eachinfinitely smallin
tutingin( 4 and( 5 andneglecting
f the infinitely smallq uantit ies w ef ind
T
( 7 ,
w
...............
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/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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inatinga andresol ingforp T wefind
T dT
+ - d- w
w iT 2 f dT 2 _. T 2 2
-i -- -+ n ---w
dr r d r r
- + - n-
o fm= 0 ormotionintw odimensions
enienttoput. .. . . .. . . .. . . .. . . .. . . .. . . .. 10 .
hichsuperimposedon = 0andrO = T
tionisirro tational andfbsin( nt- i is
sa lsotoberemar edthat w henm
superimposedmotionisirrotationa lw here if
= c o n s t. / r a n d th a t wh e ne e r i t is i r ro t at i on a l b c o s m s i n ( n t - i , w i th b a s g i e n b y ( 1 0 , i s i ts
8 by ( 9 , w eha ea lineardif ferentia le uationof thesecondorderforw . The integrationof this
u lt i n ( 9 , g i e w p a n d r in t e rm s o f
yconstantsof integrationw hich w ithm n
minedtofulf i lw hate ersurface-condit ions
conditionsofmaintenanceareprescribed
nterestingcasespresentthemsel es.
esttobegin:CASEI.
o const. . . . .. . . .. . . .. . . .. . . .. 11 ,
i 8 w h en a p pr o i m at e ly r = a
t i , , , , r = ai . 1
a b ei ng a ny g i e n q u an ti ti es a nd i
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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initelysmallsimpleharmonicnormal
es distributedo erthemaccordingto
nrespecttothecoordinatesz, 0. The
shadby supposingtheinnerboundary
, andthe li uidcontinuousthroughoutthe
ercylindric boundaryofradiusa.
ma esw= O whenr= 0 e ceptforthe
lly w ithoute ception re uiresthatcbe
orw becomes
. .. . .. . . . . . .. . .. . .. . .. ( 1 9 ,
i( or . . . .. . . .. . . .. . . .. . . .. . . . 20 ;
w he n r = a g i e s b y ( 1 ) ,
1 ,
n- iwe a
mula.
mannerofF ourier w ef indtheso lution
ionof thegenerati edisturbanceo er
o ereachof thetw oifw edonotconf ine
se andforanyarbitraryperiodicfunctionof
r edthat(6 representsanundulation
nderw ithlinearv e locityna/ ia tthe
elocityn/ithroughout.Tofindtheinterior
rationproducedatthesurface w emust
, o ranysumofso lutionsof thesametype
so lutions ina llrespectsthesame e cept
dthatgreatenoughv a luesof ima e
hereforev imaginary andforsuchtheso lutions
) ifunctionsmustbeused.
a lV orte inaf i ed
a
ed o r bi t r = a + f r a d t. .. . 2 2 ,
sturbedorbit r= a+ f rdt
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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) o f th is s ur fa ce w eh a e f ro mn P = T 2 d r/ r o f ( 6
) . .. .. . ( 3 2 .
a n d ( 2 6 , a n d ( 2 5 , a n d ( 2 ) , t h e co nd it io n
ndarygi es
o ,
i ( m a ] + ( c ( m a + [ 01 ( ma ] = 0 ......... 3 3 ) .
by ( 27 , w egetane uationto
wef ind
.. . .. . .. . .. . .. . .. . .. 3 4 ,
lyposit i enumeric.
a se i s t ha t o f a= o o w hi c h b y ( 2 7 ,
th er ef or e b y( 3 3 ) , g i e s
. .. . . .. . . . 3 5
Subcase w eseethatthedisturbance
a ellingroundthecylinderwithangular
o ( 1 - V N / i ,
superimposedononeanother tra e ll ing
ularv elocitiesgreaterthanand
thantheangularve locityof themassof the
sbye ua ldif ferences. Thepropagation
e locity isinthesamedirectionasthatin
es thepropagationof theotherisinthe
> i2( asitcertainly isinsomecases .
edinmotionwithoneor otherof
e locit ies( 3 4 , orlinearv e locities
a n d th e l i u i d be t h en l e ft t o i ts e lf i t w il l p er f or m
uatorymo ementrepresentedby ( 6 ,
. B u t i f t h e fr e e su r fa c e be d i sp l ac e d to t h e co r ru g at e d
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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A C O L U M NA R V O R T EX
henlef tf reee itheratrestorw ithanyother
elocitythaneitherofthose thecorrugation
ntotw osetsofw a estra e ll ingw iththe
aco(1 + IN/ i .
ceptional andcanpresentno
undthecylinder. Itwillbe considered
rlyimportantandinteresting.To
ma r t h at
.. . .. . .. . .. . .. . . 6 .
n mr = I o ( mnr
f ( 24 is
( m 2r 2 2 4 4+ & a m p
2 + & a mp c
ntsandSi= 1-+ 2-1+ . . . i-1. Hence
” 1 - + & a mp c .. .. .. .. .. .. . 3 7 ,
t a e n s o as t o m a e 1 o ( 0 1 .
edthere lationbetweenEandDtoma e
andfoundittobe
= + 2 - 0 79 4 42 - 1 9 6 5 1 0 = 1 15 9
( 3 8 .
41244881...........................
nterna lF rict ionofF luidsontheMotionofPendulums "
n d ( 1 0 6 . ( C am b . Ph i l. T r an s . D e c . 1 85 0 .
. W. L. Gla isherthatGauss insection3 2ofhis
r al e s ci r c aS e ri e m In f in i ta m 1 + ( a . P / l . y x + & a m p c . ( O p e r a
e s th e v a lu e of - r r 2 o r - ( - p , i n hi s no ta ti on t o
6 51002602142 47944099.
stfigureinSto es sresult( 106 ought asinthete t
lle t sTablesw ef ind
98 592825
rnumberf romthis w eha ethev a lueofE/Dto20places
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/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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nientassumptionforconstantfactor
+ 2 + 4 + & a m p c .
4 r 4
+ ' ( S 2 + 11 59 ) + & am p c .
t t he s e ri e s in ( 3 6 a n d ( 3 9 a r e
ergreatbemr thoughforv a luesofmr
micon ergente pressions w il lgi ethe
earlyenoughfor mostpracticalpurposes
llabour.
3 9 w e fi n d b y d i ff e re n ti a ti o n
5
4 + 2 4. 6 + & a mp c .. .. .. . ( 4 0 .
4 r4 & a m p c
2- + 2 2. .+ 2 + & a m p c .4 6
- 1+ / ( S + 1 15 9 1 5 ]
2 1 15 9 1 5 ] + & a mp c .
5r |
4 6 + & a m p c .
[ - 3 + 1 .2 ( S + ' 1 15 9 1 5 ]
S 2+ 1 15 9 1 5 ] + & a mp c .
2 2 42 6 + & a m p c .
eSubcasew ithi= 1 supposemato
i n g th a t S = 1 w e h a e
1 + 2 - lo gm a- + ' 1 15 9 ( m a 2 .. .. .. 4 2 .
-a + - 1 15 9. .. .. . 4 2 .
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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A C O L U M NA R V O R T EX
rofv a luesofX y ie ldedby (50 gi es
a n d ( 1 ) , a s o l u ti o n of t h e pr o bl e m of f i nd i ng s i mp l e
aco lumnarv orte w ithmofanyassumed
eharmonicv ibrationsarethusfound:
hemannerofF ourier fo rdif ferentv a lues
plitudesanddif ferentepochs gi ese ery
atinginfinitelylittlefromtheundisturbed
thatofi= 0 iscuriouslyinteresting.
1 , ( 52 gi e
m a : , , â € ” ) m al o( m ) . .. .. .. .. .. .. .. .. . 5 ) ,
E ( m a
. . . .. . . .. . . .. . . .. . . .. . . ( 54 .
5 ) , regardedasatranscendentale uation
1 st 2 nd 3 r d . .. r oo ts o f J ( q ) = 0 i n or de r of
a n d th e n e t g r ea t er r o ot s o f J , ' ( q ) = 0
dow ntotherootsof J thegreaterthey
tedbyaid ofHansen sTablesof
o a nd J 1 ( w h i ch i s e u a l to - J o f ro m q = 0
a isasmallf ractionofunity thesecond
a largenumber ande enthesmallestroot
fr a ct i on t h e fi r st r o ot o f J o ( q ) = 0 w h ic h
Table is2 4049 or appro imate lyenough
I n e e r y ca s e in w h ic h q i s v e r y l a rg e i n
hethermaissmallornot ( 54 gi es
6 , w eseethatthesummationof two
espropagatedalongthelengthofthe
n t- m ) r O = T + T C 7o s 7 1t - -M ) )
; p = P+ i c os ( n t - m ) (
tionofthesewa esisn/m.Hence when
nw ithma theve locityof longitudina l
/ o f thetranslationa lve locityof thesurface
rbit. Thisis1/ 1 2 or5of thetrans R epublishedinL6mmel sB esse l scheF unctionen Le ip ig 1868.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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hecaseofmasmall andthemodecorrespondingtothesmallestrooto f ( 5 ) . A fulle aminationof the
e a s e p r e ss ed b y( 5 5 , ( 1 ) , ( 4 8 , ( 1 5
tructi e . Itmustformamorede eloped
alSociety.
andmav erysmall isparticularly
tInitw eha e by ( 42 , fo rthesecond
p pr o i m at e ly
og I + 11 59 ] . .. 5 6 .
m a
oot q , iscomparablewithma anda ll
mparisonwithma.Tofindthesmallest
sv erysmallw eha e toasecondappro imation
4 1 . .. . .. . .. . .. . .. . .. . ( 5 7 .
1 b e co m es t o a f ir s t ap p ro i m at i on
. .. .. 5 8
dtofindthetwo un nownsX andq 2 gi e
m 2 a2
. No w w i th i = 1 ( 5 1 b e co me s
elysmall. Hence( 52 gi esfora
. . . . . .. . . .. . . .. . 59
2. . .. . . .. . . .. . . .. . . 60 .
- ~ . . .. . .. . .. . .. . .. V .
5 9 , ( 6 0 , a nd ( 5 6 i n( 5 0 , w ef in d t oa se co nd
4 3 m 2a ( 3 )
g mc + - 1 1 59 ,
- m2 2a g + 4 3 * 2 ma
a log + + -1159. . .. . . .. . . .. 61 .
s 6 1 .
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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A C O L U M NA R V O R T EX
4 ) a b o e . T h e f ac t t ha t a s i n ( 4 ) ,
, showsthatinthiscasea lsothedirection
a elsroundthecylinderis retrograde
hetranslationoffluidin theundisturbed
a s wa s t o be e p e ct e d t h e v a l ue s o f - n ar e a pp r o i m at e ly e u a l in t h e tw o c as e s wh e n ma i s s ma l l en o ug h b u t - n
y sm a ll d i ff e re n ce i n ( 4 t h an i n ( 6 1 a s
.
gt 1hasaparticularlysimple
rthesmallestq -rootofthetranscendental
u e o f i in s te a d of u n it y w e st i ll h a e ( 5 8 ,
forq small. E lim inatingq2/ m2a2betw een
t i ll f i nd X = ; b u t i n st e ad o f n = 0 by ( 5 1 , w e
w . Th u s is p r o e d t he s o lu t io n f or w a e s o f
f iguretra e ll ingroundacy lindrica lv orte ,
agowithoutproofinmyfirst article
tion .
s " P r o c. R o y. S o c. E d in . F e b . 18 1 8 67 [ s u pr a p . t I 6 .
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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S T EA D Y AN D O F P E RI O D I C F L U I D M O T I O N 1 6 7
ppositionwill helpinanin estigation
hichwillfollow.
completelyenclosedinac ontaining
eeitherrigidorplasticsothatw emayat
hape orofnaturalsolidmaterialand
tic(thatistosay returninga lwaystothe
entimeisallowed butresistingalldeformationswithaforce dependingonthespeedofthe change
eofq uasi-perfectelasticity .Thewhole
elandfluidwill sometimesbeconsideredas
disturbedbygra ityorotherforce
pposeitto beheldabsolutelyfi ed.
emaysuppose ittobeheldbyso lidsupports
iscouslye lastic materia l sothatitw il l
esenseasathree-leggedtableresting
Thefundamenta lphilosophicquestion
paramountimportance inourpresentsub ect.
e pla inedinThomsonandTait sNatural
n PartI. ~ 249 andmorefullydiscussed
oninapaper" On theLaw of Inertia the
andthePrincipleofAbsoluteClinural
ion. F orourpresentpurposeweshall
sumingourplatform theearthorthe
absolute ly f i edinspace.
hepresentcommunication so farasit
istopro eandto il lustratetheproof
ositionsregardingamass offluidgi en
arto f it: I Theenergyof thewholemotionmaybeinf inite ly
inacertainsystematicmanneron the
ngingit ultimatelytorest.
esse lbesimplycontinuousandbe
sticmateria l thef luidgi enmo ingw ithin
- esse lbecomple lycontinuousand
elasticmateria l thef luidw ill loseenergy
er buttoadeterminatecondit ionof irro tationa l
substantia lly inP roc. B oy . Soc. Edin. xnII. 1885 p. 114
r gy i n V o r t e M o ti o n. ]
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/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
s s_ u s e # p d
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atecyclicconstantfor eachcircuit
r e ma r , f i rs t t h at m e re d i st o rt i on o f t he
shapeof theboundary canincreasethe
e ly. F orsimplicity supposeaf initeoran
hapeofthe containing- esseltobe
me thiswilldistorttheinternal
ha edoneif thef luidhadbeengi en
H el m ho l t ' s l a ws o f v o r t e m o ti o n w e c an
ialstateofmotionsupposedk now n the
erypartof thef luid a f terthechange.
eshapeof thecontaining- esse lbealteredby
hatistosay dila teduniformly inone or
ns andcontracteduniformly intheother
ofthreeatright anglestooneanother.
eneouslydeformedthroughout thea is
chpart willchangeindirectionso as
ngdirectionofthe samelineoffluid
tudewillchangein in ersesimpleproportiontothedistancebetweentwoparticles inthelineof the
mplif y subse uentoperationstotheutmost
yq uic motionorbyslow motion the
ngedtoa circularcylinderwithperforated
s asshowninfig.1. Inthepresent
of the li uidmaybesupposedtoha e
tyofmolecularrotationthroughout.It
omomentofmomentumroundthea iso f
allsupposethisnottobethecase. If it
w hichcouldbedisco eredbye ternal
ecy linder roundadiameter toaf resh
eitwithmomentof momentumofthe
iso f thecy linder. Withoutfurther
w esha llsupposethecy lindertobegi en w ith
containingf luidinane ceedingly irregular
thagi enmomentofmomentumMfround
Thecy linderitself istobeheldabso lute ly f i ed andthereforew hate erw edotothepistonswe
mentofmomentumofthefluid round
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S T EA D Y AN D O F P E RI O D I C F L U I D M O T I O N 1 6 9
epistonA tobetemporarily f i edinits
dthewholecontaining- esselofcylinderandpistons to
sspi ot soas
A A thea iso f
elbeofideally
itsinnersurfacebe
o l ut i on i t w il l
remainatrest
efluid onitis
ththea is. B ut
ea lly rigid le tthe
scous-e lasticso lid
ssoftheinternalfluidmotionwill cause
ning-solidwithlossofenergy andthe
tedtomoreandmorenearlyas time
heonedeterminateconditionof minimum
momentofmomentum w hich asisw ell
ed isthecondit ionofsolidandf luid
larve locity . If thestif fnessof the
allenoughandits v iscositygreatenough
alconditionwill becloselyappro imatedtoinav erymoderatenumberoftimestheperiodof
n.Still wemustwaitaninfinite
perfectappro imationtothiscondition
mple orirregularinitialmotion.W e
cuttheaf fa irshortbysimplysupposingthe
gw ithuniformangularv e locity l i ea
g- esse l a truef igureof re o lution w hich
ras absolutelyrigid andconsistingof
iaphragmandtw omo ablepistons as
ullo rpushandlea e itto itse lf it
ce inthedirectionof the impulseand
eepalternate lypullingandpushingit
isstatementrecei esaninterestinge perimentali l lustration
e tractedfromtheProceedingsoftheRoyalInstitution
ch4 1881 be inganabstracto faF riday-e eningdiscourse
aspossiblyaModeofMotion andnow inthepressfor
therlecturesandaddressesinav o lume( V o l. r. o f the
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gularoscillation duetosuperposition
accordingtoaninfinitenumberof
thek indin estigatedinmyarticle
mnarV orte , " P roc. R oy . Soc. Edin.
not asthere l im itedtobe inginf initesimal
bev iscouslyresistedthesev ibrations
wn andiftimeenoughisallowed
beenimpartedtothe li uidby the
sw illbe lost anditw illaga inberotating
asitw asinthebeginning.
U M E N E RG Y I N V O R T E M O T I O N .
ymotionof anincompressible
ite fi edportionofspace( thatistosay
ocityanddirectionofmotioncontinue
ntofthespace withinwhichthefluidis
gi env orticity theenergy isathorough
ughminimum oraminima . Thefurther
red bytheconsiderationofenergy
eadymotionfor whichtheenergyisa
thoroughminimum becausew henthe
heenergy iso fnecessityconstant. B ut
energydoesnotdecide theq uestion
steadymotioninwhichthe energyisa
mmencingw ithanygi enmotion
edindefinitelybyproperly-designed
( understoodthattheprimiti eboundary
w ithgi env orticity butw ithnoother
horoughma imumofenergy inanycase.
ptinthecase ofirrotationalcirculationina
e s s el r e fe r re d t o in ~ 3 ( I I I a b o e b e
eenergybyoperationonthe boundary
it i eboundary , asw eseeby thefo llowing
acommunicationreadbeforetheB rit ishA ssociation SectionA atthe
urday A ugust28 1880 andpublishedintheR eportforthat
nNature O ct. 28 1880. R eprintednow withcorrections
ions.
o 9 a bo e .
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N D MI N IM U M E N E RG Y I N V O R TE X M O T I O N 1 7
p a ra l le l a n d op p os i te l y ro t at i ng v o r te
endicularlybytwofi edparallelplanes.
thecylindricboundary theymay in
otion bethoroughlyande uablymi ed
.In thisconditiontheenergyis
lt ring reducedbydiminutionof its
gtube coiledwithintheenclosure.
yisinfinitelysmall.
co lumn w ithtwoendsontheboundary
ynearlymeetstheboundary and
dedtil l it isbro enintotwoe ualand
mns connected oneendofonetooneend
nishingv orte l igamentinfinitelynearthe
herdealtwithtill thesetwocolumnsare
ua lannihila tion.
present thee tremelydif f icultgenera l
orsuggested by theconsiderationofsuch
esnow totwo-dimensiona lmotionsina
edparallelplanesanda closedcylindric
dric surfaceperpendiculartothem
f f igure(buta lw aystrulycy lindricand
es . A lso forsimplicity conf ineourse l esforthepresenttov orticitye itherpositi eorz ero ine ery
iousthat withthelimitationtotwodimensionalmotion theenergycannotbeeitherinfinitelysmall
ygi env orticityandgi ency lindric
egi encondit ions therecerta inlyare
motions-thoseofabsolutema imum
ergy.Theconfigurationofabsolute
yconsistso f leastvorticity ( orz ero
f luidofz erovorticity ne ttheboundary
orticityinwards.Theconfigurationof
yclearlyconsistsofgreatestv orticity
ndlessandlessvorticity inwards. If there
rticity a llsuchf luidw illbeatreste ither
orinisolatedportionssurroundedby
. F oril lustration seef igs. 4and5
eninsosimpleacaseasthatof the
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sentedinthediagram therecanbean
teadymotions eachw ithma imum
a imum energy anda lsoaninf inite
otionsofminimum( thoughnotleast
ninfinitenumberofconfigurations
hof themha ingtheenergyathorough
12 , w esee byconsideringthecase in
ryofthe containing-canisterconsists
unicatingbyanarrowpassage as
ch acanisterbecompletelyfilled
gf luidofuniformv orticity thestreamlinesmustbesomethingli e those indicatedinf ig. 4.
tportionof thew holef luidisirro tational it isclearthattheremaybeaminimumenergy and
ationofmotion withthewholeof
tso f thecanister orthew hole in
proportioninoneandtherestintheother.
m-lines
owninf ig. 4.
n asshown
he
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N D MI N IM U M E N E RG Y I N V O R TE X M O T I O N 1 7 5
rationofma imum energy forwhich
i uidisintw oe ualparts inthe
f theenclosure.Thereisaninfinite
ofma imumenergyinwhichthe
isune uallydistributedbetweenthe
osure.
o w hentheboundary iscircular f f^ '
ntriccirclesand thefluidisdistri-
riclayersofe ua lvorticity . In the
umenergy thev orticity isgreatestat
andislessandlessoutw ardstothe
emotionofminimumenergythe
hea is andgreaterandgreateroutw ards
presstheconditionssymbolically
thef luidatdistancerf romthea is
ectionofthe motionisperpendicularto
andletabetheradiusof theboundary . The
pressiondiminishesf rom r = 0tor= a
ndofma imumenergy . If it increases
themotionisstableandofminimumenergy.
hes ordim inishesandincreases asr
themotionisunstable .
e le tthevorticitybeuniform
of thew holef luid andz erothrough
emotionof greatestenergy the
orticityw illbe intheshapeofa
i easo lidrounditsow na is
nearlyreachedintheyear1875 byrigidmathematical
brationsofappro imatelycircularcy lindricvortices but
blicationofitbyLord Rayleigh whoconcludeshis
ty orInstabil ity o fcertainF luidMotions ( P roceedingsof
Society F eb. 12 1880 w iththefo llowingstatement: Itmaybepro edthat if thef luidmo ebetw eentw origidconcentricw alls the
idedthatinthe steadymotiontherotationeithercontinually
ecreasesinpassingoutw ardsf romthea is " -w hichw as
me(A ugust28 1880 w henImadethecommunicationto
AssociationatSwansea.
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N D MI N IM U M E N E RG Y I N V O R TE X M O T I O N 1 7 7
dofinertia. Thewholemomentof
a 2 - 2 b 2 - M
fthisamountas longastheboundary
nsumptionofenergystillgoeson
oesonisthis: thew a esofshorter
tipliedande altedtilltheircrests run
uid andthoseofgreaterlengthare
rtionof the irrotationally re o l ing
iththecentralv orte column.The
atmaybeca lledav orte spongeis
omogeneous ona largescale butconsisting
dirrotationalfluid moreandmore
stimead ances. Themi ture isa ltogether
reof thew hiteandyellow ofanegg
ell- nownculinaryoperation.Letb
dricv orte sponge andg itsmean
chisthesamein allsensiblylargeparts.
1887. - Iha ehadsomedif f iculty innow pro ingthese
nd18 o f1880. Here isproof . Denotingforbre ity1/ 27rof the
u and1/ 27rof theenergybye w eha e
T r. r dr a n d e= | T f 2 .r d r.
eleastpossible sub ecttotheconditions:( 1 thatAthasa
t ha t
0
r= a T=Pb2/ a thislastconditionbeingtheresultanto f
hetota lvorticity ise ua ltothatof ' uniformw ithinthe
sdescribedinthe lastthreesentencesof~ 17andthe
yso l etheproblemw hen
2- b2 ; o rt -& g t - a2 .
18sol esitwhen
o r = b o a 2.
18sol esitwhen
2- b2 ; or & l t ' b 2a2 .
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the radiusoftheoriginalv orte column
0tor= b ,
2 /r f ro m r= b t o r = a
b/ b
2 + -
thecylindriccasef romgoingroundin
dingituntil somemoremomentof
mthefluid.Thenlea eitto itself
ongew illsw ellby theminglingw ithito f
tationalli uid. Continuethis
cupiesthe wholeenclosure.
cessfurther andtheresultwill be
ngcanister isallowedtogoround freely
endtoacondit ioninw hichacerta in
rte coregetsf i lteredintoaposit ionne t
dadistancef romthea isw hichwesha ll
thef luidw ithinthisspacetendstoamoreand
tureofvorte w ithirrotationa lf luid. This
onrepetit ionof theprocessofpre enting
ound andagainlea ingitf reetogo
andmorenearly irrotationa lf luid andthe
becomesthic erandthic er. The
ow
r & l t c
t c
ntumis
( a 2 - C2 } .
swhichthewholetends isabeltconstitutedof theoriginalv orte corenowne ttheboundary and
e o l edirrotationa lly rounditnow
eingthecondit ion( ~ 16abo e o f
y.B eginoncemorewiththecondition
a bs o lu t e ma i m um e ne r gy a n d le a e t h e fl u id
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N D MI N IM U M E N E RG Y I N V O R T EX M O T I O N 1 7 9
hecanisterfreeto goroundsometimes
pro idedonly it isult imate lyhe ldf rom
theultimateconditionisalwaysthesame
16 o fabsoluteminimumenergy. Theenclosingrotationa lbe lt be ingtheactualsubstanceof theoriginal
nitssectionalareatorrb2 andthereforec2=a2-b2.
m isnow -7r~ b4 beinge ualtothe
theportionoftheoriginal configuration
ntra lvorte .
e eninimagination thevery f ine
nanddrawing-outoftherotational
ingwiththe irrotationa lf luid andits
mtheirrotationalfluid whichthe
18hasforcedonourconsideration. This
andw esubstituteforthe" v orte sponge
somerespectsmore interesting conception
q uitearbitrarily andmere ly tohe lpusto
energy-transformationofv orte column
eattributetotherotationa lportionof the
alattractionbetweenitsparts" insensible
andbetweenitandtheplaneendsof the
suchrelati eamountsastocausethe interfacebetweenrotationalandirrotationalfluidtomeettheend
ettheamountofthisLaplacian
ly small- sosmall f o re ample thatthe
tchthesurfaceof theprimiti ev orte
timesitsareais smallincomparison
enf luidmotion. E ery thingw illgo
17 18if insteadof " runoutinto f ine
1 7 l i ne 2 9 w e s ub s ti t ut e " b r e a o f f in t o
orte co lumns ; andinsteadof " sponge
ute" spindrif t .
mmenergy forgi env orticity
momentum(thoughclearlynotuni ue
becausemagnitudesandordersof
nsofconstituentcolumnsof the
shitf romthe" New tonian" attraction because Ibe lie e
oroughlyformulated" attractioninsensibleatsensible
dedonita perfectmathematicaltheoryofcapillaryattraction.
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varied isfullydeterminateastothe
o lumnre lati e ly totheothers andthe
esasif itsconstituentco lumnsw ere
scouslye lasticcontainingvesse l each
describedin~ ~ 17 18 f l iesroundw ith
cityasthespindrif tcloudw ithin andso
nstably w ithoutlossofenergy untilthe
nstoppedor otherwisetamperedwith.
hattheLaplacianattractionwould
e co lumnstobrea intodetacheddrops( as
w ncaseofaf inecircularj eto fw ater
nwardsfromacirculartube andwould
fw atergi enatrestinaregionundisturbedbygra ity butitcouldnot becausetheenergyof the
efluidround thev orte columnmust
mncould brea inanyplace.The
how e er ma ethecy lindricform
cludedf romallsuchconsiderationsat
12 totw o-dimensiona lmotion.
nattractionand returntoour
fincompressiblefluidactedononly
gsurface andbymutualpressure
byno" appliedforce" throughitsinterior.
to fmomentumbetw eenthee tremepossible
2 a n d 7r t b4 t h er e i s cl e ar l y b e si d es t h e 1 7 1 8
egy anotherdeterminatecircularso lution
nofcircularmotionofwhichtheenergy is
er circularmotionofsamev orticity
mentum.Thissolutionclearlyisfound
intotwoparts-oneacircularcentra l
o theracircularcy lindricshell l in ingtheconta iningvesse l theratioofoneparttotheotherbe ingdetermined
otalmomentofmomentumha ethe
hisso lution( assa idabo e ~ 14andfootnote maybepro edtobeunstable.
e amongotherillustrationsof
sub ectdemandingseriousconsiderationandin estigation notonlybypurelyscientificcoercion
acticalimportance.
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N D MI N IM U M E N E RG Y I N V O R TE X M O T I O N 1 8 1
oncludew iththecompleteso lution
e solution( onlyfoundwithinthe
r~ 10-18of thepresentarticlew ere
blemonwhichI firstcommencedtrials
nergyanabsolutema imum intw odimensionalmotionw ithgi enmomentofmomentumandgi en
anistero fgi enshape. Theso lutionis
uni ue " absolutema imum meaning
ms. B utthesamein estigationincludes
roblem: Tof ind o f thesetsofso lutions
ferentconf igurationsof themotionha ing
entum.F oreachofthesethe energy
otthegreatestma imum forthegi enmoment
nterestingfeatureofthepractical
enowattainedisthe continuoustransitionfromanyonesteadyorperiodic solution throughaseriesof
s toanyothersteadyorperiodicsolution
eof operationeasilyunderstood and
rol.Theoperatinginstrumentismerely
o lumn orrod f ittedperpendicularlybetween
andmo ableatpleasuretoanyposit ionpara llel
re . It isshow n mar edS inf igs.
entingthesolutionofourproblemforthecase
asmallpartof itswholev olume
uid tow hiche igencyof timelimitsthe
orte lininguniformlytheenclosing
n thecentreofthestill waterwithin
ocityof thew aterinthevorte increases
toSb2/ aattheoutside incontactw iththe
thenotationof~~ 15and16. Now mo e
omitscentralposition andcarryitround
elocity& lt ~ b/ aand& gt Ib/ a . A dimple
beproduced runningroundalitt le in
butult imately fa ll ingbac tobemoreand
thestirreris carrieduniformly.If
yslowedtillthedimplegetsagain in
andisthencarriedroundinasim ilar
ra lw aysa litt lebehindtheradiusthroughthe
heangularve locityof thedimplewill
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sdepthanditsconca ecur aturewill
theangularv e locity isr b/ a thedimple
atis theenclosingwall w ithitsconca ity
inf ig. 7 andtheangularv e locityofpropagationbecomes~ ' b/ a .
essvorte be ltnow becomesdi ided
thetwoac uiredendsbecomerounded
at er .
i g . 7.
arrowheadsreferstothev elocityofthestirrerand of
e locityof thef luid.
evorte re fertove locityof f luid. A rrow headsinthe irrotationa lf luidrefertothestirreranddimple . A rrow headsinabcreferto
relati elytothedimple.
F i g . 9.
ertomotionof thestirrer andof thevorte asaw hole.
ottedcirclerefertoorbita lmotionofc thecentreof the
nfullf inecur esrefertoabso lutev e locityof f luid.
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NTA LILLUSTR A TIONO F MINIMUMENER GY18
arriedroundalwaysalittlerearward
ofabreastthemiddleofthe gap.F igs.
continuingthe processtillultimately
entra landcircular( w ithonly the infinitesimal
senceof thestirrer withwhichweneed
tpresent .
tanystageof theprocess a f tertheformationof thegap thestirrertobecarriedforwardtoastation
ofabreastthemiddleof thegap or
erearof thev orte ( insteadofsomew hatinad anceof thefrontasshow ninf ig. 8 . Theve locity
gmented( by rearwardpull ) , the
llbediminished:thev orte trainwill
eachesroundtoits rear eachbeing
andbroughtintoabsolutecontactwiththe
drear uniteinadimplegradually
processmaybe continuedtillweendas
rte l in ingthe insideof thew alluniformly
emiddle ofthecentralstill-water.
LILLUSTR A TIONO F MINIMU MENER GY.
I I i. N o . 1 8 1 8 80 p p . 69 - 70 B r i t i sh A s so c ia t io n R ep o rt
g. 3 0 p p . 49 1 -2 .
a li uidgyrostatofe actlythe
describedandrepresentedby the
printed from Nature F ebruary1 1877
ththedif ferencethatthef igureof theshell is
hee perimentwasinfactconducted
w hichw ase hibitedtotheB rit ish
876 alteredbythesubstitutionof a
toria ldiameterabout-9of itsa ia ldiameter
iameter29ofe uatoria ldiameterw hich
tuswasshownas asuccessfulgyrostat.
lswereeach ofthemmadefromthe
copperwhichplumberssoldertogether
f loaters. B ya litt lehammeringit iseasy
the propershapestoma eeitherthe
.
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tthe rotationofali uidina rigid
e ingaconf igurationofma imumenergy for
uldbeunstable if theconta iningv esse lislef t
rfectlyelasticsupports althoughit
- - - - -- -
esse lwereheldabso lute ly f i ed orborne
ts orlefttoitself inspaceunactedon
ditw asto illustratethistheory thatthe
df il ledw ithw aterandplacedinthe
efirst trialwasliterallystartling
a ebeenso asitw asmere lya
nanticipatedbytheory.The framewor washeldasfirmlyas possiblebyonepersonwithhis two
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NTA LILLUSTR A TIONO F MINIMUMENER GY185
steadyashecould. Thespinningby
passingroundasmallV pulleyof i- inch
theo a lshe llandroundalargef ly -w heel
dattherate ofaboutoneroundper
edforse era lm inutes. Thisinthecaseof
sk now nf rompre iouse periments w ould
f f icientrotationtotheconta inedwaterto
ctw ithgreatf irmnessli easo lid
erimentw iththeo a lshe ll theshell
thgreatv elocityduringthelastminute
momentitwas releasedfromthecord
f ramew or inmyhands Icommenced
ontalglasstabletotest itsgyrostatic
r w hichIhe ldinmyhandsga eav io lent
dinafewseconds theshellstopped
thepi otshadbecomebento er by
intheneighbourhoodofthestiff
so lderedto it show ingthatthe li uidhad
coupleaga institsconta iningshe ll ina
theef forttoresistw hichbymyhands
shellwasrefittedwith morestrongly
hee perimenthasbeenrepeatedse era l
ecideduneasinessof thef ramew or is
nho ldingit inhishandsduringthe
asthecordiscutandthepersonho lding
perimenta ltable thef ramew or
ow riggleroundinhishands andby thetime
onthe tabletherotationisnearly all
gyrostatispreciselywhatwas e pectedfromthetheory andpresentsatrulywonderfulcontrast
w iththeapparatusandoperationsine ery
ptinha inganoblate insteadofaprolate
d.
ongcordf irstw oundonabobbin andf inallyw oundup
helargew heelasdescribedinNature F ebruary1 1877
founditmuchmorecon enienttouseanendlesscordlitt le
rcumferenceof thelargewheel andlessthanhalfround
pulleyof thegy rostat andtok eepitt ightenoughto
ntialforceontheV pulley isdesiredby thepersonho ldingthe
ftercontinuingthe spinningbyturningthefly-wheelfor
gedproper theendlesscordiscutw ithapa iro f scissors
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G I N I N IT Y I N LO R D R A Y LE I GH S S O L U T I O N
P L AN E V O R TE X S T RA T U M .
I I I. 1 8 80 p p . 45 - 46 B r i t i s h As s oc i at i on R e po r t S wa n se a
1 88 0 p p. 49 2- .
E IG H S s o lu t io n i n o l e s a f or m ul a e u i a l en t t o
ma imumvalueof they -componentof
enotesaconstantsuchthat27r/ m isthew a elength , Tdenotesthetranslationa lve locityof thev orte stratumw henundisturbed w hichisinthex direction andisafunctionofy , ndenotesthev ibrationalspeed oraconstantsuch
isstable if ononeside it isbounded
dif thevorticity (orv a lueof IdT /dy
ideally f romthisplane e ceptinplaces
nstant.
supposeafi edboundingplaneto
rpendiculartoOy andletdT/ dyha eits
0 anddecreasecontinuously orbyone
f romthisva lue toz eroaty=aand
.
t he w a e - e l oc i ty w h at e e r b e th e
mediatebetweenthegreatestandleast
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O R T E X S T RA T U M
certa inva lueofybetw een0anda
y ise ua ltothew a e- e locity or
alueofythe secondtermwithinthe
sformula isinf initeunless forthe
T/ dy2v anishes.
considerationof thisinf inity ifw eta e
constantv orticity ( dT/ dy=constant
asforthiscasetheformula issimply
einfinity whichoccursinthe more
suggestsane aminationof thestreamlines byw hichitsinterpretationbecomesob ious andw hich
hecaseofconstantv orticity themotionhas
cterat theplacewherethetranslational
ew a e- e locity . Thispeculiarity isrepresentedby theanne eddiagram w hichismosteasilyunderstood
tona lve locit iesaty= 0andy= ato
andofsuchmagnitudethatthew a e elocity isz ero sothatw eha ethecaseofstandingw a es.
- l inesareasrepresentedintheanne ed
gionof translationalv elocitygreater
alvelocity isseparatedfromtheregion
lessthanw a e-propagationalv e locityby
ernofellipticwhirls.
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andTait sNaturalPhilosophy that
nto fmomentum there isone andonlyone
uil ibrium.
minthere o lutiona lf igure isstable or
f = - - - ) i s & l t o r & g t 1 - 9 4 57 .
mentofmomentumislessthanthatw hich
oreccentricity= -81266 forthere o lutiona l
tonly stable butuni ue.
mentofmomentumisgreaterthanthat
9457forthere o lutiona lf igure there is
o lutiona lf igure the J acobianf igurew ith
w hichisa lw aysstable if thecondit ionofbe ing
ut asw illbeseenin( f be low the
houttheconstra inttoell ipsoida lf igure isin
able thoughitseemsprobablethat in
houtanyconstraint.
msonandTa it sNaturalPhilosophy ~ 778
aagreatmultipleofb w eseeob iously
o ecmustinthiscasebev erysmallin
weha eav eryslendere llipso id longin
appro imate lyapro latef igureof re o lution
a i s w h ic h r e o l i n g wi t h pr o pe r a ng u la r
esta isc isa f igureofe uil ibrium. The
w hich w ithoutanyconstra int is inv irtue
tionofminimumenergyorofma imum
momentofmomentum isaconf iguration
enmomentofmomentum sub ectto
eis constrainedlyanellipsoid.F rom
iseasilyverif ied inthe lighto f~ 778of
uralPhilosophy itfollowsthat withthe
hee uil ibriumisstable . There o lutiona l
w iththesamemomentofmomentum
( 1 , ( 2 , a n d ( 3 ) , w i ll b e f ou n d in t h e fo r th c om i ng n e w
ait sNaturalPhilosophy V o l. I. PartII. [ Theproofsw ere
neralproblemhasbeenanaly edinacomprehensi e
fLordK el in smethods byH. Po incard inaclassica l
n I . 18 8 5. C f . La mb s H y dr o dy n a i c s C h . x n I .
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F E Q U I L I B R IU M O F R O T AT IN G F L U I D
eroid forittheenergy isaminima ,
mallestenergythatare olutional
omentofmomentumcanha e butit
oftheJ acobianfigurewiththesame
fbeinge ll ipso idalisremo edandthe
it isclearthattheslenderJ acobian
stable becauseade iationf romellipso ida l
ngit inthemiddleandthic eningit
f thic eningit inthemiddleandthinningit
ouldw iththesamemomentofmomentumgi e
h sogreatamomentofmomentumasto
slenderJ acobiane ll ipso id it isclearthat
e uil ibriumistw odetachedappro imately spherica lmasses rotating( asifpartsofaso lid roundan
eof inertia andthatthisf igure isstable .
aybe aninfinitenumberofsuchstable
proportionsof the li uidinthetwo
hesamemomentofmomentumthere
e uil ibriumw iththe li uidindi ers
wodetachedappro imatelyspherical
oninmorethantwodetachedmasses
dingtothedef init ionof ( Ic be low
ranyof them e enifundisturbedby
ouldha etruek ineticstability ata lle ents
caseof thethreematerialpoints
heau( seeR outh sR igidDynamics ~ 475
mthestablek inetice uil ibriumofa
ua lorune ua lportions so farasunder
tely spherica l butdisturbedtoslightly
by thew ell- now nin estigationofe uil ibriumtides gi eninThomsonandTait sNaturalPhilosophy ~ 804 ,
prolatefigureswhichwouldresultfrom
outchangeofmomentofmomentum
latef igures now note enappro imately
stable ispeculiarly interesting. Weha e
weentheunstableJ acobianellipsoid
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YNA MIC S [ 19
ty andthecaseofsmallestmoment
withstabilityintwoe ualdetached
nofhowtofill upthisgapwith
mostattracti eq uestion tow ards
ntoffer nocontribution.
ergyw ithgi enmomentofmomentumis
imum thek inetice uil ibriumis
uidisperfectly in iscid. Itseems
allyunstablewhentheenergyisa
k now thatthisproposit ionhasbeene er
i s c o si t y h o w e e r s li g ht i n t he l i u i d o r
ye lasticso lid how e ersmall f loating
thee uil ibriuminanycaseofenergy
a imumcannotbesecularly stable
econfigurationsarethose inwhich
withgi enmomentofmomentum.
a inwhetherw ithgi enmomentof
morethanonesecularlystable configuration
ousf luid inonecontinuousmass butit
thereis onlyone.
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9 )
A L I Q U I D W I T H IN A N E LL I PS O I D A L
gsof theR oyalSocietyofEdinburgh V o l. x III.
- 7 8.
cedthepropositionsregardingfluid
lhollowwhichformthe sub ectofthe
andw hich thoughob iousenoughand
donotseemtoha ebeenpre iouslydisco ered.
nhomogeneousrotation orhomogeneous
esignatetheconditionofa fluidinrespectto
outittheamountsof itsmolecularrotation
iallinesparallel.Thisdesignationclearly
tingsolid:butitis applicableofcourse
seofaf luid inw hichirrotationalmotion
mogeneousrotationasofasolid.To
otionthussignified considerthefollowing
h i ch ( 1 a n d ( 2 a r e in c lu d ed i n ( 3 ) : 1 L e t a li u i d k e p t i n th e s ha p e of a f i gu r e of r e o l ut i on
sse l begi eninastateofhomogeneous
of thef igure . Letanimpulsi erotation
rtothisa isbegi entotheconta ining
usmotionof the li uid atthe instant
eted consistsofanirrotationalmotion
enhomogeneousrotationalmotion.The
uiddoesnotgenerallyremainhomo
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T H I N A NE L LI P SO I D A L H O L L O W
heresultantof thisirrotationalmotion
enrotationalmotion.Theirrotational
phericalhollowisof courseeasily
nownsphericalharmonicanalysisforfluid
eonlytheinstantaneousmotion which
n theimpulseiscompleted.Theinfinitely
w or ingouttheconse uencesaccording
ns asto force orastochangingshape
notfollowat present.Itwillbefully
nw hichtheboundaryof the li uidis
egin with andisconstrainedtobe
a l. Itwil lbepro edthatinthiscasethe
uidremainsalwayshomogeneous.
thegeometrica l" stra in isessentia lly
tali uidcontainedwithinachanging
pro idedthatthemotionof thef luidbe
orbeatanyoneinstanthomogeneously
ousnessofthegeometricalstrainbeing
w sf romHelmholl ' sfundamentalprinciplesof
atthemolecularrotationmustcontinuehomogeneous itsmagnitude w henthere isanystretchingorcontraction
v ary ingin erse lyasthe lengthofa lineof
ction andthea ia ldirectionvary ingso
thesamesubstantialline.
ia tionf rome actnessinthee ll ipso ida lf igure thehomogeneousnessof therotationof the li uidis
here isno lim ittotheamountofde iation
whichmaysuper eneinconse uenceof
entotheboundary w hetherinthe
orofmotionwithoutchangeofshape.
thepresentto motionoftheboundary
w ef indit interestingtoremar thatw e
easingorindefinitelydiminishingthe
by properlyarrangedactionintheway
ngv esse l. Tocontinua lly increasethe
low ingrulemaybecorrect a lthoughIdo
fofit.Supposethecontainingv essel
ndthe li uidw ithinittoha eperfectly
thinthenote actlyellipsoidalhollow
maybegintomo eoritmaynot.
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semi-a es. F rom( 4 w ef indforthe
2-a2 a-b2
a ~ Y
C 2. 6
a 2
utionof theproblem sofarasconcerns
cityatanypo into f thefluid w hichis
noughinthesolutionof ahydrodynamical
e interestingine erycase andit iseasy
eitupto thedeterminationoftheposition
uidatany time andw emaytherefore
l y to t h e a e s o f t h e el l ip s oi d l et ( x p i
e t o fanyparticularparticle$ of the
v e l oc i ti e s ( d / d t d p /d t d / d t o f t he
y tothee ll ipsoidaree ua ltothedif ferences
s ( u v , w , o f t he a b so l ut e v e l o c it y o f $ ,
mponentsoftheabsolutev elocityofan
z ) rigidlyconnectedw iththeell ipsoid and
~) atthetimet. These lastcomponentsare
p X . . .. .. . .. . .. . .. 7 .
, a t t h e i ns t an t ( x , I ) c o i n c id e wi t h
eha e
8
2 ~ + a 2 ' a
ale uationsofthefirstorder fordetermining( x , i i intermsoft. Denotingd/dtby8 wemaywrite
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I T H I N AN E L LI P SO I D A L H O L L O W 1 9 9
o . . .. . . .. . . .. . . .. . . .. . 9
ntheusua lmannerw ef ind - / = 0. . .. . . .. . . .. . . . 10 ;
hedeterminantandremo ingthesuperf luousfactor8 w eha e
.....
a bc ( x + 7 +
;
thirdof ( 9 w eha e
X . . . .. . .. . .. . .. . ( 1 4 .
may ta eastheso lutionforanyoneof the
ample asfo llowsX = A coscot
b2 ) + c - ( 2 a 2 + ( . 2 I J
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YNA MIC S [ 20
e s
b 22
.
e plicitlythepositionofany chosenparticle
seitwould beeasytofindfromthem what
siertodothisf romtheunintegrated
tiply ingthef irsto f thesebya/ a2 thesecondby 3 / b2 andthethirdbyy / c2 andadding w ef ind
1
rbit l iesintheplane
.. . . .. . 18 ,
t.
o fe uations( 8 byx / a2 thesecond
by / / c2andadding w ef ind
1 9
a e
. .. . . .. . . .. . . .. 20 ,
ant.
tli esontheellipsoid( 20 ; andwe
heellipseinwhichthis ellipsoidiscut
e plicitfully integratedso lution( 15
hataparticleof thef luiddescribes re lati e ly
hichthef luidisconta ined thee ll ipse
20 , accordingtothe law ofasingleparticle
the influenceofaforcetowardsa fi ed
proportiontodistancefrom thecentre.
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A NDSMA LLOSC ILLA TION O F A PER EC T
N E AR L Y ST R AI G HT C O R E L ES S V O R TI C ES .
rtoPro fessorG. F . F it Gerald. F romtheProceedings
my readNo ember3 0 1889.
rmedonethingIw asgo ingtowritetoyou
my lettero fOctober26 , v i . thatrotationa l
absolute lydiscarded andw emustha e
o lutionandvacuouscores. Sonotto
o f c or e le s s v o r te w o r ( ' V i b ra t io n s of a
P ro c. R .S .E . M ar ch 1 1 88 0 , H ic s P ap er ' O n
mallV ibrationsofaHollow V orte , '
1884 , willbethebeginningof the
erandmatter if it ise ertobeatheory .
ossinglinesofvorte co lumn isimpossible
utispossiblew ithv acuouscoresandpure ly
oundthem.Theaccompanyingdiagram
pla inbyanil lustration. Itshow stheshape
rica lv acuousv orte co lumnasdisturbed
f i edinaplaneperpendiculartothea is
a ingirrotationa lcirculationthroughitse lf .
acuum thespaceoneachside li uid
essectionof thetore. Thecur esrepresentingtheboundaryof thev orte areca lculatedtogi euniform
erthew holesurfaceof theho llow core .
antof thev elocitiesduetothecirculation
e andtocirculationthroughthetore . The
erseproportiontodistancef romthea is
The latterisappro imate lyparalle lto
erseproportiontothecubeof thedistance
rcularcrosssection l i eananchorring.
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orticesisstable butitsquasi- rigidity
siona lmotion w ithoutchangeofv o lume
edingly small andthecorrespondinglaminar
ingly sluggishincomparisonw iththetensile
orrespondingwa e- e locity w hichweshould
rmotioninplanes paralleltotheplane
erygreatnumberofplanesina lldirectionsasmanyw ithinanangleof1~ ofanyoneplane asw ithin1~ of
eadistributionofstraight v orte
presentedinf ig. 2 , perpendiculartoe ery
res beingthinenough theymaybe
nooneofwhich intersectsanyother.
ev orticeswillproducedisturbances
w hichw esupposedthemgi en and
ationsf rome actlycircularf igure in
dtherewillbesluggish motionsofthe
lplaced soastofulfil adefinitecondition
enif thisdefinitecondit ionisnote actly
asi- rigidity andcorrespondingve locityof
edium thusk inetica lly constituted w ill
mwhat theywouldbeifthe v ortices
eabsolutelysteadymotionforthe
themedium.
uslyconsideringtheef fecto f f reevorte
esamongthevorte co lumnsof thistensile
ggestedforcoredvorticesattheendofyour
6 1889 to lNature . Itw illbean
dynamicalq uestion thoughitseemsto
tletowardse pla ininguni ersa l
propertyofmatter soyoumay imagine
chemistryandelectro-magnetism.
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F IC IENC YOF SA ILS W INDMILLS SC R EWPR OPELLERSINWA TER A NDA IR A NDA ER OPLA NES.
o l . L. 1 8 94 p . 4 25 .
wee , onflyingmachines inthe
not forw antof t ime carriedsofaras
rica lresultso fobser ationputbefore
m thattheresistanceof thea iraga inst
omo eatsi tymilesanhourthrough
inedtotheplaneataslopeofaboutone
eaboutfifty-threetimesas greatasthe
ld" theoretica l ( ) f o rmula andsomethingli e f i eortentimes thatcalculatedf romaformulawritten
ordR ay le igh asf romapre iouscommunicationtotheB rit ishA ssociationatitsGlasgow meetingin
rewasnov a lidity e enforrough
nanyof the" theoretica l in estigations
wwildlytheyall fallshortofthetruth
ehadopportunity inthe lastfew days
aminesomeof theobser ationa lresultsw hich
introductiontohispaper. Ontheother
erdoubtedbutthatthetruetheoryw astobe
tcon ersationallybyWill iamF roude
ch thoughIdonotk now of itsha ing
hitherto isclearlyandterselye pressed
hichI q uotefromatype-writtencopy
rMa im ofhispaperof lastw ee : Thead antagesarisingf romdri ingtheaeroplanesonto
w hichhasnotbeendisturbed isclearly
ments.
es seefootnote p. 219infra .
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p le I h a e a t l as t I b e li e e s u cc e ed e d
meapproachtoaccuracy theforcere uired
ow rectangularplanemo ingthroughthea ir
e locity V , inadirectionperpendicularto
clinedatanysmallangle i to itsbreadth a .
etobeabletocommunicatetothe
intimeforpublicationinitsne tO ctober
ethe in estigation includingconsideration
andproof thatit iso fcomparati e ly small
muchmorethan1/ 10 or1/ 20 o faradian
somepractica lly smooth rea l so lidmateria l.
stheresult w iths in- resistanceneglected:Theresultantforce( perpendicular therefore totheplane is
w h ic h i s v r c o s i /s i n i ti m es ( o r f or t h e ca s e of
ndredtimes , theo ldmisca lled" theoretical
merica lfactorw as2insteadof1.
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07 )
NC E O F A F L U I D T O A P L A NE K E P T M O V I N G
REC TIO NINC LINEDTO ITA TA SMA LLA NGLE.
ca l M ag a i n e V o l . x x x v i I I. 1 8 94 p p. 4 0 9 â € ” 4 1 .
ity i itsinclinationtotheplane and
nandperpendiculartotheplane. Weha e
i n i.
ingbodytobenot anidealinfinitely
f f initethic nessverysmallincomparison
andha ingitsedgese erywheresmoothly
iscidandincompressible andthe
fectlyunyielding themotionproduced
byanymotiongi entothedisc isdeterminate ly theuni uemotionofw hichtheenergy islessthanthat
eto thefluidwiththegi enmotion
thedisctobev ery thin andtherefore
te erypo into f itsedgetobeverygreat:
nessat whichthepropositioncould
til lho ldsinthe idea lcaseofaninf inite ly
andits boundaryfulfiltheidealconditionsofthe enunciation.
ry f luidhassomedegreeofv iscous
hape andanyv iscosityhow e ersmall
erfectincompressibil ityof thef luidandun-
ary w ouldpre entthe inf inite lygreat
f thediscw hichtheuni ueminimumenergyso lutiongi esw henthediscisinf inite ly thin andw ould
ancein themotionofthefluid that
nof thediscwouldprobablybev ery
ertheactua lv a lueof thev iscosity ifnot
ththe v elocityofthediscmultiplied
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atureof theboundaryof itsarea. No
hashithertobeenmadetow ardsacomplete
nycaseofthis problem orindeedofthe
sapethroughav iscousf luid e ceptw hen
lso lutionsfortheglobeandcircularcy linder
tsconfigurationisthe sameasitwould
w andw hentherefore theve locityof
se ua lto andinthesamedirectionas
lacementofanelastic solidwhena
sheldin apositioninfinitesimallydisplacedfromitspositionof e uilibrium inthemannertranslationally
dingtothetranslationalandrotational
rigidbody inthefluid.
guidedby theteachingofWill iam
ntinuedcommunicationofmomentumto
of forcetok eepaso lidmo ingw ith
ocity throughit thatanappro imate
tance w hichisthesub ectof thepresent
robablybefoundbythefollowingmethod
~ 9 w hichIv enturetogi easaguess
mathematicalin estigation.
nitethic ness how e ersmall mo ing
ssible li uidw ithinanuny ieldingboundary
hin ingonlyof theu-componentof themotion
of~ 1 le tEandE denotethef rontand
respecti e ly . Imaginenow insteadof
ary ingso liddiscthroughthef luid that
E by rigidif icationandaccretionof the
meltsaw ay fromE by li uefactionof the
me3t thee tentof theaccretionin
fthe v -componentofthemotionof
houtdiminutionduringthisaccretion
l to ( r I - I / t m u st b e a pp l ie d f ro m wi t ho u t
Idenotingthe impulsi e forcew hich
ethev -componentv e locity totheunaugmenteddisc andI thatre uiredtogi ethesamev elocity to
pointofapplicationofthe force
orthesteady infinite ly slow motionofav iscousf luidare
e uilibriumofanelasticsolid. SeeMathematicaland
. T ho m so n , V o l . II i . Ar t . c i . ~ ~ 1 7 1 8 .
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ST A NC E O F A F L U I D T O A P L A N E
hatof theresultanto f impulsesIand- I
ntresof inertia oftheaugmenteddisc
crespecti ely.
rigidity byli uefactionofany
e ly small o fmattero f thediscatE
ousapplicationof force topre entchange
sidualsolid.Thecontinuedgradual
resupposingperformed lea esaHelmholt
f f in i te s l ip g r ow i ng o u t in t h e li u i d b e hi n d E ,
ortionsofwhicharenot easilyfollowedin
tisintheformofapoc etofw hichthe
dtothesolid disc.Thespaceenclosed
f il ledby the li uidw hichw asso lid.
ndlongerbyga inof li u idf romthe
ontof it andprobablya lsoby itsrear
arther faraw ay inthew a eof thedisc.
a f terha ingbeenperformedduringa
aprocessesof~ ~ 5 6arediscontinued and
e ualandsim ilartotheorigina ldisc but
hroughaspacee ua ltouT isle f tw ith
oughthef luidmainta ined. Thepoc etof
fartherandfartherbehindthedisc. Its
pedby theso lid w il lshrin f romitsoriginal
eofE andw illbecomealw ayssmallerand
eysmallinany f initetime. Thenec o f
eof thediscw illbecomenarrowerand
epoc etw illbedraw noutlongerand
hroughallt ime thef luidw hichwasso lid
surfaceof finiteslip orHelmholt
r om t h es u rr o un d in g f lu i d e c e pt o e r t he e e r
sc w hichstopsthemouthof thepoc et.
rotationaloutsidethepoc et and
eepthesoliddiscmo ingw ithits
ndwithnoothermotionwhetherrotational
ecessary toapply forceto it. B utthisforce
andappro imatestoz ero asthevorte * Ica llthe" hydrauliccentreof inertia o famasslessrigiddiscimmersedin
itmustbestruc perpendicularlybyanimpulse togi e
tion.
14
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r andthemotionofthe fluidinthe
appro imatesmoreandmorenearlyto
euni ueirrotationalmotiondueto
ughthe fluid.
~ ~ 5 6 7 b ee n on s ur e gr ou nd a nd
rously true notonly fora" disc o fany
fany thic nesshowe ersmall but
pe dea ltwithaccordingto~ 5
f luidisin iscidandincompressible and
Myhypothesis or" guess ( ~ 4 , w hich
f thepresentpaper isthatdefaultf rom
tofallthesethree conditionswould
eptmo ingwithuniformtranslational
1 , r e u i re t h e co n ti n ue d a pp l ic a ti o n to i t o f fo r ce
andposit ionby~ 5 pro idedv bev ery
outw ithgreateaseforthecaseofa
he length 1 isv erygreatincomparison
orthiscase by thew ell- now nhydro ineticsofane ll ipso idore llipticcylindermo ingtranslationa lly
ssible f luidofunitdensity w eha e
tionof~ 5
2 I .
7 r al u ;
intofapplicationofthis forcefromthe
is 4a.
hetica lresult w ithobser ation in
udeof theforceandits pointof
pe form thesub ectofa futurecommunication.
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F A H E TE R O G E NE O U S L I Q U I D C O M M EN C IN G
G IV E N M O T I O N O F I T S B O U N D A R Y.
gsof theR oyalSocietyofEdinbw rgh V o l. xx I.
id forbre ity todenoteanincompressible
scid butin iscidunlessthecontrary ise pressly
i u i d v i s ci d o r in i s ci d b e in g g i e n
ingv esselo fanyshape w hethersimplyor
tanymotionbesuddenlyproducedin
ry orthroughouttheboundary sub ect
dtionofunchangingvo lume. E ery
linstantaneouslycommencemo ingw ith
yandinthe determinatedirection such
fthewholeis lessthanthatofany other
couldha ew iththegi enmotionof its
tionisalso trueforanincompressible
( andforthe idea l" ether o fP roc.
andA rt. xcI . v o l. III. o fmyC ollected
lPapers .Thetruthof theproposition
li uidisvery importantinpractica l
pleof itsapplicationto in iscidand
sticso lid considerane lasticj e lly standing
ande ualbul so fw aterandofanin iscid
nMathematica lJ ournal F eb. 1849. Thisisonlya
k inetictheoremforanymaterialsystemwhate er
a lSociety Edinburgh A pril6 186 , w ithoutproof
6 p . 1 14 a n d pr o e d i n Th o ms o n an d T ai t s N a tu r al
w ithse era le amples. Mutualforcesbetw eentheconta iningv esse landthe li uidore lasticso lid suchasarecalledintoplayby
hesi ity ( orresistancetoslidingbetw eenso lidandso lid ,
sion anddonotenterintothee uationsusedinthe
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e u a l an d s im i la r t o it . G i e e u a l su d de n
ningv essels:theinstantaneousmotions
bstancesw illbethesame. Ta e asa
eof re o lutionw ithitsa isv erticalforthe
dletthegi enmotionberotationroundthis
edandafterwardsmaintainedwithuniform
itia lk ineticenergyw illbezeroforeach
Thein iscidli uidw illremainfore er
c uiremotionaccordingtotheF ourier
k nowsomethingforthiscaseby
tofgi inganappro imatelyuniform
ertica la istoacupof tea init ia llyat
uire laminarw a emotionproceeding
y.B utinthepresent communication
othecaseof in iscidli uid.
ution oftheminimumproblemthus
undingsurface issimplycontinuous is
ionoftheli uidisirrotational.
tbe irrotationaltisindeedob ious
mpulsi epressurebywhichany
etinmotionise erywhereperpendicular
and thecontiguousmatteraroundit
mentofmomentumroundany
ricalportion largeorsmall isz ero . B ut
otionofe erysphericalportionofthe
nethemotionwithin asimplycontinuous
tatedmotion isnotob iousw ithoutmathematica lin estigation.
simplycontinuous ormultiplycontinuous irrotationalitysufficestodeterminethemotionproduced
eproduced f romrestbyagi enmotion
uidactedonbynobodily force or
i t y f o r e a m pl e a s c ou l d no t m o e i t
ed themotionstartedf romrestbyany
ary remains a lw aysirrotationa l asw e
NaturalPhilosophy ~ 3 12.
suchthatthemomentofmomentumofe eryspherica l
isz erorounde erydiameter.
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hetends toha efaithinall assertions
nce.
y isanenclosingv esselo fany rea lmaterial
erfectly rigidnorperfectlye lastic andif
f tto itse lf underthe inf luenceofgra ity
perfectly in iscid w il l loseenergycontinua lly
heconta iningv essel andw illcome
econfigurationof stablee uilibrium
ensityhori onta landincreasingdensity
dt io n s as i n ( 3 ) , b u t n o gr a i t y t h e
estwillbe infinitelyfinemi ture
o f e u a l de n si t y th r ou g ho u t . C on s id e r f o r
ogeneousli uidsofdif ferentdensit iesf i ll ingthe
nglehomogeneousli uidnotf i ll ingit. A san
ttlehalf fullo fw ater andsha eitv io lently .
hew holebottle fullo fami tureof f ine
mogeneousthroughout.Thin whatthe
erenogra ity andif thewaterandair
ott lesha enasgentlyasyouplease and
cuuminplaceof theair or if f o ra ir
uidofdensitydifferentfromthat of
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E O F D I SC O N T I NU I T Y O F F L U I D M O T I O N I N
THER ESISTANC EA GA INSTA SOLIDMO V ING
D* .
l . L. 1 89 4 p p. 5 24 5 49 5 7 , 5 97 .
discontinuity " thatistosay f inite
ntw osidesofasurface inaf luid w ould
dincompressiblefluidwerecausedto
arigidsolid withnov acantspacebetween
e li e e f i rs t g i e n b y St o e s i n 18 4 7t .
owwell- nowndynamicaltheorem
iscidf luidinit ia llyatrest andsetin
dto itsboundary ac uirestheuni ue
ughoutitsmass ofwhichthek inetic
anyothermotionofthe fluidwiththe
ry.
dforthe formationofasurfaceof
dfluid wastheinfinitelygreatv elocity
andthecorrespondingnegati e - infinite
euni ueso lution unlessthef luidisa llow ed
ntactwiththesolid. Thisanin iscid
nlywoulddo unlessthepressureofthe
e erywheree ceptattheedge. In
erygreatnegati epressurearisingf rom
ncationsformedthesub ectofapro longedplay fulcontro ersy
ndhisintimatef riendSirGeorgeSto es inaseriesof letters
r ed.
o l . i. p p . 3 1 0 3 1 1 .
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afluid flowingroundacorneris always
f threedefa lcationsf romourideal: I V iscosityof thef luid pre entingthee ceedinggreatnessof thev e locity .
yof thef luid.
theouterboundaryof thef luid.
isinmanypracticalcaseslarge lyoperati e
t( II isa lso large lyoperati e insome
suchasthewhistlingofastrongw ind
rnerorthroughachin theblow ing
theenmbouchureofanorgan-pipe and
geo letoro fasmall" w histle andthe
tubeorahole inthesideof atube
etosound.
i s l ar g el y o pe r at i e a n d ( I I b u t li t tl e
mostcommonoccurrenceintheflowof
muchofthefoamseennear thesides
ew steamergoingatahighspeedthrough
ueto" v acuum behindedgesandroughnessescausingdisso l eda irtobee tractedf romthewater. A
hdiameter and1/10ofaninch thic
ulytothefigureof anoblateellipsoidof
useavacuum* tobeformedallrounditsedge
mallav e locityasIfootpersecondunder
han6 feet ifw aterw ere in iscid: and
tonw ould onthesamesupposit ion be
nov acuum andw ouldbee actly in
ueminimum energysolutiont.
ntespacev acatedbyw ater.
ydro ineticsofthemotionof anellipsoidthroughan
f luid originatedbyGreen w hof irstga etheso lutionfor
motionof thee llipso id w ek now that if0denotesthe
tothesurface atanypointandthe a isofanoblate
ofw hichthee uatoria landpolarsemia esarea b the
wngo erthispo into f thesurface is
udatgreatdistancesf romtheso lidisV , andinpara lle l
e ldf i edinthef luid w ithitsa ispara lle lto these lines.
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NE O F F I N IT E S LI P I N F L U I D M O T I O N 2 1 7
hef luidacrossthee uatoris6 ' 7feet
ocityacrosseachof thetw oparalle lcircles
nches( theradiusof thee uatorbe ing
persecond.
y rapidchangeofshapeof thef luid
toria lz onebetw eenthesecircles w ith
augmentingf rom1footpersecondto6 ' 7
ncingo eradistanceof lessthan' 85of
moneof thesmallcirclestothee uator
m6 ' 7to1f romthee uatortotheother
tionofasecondof timew ould if thef luid
i u i d g i e r i se t h ro u gh v i s c o s it y t o
thema imum v e locity andcausing
themotionofthewaterto differgreatly
energysolution notonlynearthe
e o r o e r t he r e ar s i de o f t he d i sc b u t o e r
thoughnodoubtmuchmoreontherear
thanonthef rontsideandinthef luid
atlessdepthsthan6 feet ha e
downthema imumvelocity andit is
0or20feetagreaterve locity than1foot
uiredtoma ev acuumroundthee uator
meterandthe1/2000of aninchradius
llipticmeridiona lsectiongi esit. B utit
theremust bemuchformingofv acuum
actionofa irandrisingofbubblestothe
whatsharpcornersandroughnesses of
dinary ironsa il ingshiporsteamer go ing
ek nots( thatis 20f t. persecond .
whichvacuumisformedatanedgeofa
iscidincompressible f luid underpressure
atdistances fromthesolid asuccession
formula wereduceitto 200/7r.( V sin0 appro imatelywithin
n g si n 0 = 1 a n d V = l f o o t p er s e co n d w e fi n d 6 - 7 f ee t p er
acrossthee uator. Hencethegra itationa lheadcorrespondingtothe" negati e -pressure is( 6 -72-12 / 64-4 orv eryappro imately
e s t he s t at e me n t in t h e te t .
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restingpieceofmathematicalhydro inetics
ntinuationofthepresentarticlein which
f f luidmotion e tendingfarandwide
sinmanyscientif icpapersandte tboo ssinceSto es inf initesimalrif tstartedit in1847 w illbe
ngdiagram( f ig. 1 i l lustratesthe
e inquestion toadisck eptmo ing
A
aconstantve locity V , perpendicularto
ptiontow hichIob ectasbe inginconsistentw ithhydrodynamics andvery farf romanyappro imationtothetruthforanin iscidincompressible f luidinany
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NE O F F I N IT E S LI P I N F L U I D M O T I O N 2 2 1
tterlyatvariancew ithobser ationofdiscs
s causedtomo ethroughw ater is that
representedbythetwocontinuous
ande tendingindef initely rearwards there
nuity ontheoutsideofw hichthew ater
hedisc w ithv e locityV , andonthe inside
ssmassof " deadw ater fo llow ingclose
stancyofthev elocityontheoutside
discontinuityentailsfor theinsidea
thereforeq uiescencere lati e ly tothedisc
deadw ater. How couldsuchastateof
ndw hatisit inrespecttorear are
ggestto theteachersofthedoctrine
go inginforane aminationinhydro inetics Ineednottry toansw er.
u pp o si n g th e m ot i on o f t he d i sc t o h a e
time t ago andconsideringthe
~ 9 f o r fi n it e ne s s of i t s wa e l e t ab b d b e
dtherear andbeyondoneside o f the
ssonlythroughwaternotsensiblydisturbed.
nitecaseofmotiontodea lw ith instead
teoneof~ 11. Letustry if it ispossible
efrom theedge andfromthediscon
ouldbee enappro imate ly ifnotrigorously
andindicatedby thediagram.
e l oc i ty a t a ny p o in t i n th e a i s A a a t
rearwards.Drawedperpendiculartothe
re lati e ly tothediscsupposedatrest.
eedis0 , , , , , , db , V x db , , , ba , 0
, , , , A e , O , b yh yp ot he si s.
pressionofpracticalhydraulics adoptedby theEnglish
finiteslipbetweentwopartsofa homogeneousfluid to
eati e ly tothedisc.
( T h om s on , T r an s . R. S . E .1 8 69 .
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on intheclosedpo lygonedbaA e
ationinthesamecircuit a tatime
T w henthe linebahasmo edtotheposit ion
d , w e ha e
f o r th e l at e r ti m e t + r t h e v e l oc i ty i n A a a t
e circulationinedbaAetgainsin time
oremof" circulation " + must bee ualto
imeT o fa llthevorte -sheetinits
rdingtothe statementof~ 11.Hence
0
- ( v ' - v ) dy.
ow thatthef luidhasonlycontinuous
hafinitespaceall roundeachofthe
A a n d al l r ou n d Ae e c e pt t h e sp a ce o c cu p ie d
eyonditsf rontside w eha e forthe
smotion re lati e ly tothedisc
z , t ,
ocity-potentialofthemotionrelati eto
l round:andweha ealongAa
0 y 0 t .
o f~ 14becomes
0 0 0 t + r } - { ( 0 0 0 t } .
culationinabb a isz ero andthereforethecirculationin
to t h at i n e db a Ae .
sthatthere isasmoothstreamlinef romtheneighbourhoodof
idea llthev orte motion foronly thenisthecirculationin
n " T r an s . R. S . E .1 8 69 .
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NE O F F I N IT E S LI P I N F L U I D M O T I O N 2 2
f inite ly small
t .
omttot+ T therehasbeen according
n~ 11 agrow thofv orte -sheetf rom
eingthemeanbetw eenthev e locit iesof the
ndthecirculation perlength1of the
Hencethev orte -circulationof the
intimeT byV Tx V : andtherefore
2 .
testhepressureof thef luidatgreat
e locity re lati e tothediscisV , andpthe
erearsideofthe disc beingthesame
ha e bye lementaryhydro inetics
o , t ,
thef luidate erypo into f therearside
rdingtotheassumptionof " deadwater.
asthepressureontherearsidegi enby
ionofanendlesse erbroadeningw a e
r o e s t ha t o ur s u bs t it u ti o n ( ~ 1 ) o f a f in i te
once i ablypossibleastheconse uence
onatsomef initetime t ago instead
nf igurationdescribedin~ 11 doesnota lter
deofthedisc.
tionofthe fluidforsomefinite
nbothitssides thesame orv ery
m asthatdescribedin~ 11 theforcethat
pitmo inguniformlywouldbethesame
y thesame asthatcalculatedbyLord
ofthefluid supposedtobewhollyas
enschaf tl icheA bhandlungen V o l. i. f ooto fp. 151.
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ha ew eforsupposingthev e locity
onthef rontsideof thedisc tobe
imate lye ua ltotheundisturbedvelocity
atdistancesf romthedisc NonethatIcan
dprobablethatitis inrealitymuch
nweconsiderthat w ithin iscidincompressible f luidinanuny ie ldingouterboundary theve locity in
4 ise ua ltoV ate ensofarf romthe
a n d in c re a se s f ro m V t o 6 ' 7 x V b e tw e en
e andtheedgewithits 1/2000ofan
e.
deadw ater incontactw iththe
whichthedoctrineof discontinuity
erealityandyouw illseethew aterinthe
ye erywheree ceptatthev erycentreof
dyingroundfromtheedgeand
yclosea longtherearsurface o f tenIbe lie e
city thanV , butw ithnosteadiness on
rbulentunsteadinessutterlyunli e the
erallyassumedinthedoctrineof discontinuity.
sa fe lyconcludethatonthef rontside
ss thanthatcalculatedbyRayleigh.
stanceis partiallycompensatedoris
dminutionofpressureontherear ismore
mtheoryalone inaproblemofmotion
yondour powersofcalculation:butwe
belie e bye periment. R ay le igh s
istancee periencedbyaninf inite ly thin
bytwoparallel straightedges when
ganin iscidincompressible f luid w ith
, inadirectionperpendiculartotheedgesand
eplane gi esaforcecuttingtheplane
cefromits middlee ualto
esfortheamountof thisforce ingra itation
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NE O F F I N I T E SL I P I NF L U I D M O T I O N 2 2 5
aofonesideof theblade andPthe
fluidofunitcross-sectionalarea andof
htf romw hichabodymustfalltoac uire
.
~ 11 onw hichthisin estigationis
cityof f luidmotion re lati e ly tothedisc
. Itgi esv e locity reachingthisva lue
ade andatthesupposedsurfaceof
thef luidatinf initedistancesa llrounde cept
eof " deadw ater w herethev e locity
pressuree ua lto Ia llthroughthe" dead
sit increasethroughthemo ingf luid f romII
andatthe" surfaceofdiscontinuity toa
Patta inedatthew ater-shedlineof the
difV bee ensogreatas120feet
e locityofsound * P w ouldbeonly
dingaugmentationofdensitycould
angeofthe motionfromthatassumed:
sin estigationa irmayberegardedas
ev elocityofthedisc isanythingless
hisformulafortheresistance by
carefule perimentsmadebyDinest
discsandbladesmo edthroughitat
70statutemilesperhour( 59to10 feet
fornormalincidencetheresistanceaga inst
mo ingthroughairatmB rit ishstatutemiles
to ' 0029nm2ofapoundw eight.
ecificgra ityoftheair as1/800 gi es
f~ 21
uareplateofareaA . A tthefooto f
ne1890 Dinessaysthathef indsthe
blade tobemorethan20 percent.
eplate . F ora bladewemaythereforeta e
H w hereHis" theheightof thehomogeneousatmosphere.
15
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gtoDines e periments. Thisis2 94
latedf romR ay le igh sformula( 21
moreandmoreobli ue thediscrepancy
us f romcur esgi enbyDines( p. 256
eigh sresults If indthenormalresistance
ughairinadirectioninclined30~ to its
imesthatgi enbyR ayle igh sformula . A ndby
es cur eatthepo intinw hichitcuts
e If indthat forv erysmallv a luesof i it
urtimesthev alueoftheforcegi en
orv erysmallv a luesof i w hichis
doublethatgi enbymycon ectura l
ust3 0 p. 426 andPhil. Mag. October
a l ue s o f i w hi c h i s
er mere lycon ectura l andIw asinclined
siderablyunder-estimatetheforce*. That
isperhapsmadeprobablebyits somewhat
es becausetheblade inhise periments
o faninchthic inthemiddlew ith
Aninfinitelythinbladewouldprobably
istances ata llangles andespecia llyat
the wind.
eha eherebeendoubled inadditiontootherslight
outtheappro imateagreementwhichwasfoundby
rinformationregardingtheresultso f recentin estigation
boratory DrT . E. Stantonw rites( O ct. 7 1909 as
diagramanne ed. " Theva lueofDines coeff icientq uoted
remar ablyw ellw ithourresultsherew henaccountista en
si eof theplate asyouw illseef romtheencloseddiagram.
otalresistanceperunit areawithsi eisentirelydue to
ctatthe bac oftheplateasthe dimensionsincrease.
resonlongnarrow platesandoninclinedplates our
eementw iththoseofDines andonourin estigatingthe
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NE O F F I N IT E S LI P I N F L U I D M O T I O N 2 2 7
edemonstrationthatthedoctrineof
rf romanappro imationtothetruth is
edingly interestingandinstructi emanner by
f thepressuresonthetw osidesofadisc
lati ew indof60statutemilesperhour
roducedbycarry ingitroundattheendof the
machine. Theobser ationsw eredescribed
RoyalMeteorologicalSocietyin May
neof thesameyear intheR oya lSociety
redto hestatestheresults w hichare
ontsideanaugmentationofpressure
arsidea diminutionofpressure
by1 82and' 89inchesofw ater w erefound.
sofair ofdensity1/800ofthatof water
1211and59~ feet. Theformerisina lmost
rigorousmathematica ltheory foranin iscid
hichgi es882/64 4 or1201feetforthe
efoundthatthee cessinthetota lresistanceo erthat
sformulaw as asinthecaseofnormalimpingement due
eddiesontheleewardside.
eplates.
i
10
n s ma d e by M . Ei f fe l o n fa l li n g pl a te s .. , , , , , a t N .P .L . i n a cu r re n t of a i r. , , i n t he w i nd . , , , , , b y M r Di n es o n w hi r li n g ta b le .
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YNA MIC S [ 25
epressureatthe water-shedpointor
nshapemo ingthroughitattherateof
er showsthatthereisa " suction
dev erynearlye ua ltoha lf theaugmentationofpressureonthef ront insteadof therebe ingneither
essureastaughtin thedoctrineofdiscontinuity
t o t/
r/////////////////////
ng d i ag r am s ( 2 3 , 4 5 r e pr e se n t se e r al
of discontinuityinthemotionofan
tracti etow ritersonmathematica lhydro
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NE O F F I N I T E SL I P I NF L U I D M O T I O N 2 2 9
sentedinF ig. 1( w hetherasitsstands or
incidence becauseeachisinstantly so luble
aysis andtheydonot l i e it inthe
constituteillustrationsofthebeautiful
findingsurfacesofconstantfluidv elocity
surfacesa longw hichthev e locity isnot
yHelmholt * , de e lopedinamathematica lly
byKirchhof f t andva lidlyappliedtothe
ontracta byR ay le igh+ .
t( notnecessarilyo fcircularcross-section
harpedge intoavery largev o lumeof
asthato f the j et isrepresentedinF ig. 2.
sideredbyHelmholt ~ , bothforthe
siblefluidandforreal waterorrealair.
o rbelie ingthat w ithrealw aterorreal
omthemouthasgreatasse era lt imesthe
rthe leastdiameter if it isnoto fcircular
undingf luidisnearlyatrest andthe j et
thek indofmotionithad inpassing
forethattheefflu isnearlythesame
esthesame itw ouldbe if theatmosphere
hargedwereinertia-less.Thisconclusion
nceinpracticalhydraulics hasbeen
perimentsmadeeightyearsagointhe
ni ersityofGlasgow by tw oyoung
a y MrC appsandthe lateMrHew es.
tedandconf irmedbyothere perimenters.
stapplicationofthedoctrineof discontinuitytothetheoryofthe resistanceoffluidstosolidsmo ing
entedinF ig. 3, andtheresult isno
thiscase re uiringnoca lculation might
f thee tremew rongnessof thedoctrine in
eoffluidsagainstsolids mo ingthrough
stancein thecaserepresentedby
heassumptionofaw a eof " deadw ater
eA bhandlungen " V o l. I. pp. 15 -156.
M at h em at i sc h e Ph y si , V o l . x x I .
namics " Phil. Mag. 1876 secondhalf -year.
V o l . i. p p . 15 2 -1 5 .
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D Y NA M IC S [ 2 5
ure 1I asthedistantandnearw aterf low ing
followsimmediatelyfromaneasily
tatedin thecombinedmeetingofSections
. , i n O x f o rd l a st A u gu s t t o t he e f fe c t th a t th e
thepressureoneachof theends E E , in
w ha t e e r t he i r sh a pe s a n d w h et h er " b o w o r " s t er n "
dstangentia lly inacy lindric" mid-body
egreatesttrans ersediameterofthe
A w hereA istheareaof thecross-sectionof
id.
presenttw ovarietiesofacasew holly
ableendlessnessofF ig. 1 andcarefully
nsiblebyholdersofthe doctrineofdiscontinuityifit hasanydefensibilityatall.I v enturetolea eit
ration.
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ES .
RO R INLA PLA CE STHEOR Y
calMaga ine V o l. L. 1875 pp. 227-242.
nT idesandWa esintheEncyclopediaMetropolitana gi esav ersionofLaplace stheoryof the
btedlygreatmerit ofbeingfreedfrom
pcationsof " Laplace scoef ficients " or as
monlyca lled " Spherica lharmonics " by
rattempted notq uitesuccessfully to
ngintoaccountthea lterationofgra ity
ceofthesurfaceofthe seailthesolution
ons.
temptforeach ofthe" three
M e ca n i u e C el e st e L i . I V . a r t s . 5 7 9 ,
inthecourseofw or ingoutthe
uations( art. 3) ] by theassertionthat
e t o b e ra t io n al f u nc t io n s of p a n d V / 1 U 2 " 2 t h at
arly thew holechapter( Li . IV . chap. i o f
ideof t idaltheorywasunderta enbyaC ommitteeof the
hichpublishedvariousreports( B . A . R eports 1868 70 71
epracticalmethodsofharmonicanaly siso f t idesandtheresults
ocean. O neof the longerof these( B rit. A ssoc. R eport 1872
draw nupbyMrE. R obertsunderthedirectionof theC ommittee : thene t( B rit . A ssoc. R eport 1876 pp. 275- 07 w as" draw nupby
se uently thedirectionof thisw or w asta eno ermainly
f . Thomson& Ta it sNat. Phil. ed. 2andSirGeorge
entif icPapers V o l. II. ( w hichincludefurtherB . A . R eports
boo onTheT ides.
nteddrew attentionafreshtotheLaplaciantheory
mpro edandde elopedbyv ariousw riters includingDarwin
dinparticularbyS. S. Houghintw omemoirsinPhil. T rans.
p . 2 0 1 a n d V o l . 1 9 1 A ( 1 8 98 , p . 1 9 : c f. L a mb s H y dr o dy n am i cs
ddresses v o l. III. Na igation 1891 aB rit ishA ssociationlecture( Southampton 1882 on" TheT ides isreprinted pp. 1 9-190with
C D E o f w hi c h B , C a r e pa p er s ( B r i t . A ss o c. D u bl i n 1 8 7 8 o n
ra itso fDo erontheT idesintheB rit ishC hannelandthe
ntheT idesof theSouthernHemisphereandof theMediterranean " the latterincon unctionw ithC apt. E ans w hileDisa" S etchofP roposedP lanofP rocedure inT ida lObser ationandA nalysis" f romB rit. A ssoc.
68.
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ES
de otedtothedynamica ltheoryof the
festhefo llow ingstatementwithw hichA iry
s " a r t . ( 6 6 ] i n t r od u ce s h is o w n v e r si o n of
w ouldbeuse lesstoof ferthistheory inthesameshapein
nit f o rtheparto f theMe cani ueC eleste
ofTidesis perhapsonthewholemore
arto f thesamee tentinthatw or . We
aforme ui a lenttoLaplace s andindeed
apersonfamiliarwiththe latterwill
o f thesuccessi esteps. Theresultsat
ethesameasthoseofLaplace.
stinA iry streatisethroughthe
armonicanalysisis Laplace scomplete
rotationande ualdepthof thesea
ntheearth srotationista eninto
ea iso fune ua ldepth thedifferentia l
ta esaformaltogetherunsuitedforthe
armonics ; andA iry sin estigationis
Laplace s e ceptinthe j udiciousomission
ptsreferredtoabo e.
sso lutionforthesemi-diurna lt idew ith
uetothechangeof f igureof thew ater
A irypo intsoutwhathebelie edtobe
at a f tercorrectingit itw as" needlessto
snumerica lca lculationsof theheightsof
andhisinferencesasto thelatitude
amp c. fa llto theground. WhenIf irst
nyearsagoonboardthe ' GreatEastern '
rrectionofLaplace but onthe
elfthatLaplacewasq uiteright.Not
eC elesteathand Isetthesub ectaside
oreturnto itforthesecondvo lumeof
NaturalPhilosophy ' w iththef irstvo lume
ed.
ybeenrecalled toitbyreading
R esearches ' constitutinganappendi
re lto the ' UnitedStatesCoast-Sur ey
ghsubse uently succeededinintroducingharmonicana ly sis
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GEDER R OR INLA PLA C E STHEOR Y
hefo llow ingpassagereferringtoLaplace s
ltides: Theresultsshowthatthe formwhichthesurfaceofthe
iffersv erymuchfromthatofa prolate
isinthedirection ofthedisturbing
ndepthsof theoceanthetidesat the
low w aterta ingplaceundertheattracting
how e er thetidesw erefoundtobe
nde eninthecasesinw hichtheyare
ortheyw erefoundtobedirecttowardthe
ntly there isa latitude insuchcaseswhere
ce how e er fa iledto interpretcorrectly
on sothatthenumericalresultswhich
tassumeddepthsoftheocean are
genera lresultsj uststatedarereadily
ingfailedtoseethe indeterminate
headoptedasingularandunwarranted
hev alueofaconstantwhich isentirely
f rict ion butw hichv anishesinthe
ersmall. Thiso ersighto fLaplaceand
hisconstantweresubse uentlypointed
ngularandunw arrantedprinciple thus
q uisitely subtlemethodbyw hichit
minedaconstantwhichis notarbitrary
cannotbemorethaninfinitesimally
riction. F errelfurthere tendsto
orthe" diurna lt ide theob ectionof
Airyhadraisedonlyagainsthis
rnal andhefollowsAiryin an
nbyLaplace forthe" long-periodtide "
ceofdeterminateness( strangelyinconsistentwiththeindeterminatenessassertedofthesolutionsfor
iurna l isproducedby the inad ertent
thetrueva lueofw hichistobedeterminedbyaproperapplicationofLaplace smethod. Withthese
notwaittwoor threeyearsmoreforthe
homsonandTait sNaturalPhilosophy to
ess butmustspea outonthesub ect
e t( O ctober Numberof thePhil. Mag. entit led" Note
eF irstSpeciesinLaplace sTheoryof theT ides. [ Inf ra.
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GEDER R OR INLA PLA C E STHEOR Y
dA iryassume
4 + ... + K 2 x 2 + K 2 + 2 2 + 2 + & amp c... 3 )
entia le uation( 2 . Then bye uating
hele f t-handmemberto -8H itscoeff icient
uatingtoz erothecoef f iciento fc2 +4 fora ll
oo they f ind
........ 4
6 K 2 + 4- 2 ( 2 + 3 ) 2 2 + -- = 0... 5
a luesofk [ thecaseofk = 0j ustifies
3 ) ] . T h e fi r st o f t he s e e u a ti o ns o f
H. Thesecond if f o rbre ityw eput
K 2 + 2 -i K k . .. .. .. .. .. .( 6 ,
+ 2 - ( + 3 )
ss i e l y K , K , K 0 , . . . & a mp c . a l l i n t er m s
d i ff e re n ti a l e u a ti o n ( 2 i s s at i sf i ed b y ( 3 )
K 4 a r bi t ra r y a n d th e o th e r co e ff i ci e nt s
sprocessforcompletingtheso lution
remar s: The indeterminatenessofK4isacircumstancethatadmitsof
.Itisone ofthearbitraryconstantsin
e uation. Itshowsthatw emaygi e
w e pl e as e e e n i f H = 0 a n d th e n
mpanyourarbitraryK4 w iththecorrespondingv a luesofK 6 K 8 & amp c. w esha llha easeriesw hich
atthatw illsatisfy thee uationw henthere
gforcew hate er andw hichthereforemay
yanynumber tothee pressiondetermined
enforce. Inthene tsectionw esha ll
actly sim ilartothis. Yetthisob ious
onof thiscircumstanceappearstoha e
hehasactuallypersuadedhimselftoadopt
n 3 L / 4r g i n L ap l ac e s .
a a i n La p la c e s .
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ES
ingthegenerale uationamongthe
/1
k ) - ( 2 c 2 + 6 ) K 2 + 4/ + 2
cei edthatthismustapplyw henk = 1
4 andthusapplyingthesamee uation
swhichoccursin thedenominatorofthe
6 .1 2 b m/ l ( 2 . 2 + 6 . 2 2 b m /l
22+ 3 . 2- 2. 2+ 3 . - inaninf initecontinuedf raction. A nduponthishefoundssome
aptedtodifferentsuppositionsofthe
te asathinguponw hichnoperson
ha eanydoubt thatthisoperationis
natthetimew henIf irstreadthis
nclusion andshowedmethatLaplace
+ 2/ 2 v anishesw henk isinf inite ly
annotbutbee ua ltothecontinuedf raction.
ethecase ifK 4hasanyotherv a luethanthat
K 2 + 2 / 2 c a nn o t th e n co n e r ge t o z e r o
a luesofk . B utunlessK 2 +2/ f2 is
isinfinitelygreat thesecondtermofthe
isinf inite ly smallincomparisonw iththe
mately
” 6 2 + 2
= 2 + 4 K 2
t.Nowthisis preciselythedegreeof
f t he c o ef f ic i en t s of x 2 , X 2 + 2 & a m p c . i n th e
2 . H en c e w h en x i s i nf i ni t el y n ea r ly e u a l
n d so a l so i s V ( 1 - X 2 d a /d , o r d a/ d O . N o w
o r w h e n x = 1 w e mu s t ha e d a /d O = 0
ofthedisturbancein thenorthernand
hecaseproposedforsolution byLaplace
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G E D ER RO R I N L A P LA C E S T H E O R Y 2 7
s e K - + S 2 / m u st c o n e r ge t o z e r o
ha ethev a luegi ento itbyLaplace* .
reeofcon ergenceobtainedby
ofK4 andverif y thatitsecuresda/d= 0
R . . .. . .. . .. . .. . .. . .. . .. . . 7 .
+ -. K 2 ;
f r R g i e s
3 + . .. .. .. .. .. .. . 8 .
2/ c+ 6 -R ) (
os c o n e r ge t o u ni t y ( 8 g i e s
w he n k i s g re at .. .. .. .. .. .. 9 .
nationofK 4byhiscontinuedf raction
nof theratiosby ta ingR + , =0 for
eofk andca lculating
n s of ( 8 w i th k - 1 k - 2 . . .s u bs t it u te d
t o th e s er i es ( 3 a d e gr e e of c o n e r ge n cy
sa m e as t h at o f t he e p a ns i on o f e X V + e - X V i n
s uc h th at d a/ d , d 2a /d 2 d a /d a , . .. & a m p c . a re a ll
a lueofx . Henceda/ dO , be inge ua lto
i s z e ro wh en x = 1 .
place sprocesssimplydetermines
nthatda/ dO = 0atthee uator. A ndthe
3 ) hasthere uisitecon ergency to
forthepoles. Laplace sresult isthereforethe
eproblemoffindingthe tidalmotion
ewholeearthcontinuouslyfrompoleto
rmotiontheseacouldha einv irtueofany
ot e ceptforcerta incrit ica ldepths ha e
ftheassumedtide-generatingforce.
ofsucha depththatsomeoneof
onsinwhichtheheightof thesurfaceat
leby theformulav cos2- w here r
mar sinreply Phil. Mag. Oct. 1875 re ferredspecially to
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ES
v somefunctionof the latitudeha ingthe
northandsouthlatitudes hasitsperiod
egeneratinginf luence it iseasily seen
f ferentia le uation( 2 gi esaninf inite ly
lywhenthedepthhasoneof these
bitrarysolutionsintroducedby Airyand
pplicabletoanoceanco eringthew hole
nto thesameerrorofimagining
onof thedif ferentia le uation( 2 , w ith
nts includesoscillationsdepending
f thesea asthefo llow ingpassage( Li . I .
L i n te g ra t io n d e l e u a ti o n ( 4 * d a ns l e c as g e ne r al o u i n n e s t
uneprofondeurv ariable surpasse les
maispourdeterminerlesoscil la t ionsde
asnecessairede l integrergenera lement i l
a ri lestcla irq ue lapartiedesoscil la t ions
primitifde lamer adubient6tdispara itre
ugenrequeleseau de lamereprou entdansleursmou emens ensorteq uesansF actionduso leil
raitdepuislongtempspar enueaunetat
e : F actiondecesdeu astresl enecarte
ssuff itdeconna itre lesoscilla t ionsq uien
didnotsufferhimself tobe ledintow rong
on andheseemstoha eentirely forgottenitw henhegoesdirecttotherightresult w ithoutnoteor
singular processreferredtoabo e.
A iry a f terha ing inthepassage
a llowedthesamemisconceptionto fatally
ngwiththe solution closeswitha
ntwhichissufficientto showthegroundlessnessofhisob ectiontoLaplace sresult andtheuntenability
. Thispassagehasnotonlythe
thearticle whichprecedesit butit
decidedad ance inthetheorybeyond
o f wh i ch e u a ti o n ( 2 o f o ur n u mb e ri n g ab o e i s a
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GEDER R OR INLA PLA C E STHEOR Y
erdid orsuggested andforboth
o te i t . [ A i r y " T i de s a nd W a e s " a r t.
u s in g t he m or e c om p le t e v a l ue s o f a th a t we h a e
e d to f o rm t h e v a l ue s o f a ' , b a n d u w e f in d
eriesof termsmultipliedby the indeterminateK 4. WemaydetermineK4 sothat foragi env a lueof
i s t o sa y s o t ha t i n a g i e n l at i tu d e t h e
h-and-southmotion.We mightthereforesupposeaneast-and-westbarrier( followingaparallelof
nthesea andthe in estigationwouldstil l
eha eacompleteso lutionforaseawhich
osecourseiseastand west.
sprocessby thecontinuedf raction
eterminationofK 4thussuggestedby
hichA iry smethodfa ilsthroughnoncon ergence thatistosay thecase inwhichtheproposedeastand-westbarrierco incidesw iththee uator. F orasw eha e
pa ce s d e te r mi n at i on m a e s d a/ d O = 0 w h en 0 = 7
enorth-and-southmotionz eroatthe
ousf romsymmetry orasw eseef romthe
a p l ac e L i . I . a r t. 3 ; o r A i ry a r t s .( 8 5 ,
0
a-4 sin0cos0cos2. . . 10
entofthedisplacementofthe water
e s ee t h at L a pl a ce s s o lu t io n ( 3 ) , w i th K 4
ergentfora llv a luesofx & lt 1. Therefore
rgentfora llv a luesof0& lt ~ Tr. Hence
w i th t h e fo r mu l ae w hi c h he g i e s i n hi s
u at io ns ( 3 ) a nd ( 4 , ( 5 , a nd ( 1 0 o f~ ~ 5 a n d1
completeandcon ergentnumerica lso lution
e semi-diurnaltideinapolarbasin
uallydeepfromeitherpoleto ashore
titudeonthenearside ofthee uator.
eha eseen doesthesameforahemisphericalseaf rompoletoe uator. B utforaseae tendingf rom
cidingwithacircle oflatitudebeyond
ionalcomponentofthedisplacementofthe waterin
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ES
mofsolution(sti l l how e er w ithbut
mustbesought becauseLaplace s
5 a bo e , c ea si ng t o co n e rg e wh en x ( o r th e
0 increasesuptounity fa ilstopro ide
ationof0f romzerotoanyv a luee ceeding
emethodsuggestedbyA iry w hene tended
e differentiale uationwithitstwo
mpletelysol estheproblemoffindingthe
na lseaofe ua ldepthbetweencoasts
rallelsofl atitude.
ssolutionforthewholeearth
w ef indintheMecani ueC elestethenumerica l
y (butnotq uoted becauseof thesupposed
chtheywereobtained .Theyareof
est( w henwek now themtobecorrect ;
es Imaybepermittedtoquotethem
wor ingoutnumericallytheprocess
6abo e forthreedif ferentdepthsof thesea
1 / 6 1 -2 5 o f th e e ar t h s r a di u s. T h e v a l ue s o f e
hesedepths are10 2-5 1-25respecti e ly andLaplacef indsforthesolution[ ( 3 ) ~ 5 inthethree
0 , a = H { 1 - 00 0 0. x 2 + 2 0 -1 8 62 . x 4 + 1 0 -1 1 64 . x 6
1 0 - 7 4 5 81 . x 1 2
6-0-0687. x8
22-0-0001. 24 ;
{ 1 -0 00 0. x 2 + 6 -1 96 0. x 4 + 3 - 2 4 74 . x 6
0 9 19 . x ' l + 0 - 0 07 6 . x 1 2
H { 1 - 00 0 0. x 2 + 0 - 75 0 4. x 4 + 0 - 15 6 6. x 6
0 09 . x O } .
ngLaplace sf irstdif ferentia le uation[ theonef rom
o f ~ 5 a b o e b yp ut ti ng 1 - /u 2= x 2 ( L i . i . a rt 1 0 ] ,
1 U / 2- 2e ( 1 - E /2 2 a = - - 8H 1 _- i/ 2 ,
sumption
A 2 2+ & a mp c .
sacompleteso lutionw ithtw oarbitraryconstants tobereduced
dt io n t o ma e u = 0 a t o n e p ol e ( s a y w h en = + 1 .
atedin theprecedingfootnotesufficesforthis
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GEDER R OR INLA PLA C E STHEOR Y
achcasew ef inda= 0 showingthatthere
o les. Puttingx = 1 w ef indinthethree
p t h 1/ 2 89 0 o f ra d iu s ,
1 /7 22 5 ,
1 / 6 1- 25 , ) .
firstcaseshowsthat thetideis" in erted
here islow w aterw henthedisturbingbody
highwaterw henit isrisingorsetting.
( thatistosay forpo larregions thesignis
orethetidesaredirectforthis asclearly for
c a us e i n e e r y ca s e th e f ir s t te r m is + H 2 .
uestion( depth1/ 2890 , asw eseef rom
e theva lueofa increasesf romzerotoa
andthendecreasestothenegati ev a luestated
edf rom0to1 andthe intermediateva lue
sroughly -95 orthecosineof18~ . Hence
esare in ertedinthew holezonebetw een
thandsouthlatitude whilethroughout
hof theselatitudesthetidesaredirect.
eforthesecondand thirdofthedepths
atin thesecasesthetidesare e erywheredirectandincreasecontinuouslyfrompolestoe uator.
onforthe e uatorialtideinthe
earev ery interestingasshow inghow much
hanH( thee uil ibriumheight . U pon
atforsti l lgreaterdepthstheva lueofa
minutionhasa limit namely thee uil ibriumvalue w hichitsoonappro imately reaches. Tof indw hat
" b i en t 6t ) t a e t h e ca s e of e = , o r d ep t h
raroughappro imationtoR ta e
mu la ( 9 ~ 8 w i th k = 3 . T hu s we h a e
icationsof( 8 withk = 2 andk = 1 we
= - 1 04 .
eroughly
04 . x 4 + 1 0 4. 0 6 7. x 6 + 1 0 4 .- 0 6 7. 0 18 5 8
4 + - 00 8 2. x 6 + - 0 00 07 1 x 8 ,
16
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ES
depthis aboutase entiethofthe
untofe uatoria lt idee ceedsthee uil ibrium
npercent.
secondofLaplace snumerica lformulae
1 5 w e ma y i nf e r th a t wh e n e is i n cr e as e d
o10 thev a lueofaforanyv a lueofx
yto+ o thensuddenlybecome-o
y f romthattil l ithastheva luegi enby
henehasav a luee ceedingbyhowe er
uew hichma esa= + oc theva lueofa
sofx isposit i e anddiminishesthrough0to
a luesasx isincreasedto1 thatistosay
withtwo v erysmallcirclesof latitude
recttidesw ithinthesecircles andvery
undtherestof theearth. A se isincreased
stcrit ica lva lue thenoda lcirclese pand
w hene= 10theyco incideappro imate ly
hlatitude.F romthegreatnessofthe
ace sresultforthiscasewemay j udgethat
habo e10withoutreachinga second
ichthecoeff iciento fx 4 a f terincreasingto
es-oo. It isprobablethatthenoda l
earerthee uatorthan18~ Northand
alueisreached.W heneis increased
iro fnoda lcirclescommenceatthetwo
rdsandgettingnearerto theformerpair
themsel esaregettingnearerandnearer
rearedirectt idesinthee uatoria l
nthezonesbetw eenthenodalparalle lso f
e anddirecttidesin thenorthand
Thisis thestateofthingsfor any
thesecondcrit ica lv a lue j ustconsidered
eneisincreasedthrough thisthird
pairo fnoda lcirclesgrow soutf romthe
n ertedtidesatthee uator directt ides
nodalcirclesof thefirstandsecondpair
nesbetw eenthesecondandthirdnodal
e andstill asine erycase directt ides
es a fourthcrit ica lva lueofe introduces
es andsoon.
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GEDER R OR INLA PLA C E STHEOR Y
few hichw eha ej ustbeenconsideringareofcoursethosecorrespondingtodepthsforw hichf ree
raltypesdescribedaresymperiodicwiththe
hefreeoscillationswithoutdisturbingforce
es s ed b y t he f o rm u la ( 3 ) o f ~ 5 w i th K = 0
6 + K s 8 + & amp c.
& a m p c . a re t o b e fo u nd b y g i i n g an a r bi t ra r y v a l ue
determiningtheratiosR 1 R 2 & amp c. by
ofLaplace sformula
3 R + ) . .. ( 8 o f ~ ] ,
ofk , commencingwithav aluecorrespondingtothehighestratio to-beusedincalculatingcoefficients
dR = -a thene tapplicationof
oo w hichisthetestthattheva lueofe
espondsto adepthforwhichthe period
nsise actlyhalf theearth speriodof
eratiosR , R B , R isane ceedingly
b ectofpuremathematicsorarithmetic.
apide tinctionof theerrorresult ingf rom
anitstruev a lueforR + l inthef irst
( 8 . Supposingk tobeso largethat
a s ma l l fr a ct i on w e k n o w t ha t t hi s i s so m ew h at
ue o fR , a nd t ha t e/ k + 1 ( k + 4 i s st il l
eva lueofR + l. Hencew eseeatonce
e ta e 0 i n st e ad o f R + , . I f w e ta e
e fo r mu l a gi e s 0 f or R , a n d th e n ra p id c o n e r ge n ce t o t he t r ue v a l ue s o f B R - l R - 2 & a mp c . I f we t a e R + l
6 w e ge t R = o o R _ = 0 a nd t he n ra pi d
v a l ue s f or R â € ” , & a m p c . B u t i f w e ta e f o r
an 2 b y a c er t ai n v e r y s ma l l di f fe r en c e
e lessthan2 + byacorrespondingv ery
henforR , av a lue lessthan - bya
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F T H E TI D ES [ 2 6
ence andsoon. A nyva lueofR + l
eparticularv a lue lastindicated w ill
nough leadtothedesiredv a luesof the
three fourormoresuccessi eapplicationsof theformulare uiredtodissipatetheef fectsof the
andinstructi earithmeticale ercise
1 a n d so o n d ow n t o ] R a n d th e n by s u cc e ss i e
h e fo r mu l a to c a lc u la t e R2 R , . . . R - l R .
rigorous o fcoursethe init ia lv a lueof
ttheendof theprocess butif thecalculationhasbeenappro imate(say w itha lw aysthesamenumber
nedineachstep theva luefoundfor
l v a l u e b u t 2 â € ” , o r m o re a p pr o i m at e ly
k A + 2 A n d if w e c ho o se f o r RI a n y ot h er v a l ue
k + 2 '
dbyan infinitelyaccurateapplication
thenw or upbysuccessi ere erseapplicationsof theformula w ef indforR av a lueappro imatelye ua l
asnotwarnedusof this onthecontrary
yfollowed wouldleadussimply tocalculate
fa ct i on a n d th e n to c a lc u la t e K 6 K s & a m p c .
andK 4bysuccessi eapplicationsof the
lationof theformula
merica lquantity> 1. Ta eanyv a lueatrandomforr0
r . . . bysuccessi eapplicationsof theformula . F orlarger
riw il lbefoundmoreandmorenearlye ua ltothesmaller
.
dstor0by there ersedformula
e init ia lva lueof roaga in theresult( unlesstheca lculation
ate ine erystep w illbeappro imate ly thegreaterroot
pp r o i m at e ly e u a l to 1 / ri . T hu s f o r e a m p l e t a e t h e
5 - 8 2 84 2 7.
imationstothesmallerroot ta e
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GEDER R OR INLA PLA C ESTHEOR Y
This e ceptw ithinf inite ly rigorousarithmetic
rgeva luesofk notthetruerapidly
e co e ff i ci e nt s K 2 , K 2 + 2 & a m p c . b u t sl u gg i sh l y
orespondingtotheratioR = 2 - 4. B ut
y isa o ided andatthesametimethe
uchdiminished byusingforthe ratios
ndfortheminthesuccessi estepsinthe
dfractionforRi.
o fR 1 R 2 & amp c. consideredasfunctions
undamenta limportance. Someof theremar able
esentsha ebeenalreadynoticed(~ 15
atase( w hichisessentia llyposit i e in
ncreasedf rom0to+ oo eachof theratios
, , . . . increasesf romzero eachonemorerapidly
dngorder untilR Ibecomes+ oo and
andagaingoesonincreasingtill it
suddenly -o andsoon. B utbeforeRi
me R 2becomes+ oo -oo andagain
Thesameholdsforeach oftheother
ase increasescontinuously eachoneof the
o = 6 â € ” = 5 -8 28 4
= 6 - = 5. 8 28 4
' 2 = 6 - 1 - = 5 8 27 7
' F 3
= 6 5 80 ,
r 4 6 - - = 5 06 7
' 5 6 -r - 10 7 2
6 6 - = 2 02 9
6 r 7 6â € ” = 1 72 5
8
r 8 = 1 7 16 .
tephadbeenrigorous w eshouldha efoundr7 = r7 ? ' 6 =r6 andsoon Insteadofcomingbac ontheva lue0assumedforro w ef ind
e gr e at e r ro o t of t h e e u a ti o n
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ES
ge ceptw henitsva luereaches+ ooand
orderinwhichthe v aluesofthe
ghccis asub ectofgreatinterestand
rescarefule amination. Ihopetoreturn
ny remar thattheformula(8 forca lculating
Thatnotwoconsecuti eratioscanbesimultaneously
ncreasesf rom-ooto0 R iincreasesf rom0
b u t v e r y sl i gh t ly g r ea t er t h an e / i ( i + 3 ) , a n d
ea c he s o o w h en R i + , = 2 i + 3 ) .
i s & g t 2 ( + 3 a nd t he re fo re w he n Ri + & g t 1 R i
thatintheseriesofcoef ficients
...
e cu t i e c h an g es o f s ig n . F r o m ( 2 ( 3 ) i t
ntisless inabsolutev aluethanits
esign e ceptw henthepredecessoris
fficientprecedingit andoftwocoefficientsimmediatelyfollowingachangeof sign thesecondmay
utif so onlybyaverysmallproportionof
utthroughnearly thew holerangeofv a lues
hangeofsignf rom say K itoK i+1
l i n a b so l ut e v a l u e .( F o r i ll u st r at i on o f t hi s s ee
e f o r hi s c as e o f e = 1 0 f o r wh i ch h e g i e s
18 62 K , = 1 0- 11 64 K 8 = - 1 - 10 47
1 2 = - 74 5 81 K 1 4 = - 2 -1 9 75 . .. & a m p c .
n entionw hichformsthesub ectof
reate tension asIhopetoshow in
ha enothithertofoundanytrace
entia le uations butIcanscarce ly
nsomeformorotherit isnotk now nto
eoccupiedthemsel esw iththissub ect.
it iso fe ceedingv a lueandbeautyasa
hod.AstoLaplace sDynamicalTheory
ha emuchpleasure inconcludingw ith
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GEDER R OR INLA PLA C E STHEOR Y
atementbyAirywhichI findinhis
" a rt . ( 1 1 7 .
romourthoughtsthedetailso f the in estigation w econsideritsgenera lplanandob ects w emusta llow
endidwor softhegreatestmathematicianofthe pastage.Toappreciatethis thereadermust
ldnessof thew riterw ho ha ingaclear
simperfectionofthemethodsof his
sothecouragedeliberately tota eupthe
amenta lly correct( how e eritm ightbe
terw ardsintroduced secondly the
gthemotionsof f luids thirdly the
gthemotionswhenthe fluidsco er
butcon e ; and fourthly thesagacity
anecessary toconsidertheearthasa
dthes il lo f correctly introducingthisconsideration. The lastpo inta lone inouropinion gi esagreater
eboastede planationofthelong
ndSaturn.
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S C IL L AT I O N S O F T H E F I R ST S P EC I ES I N
F THETIDES.
calMaga ine L. 1875 pp. 279-284.
ationsof theF irstSpecies aresimpleharmonicoscil la tions inw hichthesurfaceof thew aterisa lw aysa
ndthea iso f rotation. The" t ide-generating
ssuchthatthee uil ibriumtide-he ight
met adenotingaconstant( ca lledthe
sh-A ssociationT idalC ommittee sR eportfor
ctionof the latitude. ( be ingsupposedk now n
inga afunctionofthelatitudesuch
o s at i s t he a c tu a l ti d e- h ei g ht a t t im e t a n d f o r th e
deepe erywhere it istobeso l edby
ofthe differentiale uation
= 4 e .. . .. . .. . .. . .. . 1 ;
thelatitude andeandf areconstants
s
us
titssurface
oe uatoria lcentrifuga lforce be ing
f theearth srotation
sea supposedsmallincomparison
thanr/ 50.
O scillationsoftheF irstSpecies" isthe
ationa l andforita isabout1/ 14ofn
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O S C I L LA T IO N S O F T H E F I R ST S P EC I ES 2 4 9
. E enforthis andmoredecidedly forthe
andso larsemi-annua l(declinationa l and
goodappro imationtotheresultm ightbe
0. Laplacedoesnotenteronthe integrationof thee uation butcontentshimselfbypointingoutthat
f rict ionw ill w hen = 0 causethe
same asthee uilibriumtide-height
ar fortnightlytheactualheightmustbe
uilibriumheightifthereis enough
rtnightafreeoscillation toasmall
nt.Theresult ofanytide-generating
gperiodwouldob iouslybemoreand
greementw iththee uil ibriumtheory the
e itnotfortheearth srotation. B ut
otation a long-periodtidedoesnot
mentw iththee uil ibriumtide if thew ater
andthesolutionof thebeautiful" v orte
disw hatisa imedatbyA iry and
onof theprecedinge uationforthe
it isreducedtothecomparati e ly simple
ea - = 4 e .. . .. . .. . .. .. . . 2
e s ( E n cy c lo p ce d ia M e tr o po l it a na , a r t. ( 9 7 .
( A p pe n di t o U n i t ed S t at e s Co a st - Su r e y Re p or t 1 8 74 ,
ristol September2 1875. -Withoutthissimplif ication the
usceptibleofnearlyassimpleaso lutionasw ithit. A ssume
s ( K is -f2il
1 - - 2 d a
2 da = _ Mi ( i + 1 ( K i + l -K i -1 ;
K / 2
ecoeff icientsK i w eha ethee uationofcondition
1- K - _ 3 = 4e 0
.
analmoste uallysimplesolutionofLaplace sgeneral
w hichhasbeencommunicatedtotheB rit ishA ssociationat
d andw illbepublished[ inf ra p. 254 a lso intheNo ember
alMaga ine.
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ES
reesw ithA iry se uationofart. ( 97 ( w ith
edepthconstantasw enow suppose it , andw ith
u a ti o n ( 2 8 8 ; b u t is s i mp l er i n f or m p a rt l y
e snotationpforcos0 . F oreachof
intheactua lcaseof theearthunderthe
moon thefunctione isgi enby the
. .. . .. . .. . .. . .. 3 ) ,
uil ibriumvalueof thetide-he ightatthe
hisv a lueof (, f indsanintegralo f the
assuming
4 PA 4 + . .. + B i i + & a mp c .
cientssoasto satisfyit.B utthis
ngthetide-he ightatthee uatore ua l
t. Thecorrectassumptionfortheparticularproblemproposed( orforanycase inw hichO in o l es
L i s
B 4 4 + . .. + B j i i+ & a m p c .
mption
B , 2 + . .. + B i + & a mp c .. .. .. .. .. .. . 4 ,
ndincludesoscillationsinwhich the
s . W i t h i t we h a e
ea = i { ( i + 4 ( i + 1 B + 4
B + 2 -4eB i ,
o4e0. Thus forthecaseof
- 2 i = 0 i = 2 & a m p c .: 2. - 1 . B 2 = 0
4eHl. . .. . . .. . . .. . 5 ,
4 e B 2 = - 1 2 eH J
1 B + 4 - ( i + 2 ( i + 1 B i + - 4eB = 0 ... 6
a l ue s o f i e c e pt 0 a n d 2.
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O S C I L LA T IO N S O F T H E F I R ST S P EC I ES 2 5 1
5 gi esB =0 andwiththisthe
B = e B . .. .. .. .. .. .. .. 7 ;
u cc e ss i e l y i = 4 i = 6 i = 8 . .. a n d us e i n
sofound w ecanca lculatesuccessi e ly
B o 0 B 1 2 . . . ea c h in t e rm s o f B , ; a n d w e
( 2 w i th o n e ar b it r ar y c on s ta n t B 0 w hi c h
+ B o .F ( z , e . .. .. .. .. .. .. .. 8 ,
d e no t es t h e fu n ct i on o f u a nd e e p r es s ed b y t he
hecoef ficientsca lculatedforthecaseB = 0and
e t h e fu n ct i on s i mi l ar l y fo u nd b y t a i n g H= 0
iryhaspointedout [ " TidesandW a es "
referencetoacorrespondingq uestioninthe
tides maybeassignedsoastoma ethe
entmotionof thew aterz ero inagi en
ase( thatis thecaseofsymmetry round
w e ha e [ A i r y a rt . ( 9 5 , o r La pl ac e L i . I V .
o f wa t er = 4 / 1 t - da . .. 9 ;
henorthandsouthmotionzerow emust
.. . .. . . .. . . 10 ;
( , e
~ * * - 11 .
b y t hi s e u a ti o n fo r a ny g i e n v a l u e of u w e
eterminateproblemoffindingthemotion
entide-generatinginfluencewhen
w holeearth theseaco ersonlyane uatoria lbe ltbetw eentw oe ualcircularpo larislands.
isin aseriesessentiallycon ergent
caseof thepolarislandsvanishing. F or
6 i n t he o r de r i nd i ca t ed a b o e a n d so
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F T H E TI D ES [ 2 7
essi e ly f romsmallertogreaterv a luesof i
. . .. . .. . .. . . 1 2 ,
i + 1 ( 1 2 ,
reaterandgreaterv a luesof i
m o re n e ar l y th e g re a te r i s i. .. 1 ) ,
u e z e r o or o t he r w e gi e t o B o ( u n le s s we
luefoundbyLaplace smethodbelow and
he calculationwithinfiniteaccuracy .
theva lueofe theseriese pressingthesolution
a lueofA & lt 1. Thustheso lutionisthoroughly
edcaseoftwo e ualpolarislandsofany
eult imatecon ergence isshow nby ( 1 )
theseries . 2 . 4 . 22
1- 2
theseriesfora becomesinf inite lygreat and
ite lygreatva lueforda / da unlessit
ciselythe particularv alueofB osought.
fa ilstodeterminethisva lue. Thusthe
caseforw hichitw assought thecase
place andta enbyA iryandF erre las
estigation- thatis thecaseof thew hole
er. HereLaplace sbril l iantprocess re ferred
dingNumberofthe Philosophical
oura idmar e llously .
................... 14 .
2 4 ( i + 1 } . ... .( 1
ppliedtoanymoderatelygreate env a lue
eataccordingtothedegreeofappro imation
g N+ 2 = o c a lc u la t e Ni a n d th e n b y s u cc e ss i e
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O S C I L LA T IO N S O F T H E F I R ST S P EC I ES 2 5
. N6 N4successi e ly . E uations( 7 ,
. .. . . .. . . .. 16 ,
2H
B - 4= e + 2N 3 + 2 A7. 7
tionis
4U 6 A
6 N 6. N 8
in d 1 8
” . . .. N
, . . .a r e fu n ct i on s o f e de t er m in e d by ( 1 5 .
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2 8
NO F LA PLA CE SDI F ER ENTIA L
H E TI D ES .
calMaga ine L. 1875 pp. 388-402.
ceanas arotatingmassoffrictionlessincompressibleli uidco eringarotatingrigid spheroidtoa
tely smallinproportiontotheradius and
nsundertheinfluenceof periodicdisturbingforces withthelimitationthattherise andfallisnowhere
allfractionof thedepth thecondition
locityofe eryparto f the li uidisthe
andtheassumptionthatthedistance
he disturbedwater-surfaceisnowhere
ofthedepth.Thislast assumptionis
atedbyLaplace impliedin andisv irtua lly
as s um p ti o ns ( M e ca n i u e C el e st e L i r e I . No . 3 6
ofthewateris smallincomparisonwith
andthatthehori onta lmotionissensibly
ationof thew ater-surfaceabo e
ndDsin0thesouthw ardandeastw ardhori ontalcomponentdisplacementsof thewaterattimet andatthe
isTr -0( ornorth-polardistance0
e" e uationofcontinuity [ M ec. dl.
ir y " T i de s a nd W a e s ( E n cy c lo p ce d ia M e tr o po l it a na , a r t . ( 7 2 ] i s
r
. .. . .. . .. . .. . ) ,
) d yE . 1 b is
+ h = O . .. . .. . .. . .. 1 6
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G RA T IO N O F L A PL A CE S E Q U A T IO N
oof thedepthof theseatotheearth s
le uations[ Mec. Ce l. Li . I. No . 36 ( M ,
h-e
h - e ( 2 ,
d 2
h sradius ntheangularv e locityof its
gra ityatitssurface andethe" e uil ibriumtide-height att imet andco- la titudeandlongitude0
ThomsonandTa it sNaturalPhilosophy
atw hichthew aterw ouldstandabo ethemean
datrestre lati e ly totherotatingso lid
stifthedisturbing forcewerek ept
lityattimet.
atthegenera lintegrationof these
atdif f icult ies andheconf ineshimself to
se thatinw hichy isafunctionof latitude
ameina ll longitudes. Inthiscasethe
beeffectedby assuming
j
. . .. . .. . .. . .. . .. . .. . . 3 ) ,
fo rce issuchthat
r . .. . .. . .. . .. . .. . .. . .. 4 ,
r e fu n ct i on s o f th e l at i tu d e o f w hi c h E is g i e n
efoundby integrationof thee uations. With
b i s an d ( 2 g i e
H-0. . . .. . . .. . . .. . . .. 5 ,
= E
. d E - Er ' d O
( 6 ,
. .. . .
” s i
.....( 7 ,
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ES
b H f ro m ( 5 , b y ( 7 a nd ( 8 ,
)
4n 2 c os 2 0
0
ntiale uationofthetides[ Mecani ue
o3 , e u at io n ( 4 ; o r Ai ry " T id es a nd
p ce d ia M e tr o po l it a na a r t. ( 9 5 ] . I t i s a li n ea r
hesecondorder thecompleteintegration
t h en c e b y ( 8 , a a n d b i n t er m s of 0 w i th
bedeterminedso astofulfilproper
11-17 be low . It isessentia lly inthe
it be ingthatinw hichitcomesdirect
ngitin thein estigation.Itoriginally
ni ueC eleste mas edsomew hatby theaddit ion
intermw hichgi esitadif ferentform ,
betterorsimpler butthisasitw ere
ggeststhefollowingv erysubstantial
a nd ( s in 6 0 E = . .. .. .. .. 1 0 ;
n O d O
1
0 a 2 s in 2 0
pu t c os 0 = / L a n d fo r b re i t y
= f . . . .. . . .. . . .. . . 12 ,
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 5 7
b e co m e
8 intheprocessbyw hich( 9 w asfound
ngtheresult inge uationby4m( sinO) slf+2
1 U " 2 -do
2 S y 1 2d
f ' ) U J 2 -f L2 d̂ u
4m ( 1 - L2 4 ( . .. 1 4 .
e f irstthecaseof ( = 0( f reeoscil la tions , andassume
+ + K K p + . .. + K I i + & a m p c .. .. .. .. 1 5 .
f 2 + ( f 2 2 - o + . .
+ & a mp c .. .. 6 ,
tofintegration.Nowlet odenotea
hat
ra ll yw ( i = F ( i - 1 . .. 1 7
ctionof i. B ya idof thisnotationw emay
i . .. . .. . .. . .. . .. . .. ( 1 8 ;
eni= 0
9 ,
ra llnegati ev a luesof i.. . . .. . . .. . . 20 .
17
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 5 9
f f iculty inde e loping( ~ 7below the
andw or ingoutapractica lsolutionof the
stinterestingandinstructi eresults
edtothecaseofanoceanofuniformdepth
n q = 0 o r y = c on s ta n t . T a i n g th i s
.... 26 ,
m e mb e r of ( 2 5 w e h a e
+ ( - 2i + - 4f ) K ^ J [ I 2_ 18 + 9 - i & l t / K i - + -i - _- O ( 2 7 .
uesof iupto-2thise uationisanidentity
a n d fo r i = - 1 i t b e co m es
. .. . ( 2 8 ,
ry . F o r i = 0 i t b ec o me s i n v i r t u e of ( 1 9
. .. . . .. . 29 ;
o( 2 0 ,
a f i K o = 0 . .. .. . .. .. . . 3 0 ;
tu e of ( 1 9 ,
s 2- aI f K 1 - a C= 0. .. .. . 3 1 .
, i= 4 etc.
K 2
K O
0
1 = 1 7- 2
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 6 1
w e s up p os e d 1 = 0 a n d so m ad e f o r th e t im e
usub ect. Now suppose( ) tobeanygi en
tualproblemoftides ofanyspecies
nc t io n o f M/ o r o f a n d V / - U 2 , i f w e
ucedbythechange ofattractionofthe
f f igure . A properw ayof ta inginto
successi eappro imationsw illbe
me w ithoutlosinggenera lity Iassume
c I + ( I 2 + . .. + cD i i + & a mp c .... ... . 3 8 ,
( i 2 & a m p c . a re g i e n c on s ta n ts e i th e r fi n it e i n nu m be r
torendertheseriescon ergentforv a lues
sedineachparticularcase. Withthisfor
be r o f ( 1 4 b e co m es
.. . .. . .. . .. . 3 9 ,
w e ha e
2 s+ + l 2 ( 1- W 2 , ( W )
1 - 2 ( / 2 - 2 2 K + 1 = - = m - ( i - 2 . .. .. .. .. ( 4 0 .
accordingtothisformula mustbemade
h of t he p ar ti cu la r e u at io ns ( 2 9 , ( 3 0 , ( 3 1 ,
, ( 3 6 , ( 3 7 when re uired.
econditionswhichmaybefulfilled
fthetwoarbitraryconstantsC andK 0
estigatethecon ergencyof theseries( 16
orthecompleteso lution. F orthispurpose
g ( 2 5 a s t he c a se f o r wh i ch 4 = 0 i n to t h e
2 ) ( ) K / + 1
1 - _ 2 2 . ( f 2 _ -K , _ 3 ) - 4 m ( D hi - i- _2 } . . .. .. ( 4 1 .
antclassofcases o fw hichthef irst
mathematiciansisthatsosplendidlyand
placeinthe processdefendedandcontro ertedinthetwoprecedingNumbersof thisMaga ine terms
thise uationare forinf inite lygreat
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F T H E TI D ES [ 2 8
parable inmagnitudew ithtermsof thef irstmember
ginfinitelygreatof theorder2. These
ny ise itherconstantore pressedin( 3 4
70+ 7eya. R eser ingthemforconsideration
t h at e c e pt i n t ho s e sp e ci a l ca s es K i m u st
sof ifulf i l moreandmorenearly thegreater
+ = 0 . . .. . .. . .. . .. . .. . . ( 4 2 .
eandrigorousso lutionof thise uationin
e
+ " ' i ( - l i + ~ + + c.... 4 ) ,
m p e . de n ot e t he r o ot s o f th e e u a ti o n y = 0 a n d 1 1 ,
' , & a m p c . co ns ta nt s. H e nc e fo r gr ea t v a lu es o f i K i m u st
a lto ( 4 ) w ithsomeparticularv a luesfor
& a m p c . B u t f o r v e r y gr e at v a l ue s o f i al l t he t e rm s
n e le a di n g te r m o r [ b e ca u se o f t he e u a l ro o ts o f
e leadingpairo f terms v anishincomparisonw ith
Hencew emustha e forverygreat
, o r K i = [ ' + I ' ( - 1 ] i
o. .. 4 4 .
- r a n d so o n
ftherootsp p , & amp c.isgreaterthan
and( 16 arenecessarily con ergentfora ll
1 t o / = + 1 a n d th e y ar e d i e r ge n t fo r
hese lim itsunlesscondit ionspropertoma e
= 0 " ' = 0 a r e fu lf il le d. B u t i f on e or m or e of t he
mp c. islessthanunity andptheabso lutely leasto f
ionally theseries( 15 and( 16 arenecessarily con ergentfora llva luesof / f rom-pto+ p andtheyare
sof / beyondthese limitsunlessacondition
0isfulf i l led . When7y=0hasimaginary
asIaminformedbyhimselfandProfessorC ay ley has
enera lterms thiscriterionforthecon ergencyof theseriesin
fortheintegralof
- + x ( x ) = 0
totheR oya lSociety o fw hichcerta ine tractsha ebeen
ngsfor1870 1871 1872.
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 6
1 theabso lutemagnitudeofe itherof thepair
2+ 2 , andw iththisunderstandingthe
n ergencyanddi ergencyho ldsasfor
s distinctioninthecircumstancesof
inthetwocases o f transit ionthrougha
absolutev alueofapair ofimaginary
ereisnodiscontinuitywhenpu is
houghthecrit icalv a lueV / a2+ / 2 ; in
ferentialcoefficientsbecomeinfiniteand
asedcontinuouslyuptoandbeyondany
erpretationofthecircumstanceswhen
inf luencethesolutionisane ceedingly
owhichIhope toreturnina futurecommunication.Theremainderofthepresentarticlemustbe
0ha ingtw orea lroots eachless
ofy= 0 andputu= z + p. Then
sofz , thedif ferentia le uation( 14
+ ~ ( c + d ) + e + z . .. .. .. .. ( 4
e fdenoteconstants. Thecompleteso lutionof
ationmaybefoundbyassuming
z + H 2 + & amp c . } ( 4 6
K 2 + K z ' + & amp c.
, & a m p c . i n te r ms o f H 0 a r bi t ra r y b y
f og z , z l og z , z 2 l og z , & a m p c . to z e r o a nd
K 2 K , i n t er m s of K , a n d H , e a ch a r bi t ra r y
& a mp c . p re i o us l y fo u nd . T hi s s ho w s th e k i n d o f
ompletesolutionof thee acte uation
sentsw henthev a lueof / passesthrougha
dhow thisdiscontinuity isa ertedbyan
stantsofintegrationin therigorous
Ho= 0intheappro imateso lution( 46 .
uestion( ~ 9 o fassigningthetwo
asto fulfilanyproperphysicalconditionsofour problem.F irst towor outthegeneralsolution
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ES
u s e ( 4 0 , a n d c a lc u la t e K 1 K 2 & a m p c .
K oarbitrary . Thusw ef ind
Do X i 2 + X i ) + ( ) + . .. 47 ,
~ ) , X ( l X , X i 2 , & a mp c .a re nu mb er s ca lc ul at ed by th e
s m r 7 0 i Y Y 2 & a m p c . t o ha e h a d an y p ar t ic u la r
ne d t o th e m a n d & l t o I & g t 2 , . . . to d e no t e
forew ebeginthearithmetica lprocess
si g ne d t o 4& g t o l D , D 2 & a m p c . s o th a t we
L D 2 = n 2 L & a mp c .. .. .. .. .. 4 8 ,
a n d no n 1 n 2 & a m p c . g i e n n um e ri c al
o f theprocessofca lculationofKi f rom(40
X i L . .. . .. . .. . .. . .. . .. ( 4 9 ,
r e c a lc u la t ed n u mb e rs . T he n w e ha e b y ( 1 5
C + /( , ) . K o + X ( H . L
a o + a lp + a 2I 2 + & a m p c .
3 2 2 + & amp c.
X + 2 + & amp c.
C + B ( , t . K o+ x ( / . L
2 ao + I / f2 al 2 + 3 ( / f 2- c ia 3 + & a m p c .
f 2 + & a m p . / S+ ( -o + .
o + -+ f 23 2 + ( f 2X 2 - ) 3 + & a mp c ... 50 ,
constantsof integrationsoasto
renderingtheproblemdeterminate.
ortwotypicalcases:-first thesea
by twov erticalclif f s secondly by tw o
ualdeepeningfromeachtoa single
nintermediateparalleloflatitude.
nbeabelto fw aterbetw eenvertica l
des e itherbothinthesamehemisphere
rsouth.Theconditionsofthiscase
ndsouthmotionof thewaterateither
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 6 5
flatitude andtheyareto befulfilled
b y pu tt in g
........... 52
a luesofF A ( thatistosay thesinesof
Ifeachof these islessinabso lutev a lue
0 e a ch o f t he s e ri e s in ( 5 0 a n d ( 5 1 i s
wholerangeofv aluesoffucorresponding
F V " t h e si n es o f t he t w o bo u nd i ng l a ti t ud e s
ati e forsouthlatitude ife itherorbothbe
b yu si ng ( 5 2 , i n( 5 0 ,
) . o + x ( ' ) L= ......... 3 ) ,
C+ 1 ( / ) . K o + X ( " ) . L = 0
/ - / ( ) . ) L LA
- as ( 0 ) a ( / . .. .. .
X ( ' ) - a ( " ) . X A ( u ) LI
- / ( ' ) . a( / )
C a nd K o ( 5 0 a nd ( 5 1 g i e 1 f_ - d
ofA throughtherangeof thesupposed
llow ingformulae[ w hichit iscon enientto
( 7 , ( 3 ) , ( 1 0 , ( 3 8 , and ( 48 abo e gi e hthe
4thesouthw arddisplacement and
eastw arddisplacementof thew aterattimet
ongitude r:
s/ - ' - f 2 2 df c os ( C t + sr
}
f / l2 da f2 l - ) s 2 . .. .. .. .. ( 5 5 ,
andm/ rtheratioofe uatoria lcentrifuga l
ione pressesthemotionofthe
duetoadisturbinginf luence o fw hichthe
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ES
sEcos( at+ sf , Ebe inge pressedby
o + n l + n 2 pY + & a mp c . . . .. .. 5 6 ,
a m p c . a re a n y gi e n n um b er s .
io n ( 5 4 g i e s e c ep t in a c er ta in c ri ti ca l
esently C = 0andK o= 0 andtherefore
eterminate ly thatthere isnomotion that
beany" f reeoscil la tion o f theassumed
( 3 ) * , e c e pt i n t he c r it i ca l c as e a ll u de d t o.
einwhichthe denominatorofthe
0 v a n is h es o r
nitev aluestoCandK ounlessLis z ero and
g i e s
( )
.. ./ / . ( 5 8 ,
* ( ) * *
oofC toKo butlea ingthemagnitude
all thefundamentalmodesof
posedz ona lsea isso l edbygi ingtos
2 & a m p c . a n d fo r e ac h v a l u e o f s tr e at i ng ( 5 7 a s
onforthedeterminationofor.After the
dSturmandLiou il le itmaybepro ed
uationcannotha e imaginary roots
nitenumberof realrootsmoreand
ntw henta eninorderofmagnitude
eto largerandlargerposit i e orf rom
largerandlargernegati e . Inthecase
n d ne g at i e r o ot s a re e u a l u n e u a l in a l l
s = 2 & a mp c . .
n cy o f t he s e ri e s in ( 5 0 a n d ( 5 1 i t i s
~ 9 thattherebenoroot rea lor
oseabsolutemagnitudeislessthan that
inesperfectly theconf igurationof theassumedmotion and
odis2w r/ a orits" speed a- .
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 6 7
of thetwoq uantit iesA andp . B ut
ebraicsignsta enintoaccount there
weenpu and' u ( thatistosay w hen7
v a lueofponthedirectrangef romtp to
si f un ct ion sa , ) , / ( p , X ( , ) , a / , ( / u ,
pr oc es se s( 5 ) , ( 5 4 , ( 5 7 , ( 5 8 , a nd in th ef in al
5 0 , ( 5 5 , i s d i s c on t in u ou s . W h y so me o r a ll o f
iscontinuousinthis caseisob ious:
eroa longanypara lle lo f latitude lim its
hesideonw hichthedepthispositi e
sof7y= 0 separatestheproblemintotw o
ndthemotionsof thewateronthetwo
w ash. A nimaginary rooto f7= 0ha ing
betweent and ' " , o rarea lrooto f
ute lygreatero fp andj A , andof
w eenthem renderstheseriesfora( ip ,
m p c . i n as c en d in g p ow er s o f A di e r ge n t fo r t he p o rt i on o f o ur
sbeyond+ sin- R . Stilltheso lutionof
n b y ( 5 5 i n t er m s of s i f u nc t io n s a ( , u , , / ( t u , & a m p c . e a ch c o nt i nu o us t h ro u gh o ut t h e ra n ge b u t ca l cu l ab l e b y
wersof psetforthin ourpreceding
tof therangeoflatitudewhichli es
sin-1R.Themodeofdealingwith the
as toobtaincon enientformulaeforthe
( t & c. isaninterestingandimportant
toreturn. B e ing( ~ 9 atpresentlim ited
it isenoughtoremar thatinthiscase
t io n s a ( / u , 8 ( / u , & a m p c . a s er i es c o nt i nu o us l y
therangef rom// to / u maybefound
onsecuti erea lrootsof = 0 andlet
b e i n or d er o f a lg e br a ic m a gn i tu d e. L e t a be a n y
raically
a nd a& l t ( , A + p ) . .. .. .. .. .. .( 5 9 .
........ 60
d wo r i n g pr e ci s el y a s in ~ 5 b u t wi t h z i n s t ea d
mberof ( 15 andthepropercorresponding
amp c. w eobtainaso lutioninascendingpow ers
hisnecessarily con ergentthroughoutthe
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ES
degreeof con ergenceoftheseriesso
functions
( z + a , X ( z + a ,
( + a , X ( z + a ,
, thesameasthatof thegeometricalseries
c .
twoq uantit iesa-p p-a .
os ed c as e ( ~ 1 1 l et p , p p , p b e
o f ( 1 - / 2 y = 0 l e t p p b e e ac h b et w ee n
eposit i e forv a luesof / A betw eenp
determinetides andthef reeoscilla t ions
espondingtotheseintermediatev alues
ty a b e tw e en p a n d p , s u ch t h at p - a
hanthe lessof thetw odif ferencesa-p ,
a a n d so l e i n a sc e nd i ng p o we r s of z , a s i n
bethecoef f icientsofzi intheseriesthus
p , X 8 ) , ) i n fo rm ul ae c or re sp on di ng t o ( 5 0 , b ut
ondmembers sothatw eha e
C+ ( i ) o ( X i i L ... 6 1 .
a l ue s o f i a n d pu t
0 . .. . .. . . ( 6 2
q L =
b e e ac h i nf i ni t el y g re a t t h e v a l u es o f C a nd
e e u a ti o ns a n d us e d in ( 6 1 a n d ( 5 5 , g i e
eneratinginfluence
/ + n2 / 2 + & a mp c . .
entalfreeoscillationsof thesupposed
minedby f indingasoastoma e
. .. . .. . .. . .. . .. . 6 ) ,
ene pressedby ta ing
. .. . . .. . . .. . 64
of p - a= - p - a , w he n we m us t ta e p â € ” q = , o r an y
isbestinthiscase.
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T EG R AT I O N O F L A PL A CE S E Q U A T IO N 2 6 9
. B y g i i n g mo d er a te l y gr e at v a l ue s t o p q ,
orousso lutionmaybesatisfactorilyappro imated
andnotvery laboriousarithmetic. The
9 ( 4 4 .
stigationstofindsolutionsforthe
orbothpo lesmustbereser edfora
l ehighly interestinge tensionsof
ocessreferredto in~ 9of thepresent
acesof the lasttw oNumbersof the
.
sby re uestk indly suppliedthefollowingnoteon
nhis discussionoftidaloscillationsis now
scorrect andthepaper( No. 26 on" ana llegederrorin
ac now ledgedtoha ef ina lly settledthecontro ersy
AiryandF errel.
r( No. 27 onthe" oscil la t ionsof thef irstspecies "
sof longperiod Ipo intedout( P roc. Roy . Soc.
p . 3 3 7 o r V o l . I o f c ol l ec t ed p a pe r s t h at L a pl a ce s a r gu m en t
amely thatf rict ionw asade uatetocausethesetidesto
umtheory . F o llow ingLordK el inIfoundnumerical
place smethod otherso lutionsha ebeenfoundby
~ 2 1 6 a n d by H o ug h ( s e e r e fe r en c es o n p . 2 1 . I t
ionsthatonan ocean-co eredplanetthee uilibrium
rror.
place sargumentasto f rict ion Ie a luated( Thomson
840 , ormyco llectedpapers V o l. I thee lasticy ielding
edf romobser ationof theoceanictidesof long
r ( B e i tr . z . G e o ph y si , V o l . 9 ( 1 9 0 7 , p . 4 1 , u s in g f ar
arri edataclosely sim ilarresult. Sincetherigidityof
sw ayagreesadmirablyw iththatfoundf romobser ationsw iththehori ontalpendulum w emay fee lconf identthatLaplace
ngtheseoceanictidesto conformtothee uilibrium
we erbee pla inedby f rict ion andatlength
gV o l . v ( 1 9 0 ) , p . 1 6 s h ow e d th a t la n d ba r ri e rs a s
nulthosemodesof f luidmotionw hich inthecaseof
et causesowideadi ergencef romthee uil ibriumlaw.
aplacewascorrectinfact asregardstheearth
g.
hesub ectw illbefoundinV o l. v iio f theEncy lopidiederMathematischenWissenschaften A rt. " B ew egungderHydrosphare .
n" thegeneralintegration ofLaplace stidale uation
stthanthetw oprecedingones since itssub ectisto
ed b y t he t w o me m oi r s of H o ug h ( s e e p . 2 1 w ho a l so
heeffectsof themutualattractionofthewater.
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[ 2 9
R .
A V E S I N F L O W I N G W A TE R .
ca l M ag a i n e x x i i . O c t ob e r 18 8 6 p p. 3 5 - 5 7 h a i n g
A of theB rit ishA ssociation B irm ingham
ebeautifulwa e-groupproducedbya.
hroughpre iously sti l lw ater butthe
islimitedtotwodimensionalmotion.
flowinginuniformregimethrough
hv erticalsides andbottomhori ontal
by trans erseridgesorho llow s orslopes
onta lbottomatdifferentle e ls. Included
sw emaysupposebarsabo ethebottom
tw eenthesides. Letthese ine ua lities
on A B , o f the length andletfdenote
of thebottom onthetw osidesof this
ebottombeyondA ishigherthanthe
i enataninf inite orv erygreat
petuallyflowingtowardsA withany
ocityu andf il lingupthecana ltoa
a.Itis re uiredtofindthe motion
throughA B , andbeyondB asdisturbed
weenA andB . Thisproblemisessentia lly
inasuf f icientlypractica lform theso lutionforthew a egroupproducedby theship w hichIhopetocommunicatetothePhilosophica l
nintheNo embernumber.
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A R Y W A V E S I N F L O W I N G W A T E R- I
onlyone solutionifweconfineitto
ca lcomponentof thew ater sv e locity is
mparisonw iththeve locityac uiredbya
heighte ua ltoha lf thedepth. Letb
v themeanhori ontalv e locityatv ery
and( toha ew todenotew a e-energy
w .. . .. . .. . .. . .. . .. . ( 1
ineticandpotentia l perunito f thecana l s
nto f itslength. Incasesinw hichthewater
eatdistancesf romB , w isz ero . B ut
isruf fled andthewaterf low s" steadily
andacorrugatedfreesurface asinthe
ofwaterflowinginamill-lead orHighland
uletontheeastsideofTrurnpingtonStreet
aceofPortland orIslayo erfa lls. The
esw hichweseeinthew a eofeachlitt le
w ould o fcourse e tendto inf inity if
long andthewaterabso lutely in iscid
d a si n gl e i ne u a li t y o r g ro u p of i n e u a li t ie s i n
mw ouldgi erisetocorrugationinthe
ssingthe ine ua lities moreandmore
hridgesandhollowsmoreand morenearly
ofthe canal thefartherwearefrom
tes. Obser ation w itha litt lecommon
lk ind showsthatatadistanceof tw o
omthelastofthe irregularitiesifthe
allincomparisonwiththewa e-length
en breadthsofthecanalif thebreadth
ththew a e- length thecondit ionof
straightridgesperpendiculartothe
dbefa irlyw ellappro imatedto e en
reasingle pro ectionorhollowinthe
tthesub ectof thepresentcommunicationissimpler asit isl im itedtotw o-dimensionalmotion and
s orridges orho llow s perpendicularto
us inourpresentcase w eseethatthe
rmityofthestandingwa esinthe
esisclose lyappro imatedtoatadistance
gthsf romthelasto f the ine ua lities.
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R
w ofi edv erticalsectionsof thecanalat
beyondA andbeyondB ; andp q the
seplanes.It willsimplifyconsiderations
SB atanode( orplacewherethedepth
a n de p th , a n d we t h er e fo r e ta e i t s o
saryforthefollowingk inematicaland
Thev olumesoffluidcrossingSA andSB inthesameor
l o r i n s ym b ol s
.. . . .. . . .. . . .. . . . (2 ,
lumeofwaterpassingperunit oftime.
e o r n eg a ti e o f t he w o r d o ne b y
w aterenteringacrossSA abo ethew or
lvo lumeof thew aterpassingaw ayacross
cessof theenergy potentia landk inetic
abo ethatofthe waterentering.
t a i n g th e v o l u me o f wa t er u n it y w e h a e
g b + ) ] . ... .. 3 ) .
( 3 .
tthef reesurfacezero w eha e
gb â € ” . . .. .. .. . ( 4 ;
ntitydependingonw a e-disturbance. Hence
w - w
+ = O . .. .. .. .. 5 .
2 = D ; a n d M = V D . .. .. . .. . .. . .. . 6 .
ndepth( intermediatebetweenaandb
ualtotheirarithmeticmean w henthe ir
parisonw ithe ither andV w illdenote
elocityofflow( intermediatebetweenu
mate lye ua ltotheirarithmeticmean w hen
comparisonwitheither .
5 g i e s
V 2
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A R Y W A V E S I N F L O W I N G W A T E R- I
ua lto f andif therew erenoberuf f lement
themeanle e lo f thew aterw ouldbethe
ea ingwateratgreatdistances onthe
thisisnotgenerally thecase andthere isa
e r i se o f l e e l g i e n b y th e f or m ul a
- ) - ( I . .. .. . 8 .
/ gD
nocorrugation( thatistosay o f
ormflow atgreatdistancesbeyondB .
0 a nd t he re fo re
/ i ) V \ ( 9 ;
byM2/ D2
) = ..........( 10 ,
D = . . .. .. . .. . .. . .. ( 1 1 .
betweenthesethreee uations
f It isclearthatthechangeof le e l
cientlygradualto ob iateanyofthe
dw henthisisthecase thee uationof
undf romy intermsof f fbe inga
ri ontalcoordinate x .
incomparisonw itha Disappro imatelyconstant[ muchmoreappro imate lye ua lto~ ( a+ b ] ,
nconstantproportiontof.
le thatV issmallincomparisonw ith
pr o pe r f ra c ti o n a n d y is a p pr o i m at e ly e u a l
asewhenV & lt V gDtheuppersurfaceof
ttomfalls andthewaterfallswhen
h en V & g t V I g D t h e wa t er s u rf a ce r i se s
r y pr o e c ti o n of t h e bo t to m a n d f al l s c on c a e o e r
andtheriseandfall ofthewaterareeach
rise andfallofthebottom sothat
18
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R
re le ationsof thebottom andisshallow er
bottom.
ecto f standingw a es( orcorrugationsof thesurface o f f rictionlessw aterf low ingo erahori ontal
ert icalsides Isha llnotatpresententer
ysisbywhichtheeffect ofagi enset
mitedspaceA B of thecana l slength in
onsinthew ateraf terpassingsuchine ua lities canbeca lculated pro idedtheslopesof the ine ua lities
ionsare e erywherev erysmallfractions
ongtocommunicateapaperto the
onthissub ectforpublication. Isha ll
hefo llow ingremar s: 1. A nysetof ine ua lit ieslargeorsmallmustingenera l
orrugationslargeorsmall butperfectly
arge shorto f the lim itthatw ouldproduce
u r a t ur e ( a c co r di n g to S t o e s s t h eo r y an o b tu s e
y trans erse lineof thew atersurface.
thewaterflowingawayfromthe
fectly smoothandhori onta l. Thisis
fo llow ingreasons: i Ifw aterisf low ingo eraplanebottomw ithinf initesimal
ualitywhichcouldproducesuchcorrugations
tomsoaseitherto doublethosepre iously
thesurfaceor toannulthem.
th( thatistosay the lengthf romcrestto
unctionofthe meandepthofthewater
orrugationsabo ethebottom andof
wingperunitof time.Thisfunctionis
nSto es stheoryof f initewa es. It is
t andisgi enby thew ell- now nformula
mal.
i t f ol l ow s t ha t a s i t is a l wa y s po s si b le t o
corrugationsbyproperlyad usted
itisalwayspossibletoannulthem.
cipleinthis modeofconsideringthe
erdisturbancetheremaybeinaperpetua lly
motionbecomesultimatelysteady all
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A R Y W A V E S I N F L O W I N G W A T E R- I
waydownstream.Thee planationof
e lopedinPartIII. tobepublishedin
edintheNo embernumberof the
a lhori ontalcomponentof f luidpressureon
tesinthebottom orbars w il lbefound
wor doneingeneratingstationary
iousapplicationtothewor donebywa ema ingintowingaboatthroughacana lw illbeconsidered.
onofthewa e-ma ingeffectwhenthe
maregeometrica llydef ined tow hich
red w illf ollow andIhopeto include in
tsinPartIII. tobepublishedinDecember
n i l lustratedbydraw ings o f thebeautiful
edbyaship propelleduniformlythrough
ca l M ag a i n e x x i r . N o e m be r 1 88 6 p p . 44 5 -4 5 2.
PartI. thesumofhori ontalpressures
bottom oronabar oronaseriesof
onsiderthehori ontalcomponentsofmomentumofdifferentportionsofthewaterin thefollowingmanner.
teady themomentumof thematterat
edv o lumeofspaceSremainsconstant
li eryofmomentumfromSbywater
egainofmomentumbywaterflowing
mustbee ua ltothetota lamountof
nthe waterwhichatanyinstantis
fthisforce beingthatoftheflow when
ingw atere ceedsthatof theentering
aceboundedbythebottom thefree
ndfourv ertica lplanes tw oof them ca lled
rtothestream andtw oof thempara lle lto
tancef romoneanother. Let$PB and
alinesonthetwotrans erseendsA andA oof the
b e i n g po i nt s o f th e s ur f ac e a n d B , B A p o i n ts o f t he
y
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R
ta lcomponentve locityatP . Therate
m( perunito f t imeunderstood f romSby
e ualto
1 ;
eryofmomentumfromSacrossA abo e
ossA 0ise ua lto
...( 2 .
thew aterbetweenA 0andA muste perience
ueinthedirectionf romA otowardsA made
ssuresonthe endsectionsA0andA
aterby f i edine ua lities if thereare
A . Hence ifX , X odenotethe integra lf luidpressuresonthe idea lplanesA A 0 andF thesumofhori onta l
l it iesonthef luid regardedasposit i e
tota lisf romA tow ardsA o ( 2 mustbe
. . .. . . 3 ) .
- ( X + u dy .. .. .. .. .. .. ( 4 .
P i s e u a l to g y + 1 ( q 2 - q 2 , b y t he
essureinsteadymotion( thepressureat
enasz ero , q andq denotingtheve locity
especti e ly .
2 -q 2 ] d y = I ( gD + q 2 D - q 2 dy.. . 5 .
D + q ) D + - v 2 d y. .. .. .. .. .. 6 ,
mponentv e locityatP .
ge pressionrelati elytoA0 gi e
mofhori ontalpressuresona ll ine ua litiesbetween
roblemof thef luidmotioninthecircumstancesissofarso l edastogi eD q , andu 2-v 2foreachof the
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A R Y W A V E S I N F L O W I N G W A T E R- I I
besofarontheup-streamsideof the
onofthe wateracrossitis sensibly
w ithve locityw hichw eshalldenoteby
0 ( 6 b ec om es
g D2 + U 0 D o . .. .. . .. . .. . .. . ( 7 .
a nd ( 4 ,
+ U 2 Do - q 2 ~ -D I- ( u 2 - d y. .. 8 .
locityatthef reesurface insteady
D o -D . . .. . .. .. . .. . .. . .. ( 9 ;
o B o f thebottombeingonthesamele e l
e lsbetweenthesurface-po ints$ 0
e co m es
U 0 ( D o -D - ( 2 - U 02 D
y. .. ( 1 0 ,
stantw hichmayha eanyva lue. It is
themeanhori ontalcomponentv e locity
reta e
.. . . . 11 :
ntit iesflow inginacrossA oandoutacross
m ot i on i s s te a dy w e h a e
. .. . . .. . . .. . 12 .
o f r o m ( 1 0 w e f in d
o- D 2 ~ ( + ( 2 + U 2 -U 2 dy .. 1 ) .
wek nowenoughaboutthemotion
eisrelatedtoother characteristicq uantities
9 , a nd i n it t a e
.. . .. . . .. . . .. . . .. 14 .
ta pointofthewater-surfacewherethe
e locity isrigorouslyorappro imately
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R
srigorouslyorappro imately thevertica l
. U s i ng n o w ( 1 4 i n ( 9 , w i th U D / D o
.. 15 ;
, g i e s
o + U 2 2
DU ] f V 2 + U 2 -U , 2 dy
2 . .. . .. . .. 1 6 .
eof le e l Do-D isbutsmall incomparison
e
( v 2 + U - 2 u dy .. .. .. 1 7 ,
matee ua lity . Go ingbac to( 16 , le t
ater-surfacethat
.. . .. . . . (18 ,
o becauseatacrestthef irstmember
andataho llow greater. Whenthe
mpleharmonic( thestream-lines
heposit ionof$ thuschosenw illbee actly
ndhollow.W henthemotionis
re a t u p t o St o e s s h i gh e st p o ss i bl e w a e
alessormoreroughappro imation
faw a e: it isa lw aysrigorouslydeterminate. F orbre ityw eshallca ll it thatistosayapoint
n o da l p oi n t. T h us w h en $ i s t a e n a s a no d al
messimplif iedto
D 2 ] + h i if v 2 dy .. 1 9 .
us.Init A whichisgi enrigorously
i m at e ly ( n o t ri g or o us l y e u a l to t h e v e r ti c al
: andifw esupposeDgi en Do isfoundby
b ic e u a ti o n in D o m o st e a si l y so l e d b y su c ce s si e
ngtotheprocessob iously indicatedby
uationappearsin( 15 . ( A saf irst
forDo inthesecondmemberandsoon.
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A R Y W A V E S I N F L O W I N G W A T E R- I I 2 7 9
19 forthecaseof infinitesimal
y ta eaatagreatenoughdistancef rom
rfaceinitsneighbourhoodbesensibly
themotionsimpleharmonic. The in estigationisfacil itatedbya lsota ing$ atanode asinthediagrams.
.. . .. . . . 20
reesurface thek now nsolutionforsimple
erofdepthDgi es
E -m D -y )
- s i n me
-D c o s m , . .. .. . 2 1 .
D+ mD
a s i n th e n od a l se c ti o n 3 P B ,
m t h m _ -m D. .. .. . 2 2 ;
2 mh 2. 2 )
. .. .. .. .4 D * . 2 4 .
9 w eseethatw henU approachesthe
D
eimportant e enthoughthecorrugationsata greatdistancedown-streamfromtheine ualities
ingconsiderationsofthiscase and
tobeconsiderablysmallerthanthe
ayneglectthef irsttermincomparisonwith
ingthatinfactq uantitiescomparablewith
dintheappro imation( 24 tothe
ndw eha e asourf ina lappro imate
2mD. .. . . .. . . .. . . .. 25 .
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R
derstandingthepermanentsteadinessofthemotionwhichweha enowbeenconsidering:toany
ergreat one ithertheup-streamordow nstreamsideof the ine ua lities if thew aterinthef initespace
hisstateofmotion andifw aterisadmitted
dawayontheother sideconformably.
ingandinstructi etoconsiderthe init ia tion
omanantecedentconditionof uniform
m. Suppose astheprimarycondit ion an
e le ationordepression toe istinthebottom
ththew ater sothattheflow of the
nformandinpara lle ll ines. If the ine ua lity isane le ationabo ethebottom oursupposit ionisthat
ece mo ingw iththew ater slipsa long
a litybeadepressioninthebottom the
ionmustbemadeofa plasticityofthe
f the ine ua lity carrieda long w hilethe
nebeforeandafter thisdepression.
ne ua lity isgradua llyorsuddenlybrought
eresult ingmotionof thew ater The
that offindingthemotionofwaterin a
ternalforce suchasthatofatow ing- rope
denlysetinmotionthroughit or
calif theboatwereabeamfilling the
anal sothatthemotionofthewater
sional.Ihopeina laterarticle( Part
sentseries to in estigatetheformation
dngw a esinthew a eof theobstacle
nfartherandfartherdown-streamfrom
nha ingbecomesensibly steady inits
comingsotogreater andgreaterdistances
letionofthegrowthoffresh wa es.
reamfromtheinitiatingirregularity
E uation( 15 showsthatw hetherthe
tion asinourf irstdiagram( f ig. 1 , o ra
a risingof le e lmusttra e lup-stream
ytothew aterw hichw ek now mustbe
sintermediatebetweenDoandthesmallerdepth
intheundisturbedstreamabo e. B ut
nit iatingirregularitymayha ebeen
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A R Y W A V E S I N F L O W I N G W A T E R- I I
l ingofane le ationup-streammustde e lop
locityofpropagationis asitw ere dif ferent
ope be ingV gD atthecommencement
ngfromthis throughV gDo , toV gD0asthe
o sothat asitw ere thebrow of the
up-streamo erta estheta lus t i l lthe
forourappro imation. The ine itable
w ater( ine itablew ithoutv iscidityof the
eactionpre entingthee cessi esteepness
-streaminamannerwhichitis difficult
therefore interestingtoseehow itmay
bysurface-action orbygi ingsomev iscosity
erestingtodothisby surface-action
beperfectly in iscid sothatourstanding
ybeperfectlyunimpaired.Andwemay
eringthef reesurfacea llo er( up-stream
haninf inite ly thinv iscouslyelasticfle ible
rans erse ly ( a f terthemannerof thesa il
by rigidmasslessbarswithendstra e ll ingup
desonthesides ofthecanal.Ifwe
eseendstoberesistedby forcesproportionaltothe irve locit ies andthemembranetoe ercise(posit i e
tractile tensiona lforce insimpleproportionto
eof itslengthineach infinitelysmall
hanica larrangementbywhichisrea li edthe
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R
fasurfacenormalpressurev arying
onentv elocityoftheotherwisefree
roportiontothis normalv elocitywhen
yma ingthev iscousforcessuf f iciently
theprogressof theriseof le e lup-stream
andperfectlya o idthebore . Wemay
fthe processionofstationarywa esdownstreamasslowasweplease.Theform ofthewater-surfaceo er
ua lit ies andtoanydistancef romthem
stream isnotultimatelyaffectedatall
n g a n d it b e co m es a s t im e ad a n ce s m or e
mathematicalsolutionforsteady
gi e w ithgraphicil lustrationsdraw n
mthesolution inPartIII.
ca l M ag a i n e x x I I . D e c em b er 1 8 86 p p . 51 7 -5 0 .
wemaynowconsidertheapplication
pedinitandinPartII. to thequestion
dweshallfindalmostsurprisinglya
nde planation 49~ yearsaf terdate o f
E perimenta lR esearchesintotheLaw s
lPhenomenathataccompanytheMotion
andha enotpre iouslybeenreducedinto
w nLawsof theR esistanceofF luids , "
ishsystemof" f ly -boat carrying
wandArdrossanCanalandbetween
ntheF orthandC lydeCanal a tspeeds
lesanhourtbyahorse orapa iro f
theban .Thepracticalmethodoriginated
itedhorse whosedutyitwastodrag
ed( Isupposeaw al ingspeed , ta ing
rawingtheboataf terhim andsodisco eringthatw henthespeede ceeded4/ gDtheresistancew as
ssell Es . M. A . F . R . S. E. R eadbeforetheR oyalSociety
18 7 andpublishedintheTransactionsin1840.
EnglishandA mericanrec oningofv e locity w hich
es1-609 3 k ilometresperhour or-44704metrepersecond.
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A R Y W A V E S I N F L O W I N G W A T E R- I II
Mr ScottRussell sdescriptionofthe
MHoustontoo ad antageforhisC ompany
ry isso interestingthatIq uote it ine tenso: C analna igationfurnishesatoncethemostinterestingil lustrationsof the interferenceof thew a e andmostimportant
cationofits principlestoanimpro ed
thediminishedanteriorsectionof
dbyraisingav esselwithasuddenimpulse
ressi ew a e thataverygreatimpro ementrecently introducedintocanaltransportow esitse istence.
n the iso latedfactw asdisco ered
wandArdrossanCanalof small
seinthe boatofW illiamHouston
prietorsof thew or s too f rightandranof f
anditw asthenobser ed toMrHouston s
hefoamingsternsurgew hichusedtode astate
andthevesse lw ascarriedonthroughwater
w itharesistancev erygreatlydim inished.
operce i ethemercantileva lueof this
withwhichhe wasconnected and
ducingonthatcana lvesse lsmo ingw ith
resulto f thisimpro ementw asso
epointo fv iew astobring f romthe
ersatahighv e locity a large increaseof
prietors.Thepassengersandluggage
oats aboutsi ty feetlongandsi feetw ide
ddrawnbyapair ofhorses.Theboat
behindthew a e andatagi ensigna l
o f thehorsesdrawnuponthetopof the
esw ithdiminishedresistance attherateof7
.
orsectionofdisplacementproducedby
ddenimpulseto thesummitofthe
snodoubtacorrectobser ationofanessentia l
n butit istheannulmentof " thefoaming
helowerspeeds usedtode astatetheban s
planationof thediminishedresistance.
atwhenthemotionissteady now a es
v o l . x i . ( 1 8 40 , p . 7 9.
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R
ttowedthroughacanal ataspeed
evelocityofaninfinitely longw a einthe
thew aterbe ingsupposedin iscid the
tbenilw hentheve locitye ceedsV / gD.
ouslyfortowageinani nfinitee panse
o eraplanebottom.
artII.for thewholehori ontalcomponent
torsuccessionof ine ua litiesonthe
culatethe resistanceonaboatof any
ppro idedw ek now theheightof the
owitsteadilyat itsownspeedinthe
reatdistancebehindit tobesensibly
hofthe canal accordingtotheprinciple
ofPartI. Theprinciple uponw hichthe
of f o rm u la ( 2 5 P a rt I I . m a yb e c al c ul a te d a re
inderof thepresentarticle andw illbe
PartIV .
f waterflowinginarectangular
w ithgeometrica lly specif iedine ua lities it
hemannerofF ourier to f irstsol ethe
hichtheprofileo f thebottomisacur e
esimally f romahori onta lplane.
a e O X a l on g t he m e an l e e l o f th e b ot t om
nofU themeanv elocityof thestream
sit i eupw ards. Let
. .. . . .. . . .. . . .. . . 1
bottom and
. . . .. . . .. . . .. . . .. . . . (2
reesurface f be ingheightabo eits
eve locitypotentia l u v theve locity
pressureatanypoint( x , y o f thew ater
e
. . . .. . . .. . . .. . . .. . . . 3 ) ,
+ ) . . .. . .. . .. . .. . . 4 .
uniformhori ontalv e locity isinf initesimal
are inf inite ly small. Hence( 4 gi es
- ( u -U ) . .. .. .. .. .. .. .. ( 5 .
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A R Y W A V E S I N F L O W I N G W A T E R- I II 2 8 5
f thee uationofcontinuity
rpresentcaseclearlyis
nm ( K e m Y+ K ' e -m y . .. .. .. .. .. . 6 ,
otionissteady K andK' areconstants.
) , gi es
m y + K ' e -m y . .. .. .. .. .. 7 ;
- K e -m Y . .. .. .. .. .. 8 .
o fyatthebottomandatthesurfaceare
especti e ly w ef indrespecti e ly forthe
elocityatthebottomandat thesurface
) , a nd m si nm ( K e mD - K ' / -m D .
ottom-stream-linesandsurface-stream-lines
theassumedforms( 1 and( 2 , w eclearly
H U . . . .. . .. . .. . .. . .. . .. ( 9 ,
D- K ' e - mc D = m s U . . . .. . .. . .. . 1 0 ;
H e - n D
. . .. . . .
hepressure isconstant andhence by ( 5 ,
c o ns t an t .. . .. . .. . .. . .. 1 2 :
( 7 , a nd ( 1 1 , w ef in d
( e + _ - - 2 H
1 ) ,
e 1 D - eD
rproblem forthecaseofthe bottoma.
ionof thebottomtobe
c 2c os 2m + K C c os 3 m + & a m p c . m A/ 7r .. . 1 4 ;
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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R
face foundbysuperpositionofso lutions
a llowablebecausethemotionde iatesinf initely
formmotionthroughoutthew ater is
mA / 7r
e - .. . 1 5 .
( imD_ - imD
n( 14 bywhichthebottomis defined
w ell- now nsummationof itssecondmember
/ n A/ r. K ( c os m - K )
n A = 1 - 2 c os m + 2 +
ergentforallv aluesofK lessthanunity .
o fF ourier C auchy andPo isson the
inite ly litt le lessthanunityw illbemadethe
lso lutions. B y ( 14 w eseethat
.. . . .. . . .. 17 ;
uations( 16 w eseethat
2 _ _ .
.. . .. . .. . 1 8 .
K 2 . .. . ..
ittleshortof unitythefactorofd
8 isz ero fora llv a luesofx dif fering
r/ m(i be inganinteger ; andit isinf inite ly
7r/m. Hencew einferf rom( 17 and( 18
nalsectionofthebottompresentsa regular
nsanddepressionsabo eandbelow itsmean
nsbeingconf inedtoverysmallspacesonthe
o intsx = 0andx = 2i7r/m andtheprof ileareaofeachele ationbeingA . Thedepthsof thedepressions
e linthe intermediatespacesbetweenthe
ursee tremelysmallbecauseof thee ceeding
erw hicharethee le ations. F orour
on notonlymustA beinfinitelysmall
opeuptothesummit ofhmuste erywherebeaninfinitelysmallfractionofa radian andofcourse
loweringofthebottombetweenthe
s pl o tt e d in ~ 4 o f " D e e p W a t er S h ip W a e s " i n f r a.
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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A R Y W A V E S I N F L O W I N G W A T E R- I II 2 8 7
optionofameanbottom- le e lforourdatum
uced maybeleftoutof accountinour
ot aninfinitelysmallfractionofa
lho ld pro ideditsheightisverysmall
epthof thew atero erit . B utthe
dge wouldthennotbeits profile-areaA
er ofwhichtheamountwouldbefound
o erit f a renoughabo eittoha enow here
slope andfindingtheprofile-areaof
e itsow na eragele e lconsideredasthe
esee planationswesha llspea o faridge
n" irregularity or" obstacle " andca ll its
he" magnitudeof theridge" ; thisbe ing
themeasureof itspotency indisturbingthe
faridgeweha eaho llow A isnegati e
emay o fcourse ca llaho llow anegati e
n erges anddoesnotdependforits
glessthanunity sothatinitw emayta e
nity andw esha lldosoaccordingly .
singleridge remar thatifI bethe
....( 19 .
uriernow supposeIinf inite ly large w hich
l andput
. . . .. . . .. . . .. . . .. . 20 ;
5 b e co m es
s q (
- - .. . . .. . . .. 21 ;
U2/ g. . .. . . .. . . .. . . .. . . .. . . .. . . . 22 .
beshortened andforsomeinterpretations
q D = a w h en i t b ec o me s
/ D
.
Eâ € ” )
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/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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R
1 or( 2 ) seemedratherintractable
uiredtoe a luate it f o rmanyandw idespreadenoughv a luesofx toshow theshapeof thesurfaceforany
D/b w ouldbev ery laborious. B utI had
uatingitfromtheperiodic solutionforan
uidistante ua lridges( 15 , w ho llyana logous
omcorrespondingsolutionsforcasesof
signallingthroughsubmarinecables to
and56ofmycollected Mathematical
nd tow ardsapply ingthismethodtoa.
bancedue toasingleridge Ihadfully
olutionfor thecaserepresentedbythe
g. 3, p. 295 , w henIfoundadirectandcomplete
single - ridgeprobleminaforme ceedinglycon enientforarithmetica lcomputation e ceptforthecase
o r f ro m z e r o t o a q u a r te r o r a ha l f of t h e de p th .
ppilygi esthesolutionforsmall v alues
a luesuptotw oorthreetimesthedepth by
ngseries andthusbetw eenthetw omethods
ysatisfactorysolutionof thew holeproblem.
ecur esandtheirre lationtotheproblem
allgi ethenew directso lutionof this
well- nownanalyticalmethodof
amplesaregi enintheEighteenthnote
i r on t h e Th e or y o f W a e s .
minatoro f (2 ) to theformof theproduct
uadraticfactors asfollows:-Let
- ( - e- . .. .. .. 2 4 .
o - w e ha e
l 2 ( 3 b
4 + & am p c . .. . 2 5 .
terthanD W isposit i e fora llrealv a lues
y po s it i e v a l u e l es s t ha n D W ( w hi c h
demieR oya lede ' InstitutdeF rance sa ansetrangers
P u b l i c D o m a i n
/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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A R Y W A V E S I N F L O W I N G W A T E R- I II
mallv a luesofa2 isnegati e forlargev a lues
t le a st o n e po s it i e v a l ue o f 7 2 m a e s W z e r o.
hatonlyoneposit i ev a lueof -2doesso .
erosofW w henbisgreaterthanD and
thanD correspondtorea lnegati e
disob iousif fo rQ2w eput-02 w hich
0 ( 2 6 ;
0. . .. . . .. . . .D
zerosofW aregi enby therootsof the
nta le uation
Dthise uationhasa ll itsrootsrea l
f if th & amp c. q uadrants. Whenbislessthan
drantislost andinitssteadw eclearly
w h il e t he r o ot s i n th e t hi r d f i ft h & a m p c .
Le t 0 i 0 0 & a m p c . b e th e r oo t s of t h e fi r st
c . q u a d r an t s. A s t he f i rs t t er m o f e u a t i on ( 2 5 i s
1+ f. ) ( + ) .
a m p c . a re r e al p o si t i e n u me r ic s w h il e 0 12 i s r ea l
eaccordingasbisgreaterthanDorless
proca lo fW intopartia lf ractions w e
& a m p c . . .. .. .. .. 2 9 ;
-D/b cos0i
D / b- T co s i
0
' (
( 0
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R
b O i i s a s we h a e s e en i ma gi na ry ( i t s s u ar e
forthiscasetheformula( 3 0 maybecon enientlyw ritten
I - I ( t 2 a 1+ e -2 1
dingo- is
. ( e - e -u = O . .. .. .. .. .. .. . 3 2 ,
ne andonlyone rea lrootwhenD> b
lt b.
t i s ea s y to f i nd a s t he c a se m ay b e a o o f
t- u a dr a nt r o ot o f ( 2 7 b y a ri t hm e ti c al t r ia l a nd
e r o ot s 0 2 0 , & a m p c . m or e a nd m o re e a si l y
. I t i s to b e r em a r e d t h a t w h at e e r b e
serootsapproachmoreandmorenearly to
uadrantsinwhichtheylie:thus if
.. .. . .. . .. . .. . ( 3 3 ) ,
- D / b s in a i
. 9- /- / sc I ( 3 4 ;
1 7 r - a ] = D / b. c o s a. . .. . .. . .. . . 3 5 ;
orappro imationw heniisv ery large
= D / b. a s/ ta n a . . .. .. .. .. .. ( 3 6 ,
creasedto inf inity theva lueofa i
y toD/ b( i- ) 7r . Hencewheniis
ndmemberof (3 6 becomesappro imately
d th e e u a ti o n be c om e s
' 7 r _- D/ b. .. .. .. .. 3 7 ;
hthesmallerrootw henDislessthan3b and
Disgreaterthan3b isthere uiredva lue
) a n d m od i fy i ng i t b y ( 2 4 a n d ( 2 9 , w e
a / D
;
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/ h t t p : / / w w w . h a t h i t r u s t . o r g / a c c e
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A R Y W A V E S I N F L O W I N G W A T ER - II I 2 9 1
ll- now ne aluation( attributedbyCauchy
teintegralindicated
. . . .. .. . .. . .. . . 3 9 ;
b y ( 3 3 ) a nd ( 3 4 ,
( i - ]
D .. .. . .. . . 4 0 ,
d en o te a l l th e p os i ti e r o ot s o f ( 3 5 .
the ceedingrapiditywhenx isany
ndw ithv erycon enientrapidity forca lculationw henx ise enassmallasatenthofD. Whenx = 0 the
ally thesameorderasthato f1-e+ e2-& c.
f indthesumbyta ingasremainderha lf the
cluded.Thetruev alueofthesumis
v alueswhichweobtainbythis rulefor
s andthenforonetermmore. Whenit
sultwithconsiderableaccuracy alarge
ere uired anditw illnodoubtbe
ethodas indicatedabo e.
ef irsttermforthecaseD& gt b w hich
h e fo r m ( 3 9 b u t re a l in t h e fo r m ( 3 8
2. F orthiscasew eha e by thew ell now ndef inite integra l f irst Ibe lie e e a luatedbyC auchy
. . .. . .. . .. . .. 4 1 ;
e n by ( 3 2 a n d ( 3 1 * .
t i n as m uc h a s ( 3 8 h a s th e s am e
e a n d ne g at i e v a l ue s o f x , t h e e a l ua t io n s
a n d ( 4 1 a r e es s en t ia l ly d i sc o nt i nu o us a t x = 0
e - mustbesubstitutedforx inthe
rmulas. Ihope inPartIV . togi e
butwithorwithoutnumericalillustrations
re inthe integralsthe" principa lva lue o fC auchy is
ectstheinfiniteamplitudesinthe integrand which
ithf reev ibrations innaturesuchvery largeamplitudes
ictionalagencies andwhenthefrictionis slightthe
narrow confinedtothev erynearneighbourhoodofthe
ractua lcontributionisnegligible andthe" principa lva lue
fed.
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R
9 , w ith(41 foritsf irsttermandthe
ghoutw henx isnegati e isparticularly
uouse pressionforacur epassingcontinuously f romonetotheotherof thetw ocur es
--forlarge positi ev aluesofx
argenegati ev a luesofx . . . .. . . .. 42 .
D e e r y te r m of ( 3 9 i s r ea l a n d ( r e m em b er i ng t h at t h e si g n of x i s c ha n ge d wh e n x i s n eg a ti e w e
u a l fo r e u a l po s it i e a n d ne g at i e v a l u e s
symptotically toz eroasx becomesgreater
on. Ite pressesunambiguouslythe
uew henb> D o f theproblemofsteady
rmrectangularcanalinterruptedonly
tudeAacrossthebottom.This isthe
greaterthanthatac uiredbyabody in
ua ltoha lf thedepth.
ouni uenessofthesolutionwhen
ssthanthatac uiredbyabody infa lling
to h a lf t h e de p th ( b & l t D . F o r t hi s c as e
and( 41 e pressaparticularso lutionof the
througharectangularcanal when
nlyinterruptedbythesingleridge of
earlyha eaninf initenumberofso lutions
e instil lw aterinacanalo fdepthDw e
o f an y v e l o ci t y fr o m z e r o t o V / g D w hi c h i s
te ly longw a einw aterofdepthD. In
perimposeupontheso lution ( 3 9 ( 41 ,
trarymagnitude andarbitrarilychosen
eros w ithw a e- lengthsuchthatthe
agationisU , andthedirectionofmotion
essionofthewa etobeup-stream.
nstitutedconstitutesasetof freestationary
positionofthisupon thecaseofmotionrepresentedbyour symmetricalsolutionconstitutesthegeneral
ngle-ridgesteadymotion.To findthe
emustthusma etooursymmetrical
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A R Y W A V E S I N F L O W I N G W A T E R- I II 2 9
also lution put( 1 ) intothefo llowing
g ( D e â € ” D . .. .. .. .. 4 ) .
' . .
~ mayha eanyv alue( thatistosay
yw a esofanymagnitudeo eraplane
e m D = . . .. . .. . . ( 4 4 .
nowne uationtofindthev elocityU
ofperiodicw a esofw a e- length27r/ m in
satpresente uation( 44 istobe
cendentale uationfordeterminingthe
dingtoU agi env e locityofprogress and
en o n ly o n e re a l ro o t wh e n U & l t / g D b u t no
V / g D. P u tt i ng n o w in ( 4 ) U 2 = g b a n d co m pa r in g wi t h ( 3 2 , w e se e t ha t m D = o -r a n d go i ng b a c t o e u a ti o n
ha t
.. . .. . . .. . . .. 45 ;
raryconstants istheaddit ionw hichw e
t o g i e t h e ge n er a l so l ut i on f o r th e c as e b & l t D .
3 9 a n d ( 4 1 w e a cc o rd i ng l y ha e f o r
single-ridgesteady-motionproblem
V g D
C ~ W A/ D i N s n D + ~ i D
a n d
( ~ ^ A /D I s x A / ID o N I
.7 + E s C - D B N e D
D l b 2 %
.. . .. 4 6 ;
arbitraryconstants andAis theprofilesectionalareaoftheridge onthebottom.
ythissolution withanyv aluesofC
andstablethroughoutany f inite lengthof the
idge pro idedthewaterisintroduced
onsideredandta enawayattheother
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S O N W A T E R [ 2 9
tendstoinfinityinboth directions
utbegi eninthestateofmotioncorrespondingtothesolution( 46 themotionthroughoutany f inite
heridgewillcontinuefor aninfinite
6 . Thew ater ifgi enatrest m ightbe
otioninthefollowingmanner:- irst
eshaperepresentedbye uation( 46 , and
dtok eepite actly inthisshape so
twerein arectangulartubewithone
desplane andthefourthside( thebottom
placeof theridge. Ne tbymeansofa
allyin motioninthistube.To begin
he lidw ill inv irtueofgra ity benonuniform lessatthehighpartsandgreateratthe low parts. If
gi entothew aterby thepistonthe
ueof fluidmotion begreateratthehigh
arts. If thea eragev elocitybemade
surew illbeuniformo erthe lid w hichmay
the li uidisle f tmo ingsteadilyunder
ye uation( 46 asf reesurface. B utit
motionbeinggi entothef luidthroughout
ana loneachsideof theridge thatthe
oneachside oftheridgeconformableto
theparticularcaseof thisgenera lsolution correspondingto
/D . .. .. . .. . .. . 4 7 ,
4 6 t o
D w h en x i s p o si t i e
j E D w he n x i s ne ga ti e
sthemathematicalso lutionpromisedinthe.. . . .. . . . (48 ;
tion forthecaseofwater flowingfrom
o erthesingleridgeandtow ardstheside
mathematica lreali a tion forthecaseofa
cumstancesdescribedinPartI. abo e( ante
sthemathematicalso lutionpromisedinthe
Thedemonstrationthatthisis the
onforin iscidw aterf lowinginacana l
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A R Y W A V E S I N F L O W I N G W A T E R- I II 2 9 5
dthee planationofhow anyotherstateof
a m pl e a s t ha t r ep r es e nt e d by ( 4 6 w i th a n y
butgi entothew aterthroughoutonlyaf inite
heridge settlesintothepermanent
edby ( 48 , mustbereser edforPartIV .
theJ anuarynumber.
ingdiagramrepresentsbytwocur es
46 fortheparticularv a lue2 456for
o r v e l o ci t y = ' 6 8 1 o f th e c ri t ic a l v e l o ci t y V / g yD .
ntsthesolution( 46 withC= 0 andC = 0.
esentsthepracticalsolution( 48 .These
ca lculationsofaperiodicso lution accordingtothef irsto f thetwomethodsindicatedabo e before Ihad
on( 3 9 byw hichthedesiredresult
edatw ithmuchlesslabour. Thefa intcur e
alculationfromtheperiodicsolution:
- 1 s h ow o n t he t w o si d es o f o ne r i dg e
romridge toridgeinthe periodicsolution
themiddleof thediagram. Thehea y
gtotheordinatesof thefaintcur ethe
ines foundby tria ltoasnearlyaspossible
andtodoubleontheotherside theordinates
wnearlyperfectwastheannulmenton
blingontheotheris illustratedbythe
ed( f ig. 3) , w hichhasbeendraw nby the
t imeslargercopy . How nearlyperfectthe
ngoughttobe atanyparticulardistance
easilycalculatedfromthesecondline of
ndw illbeactua lly ca lculatedforthecaseof these
ya lso forsomeothercasesfornumericali l lustrations w hichIhopetogi e inPartIV .
thepresentin estigationtotheef fecto fanine ua lityo fany
ea m i s gi e n b y V . E m a n A r ch i f i r Ma t em a ti ,
, B a n d 3 , N o . 2 1 9 0 6 .
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R
Y W A V E S O N T H E S U R A C E PR O D U C E D B Y
S O N T H E B O T T O M .
ca l M ag a i n e V o l . X X I I I . J a n ua r y 18 8 7 p p . 52 - 57 .
fsol ingthisproblemis bytheuseof
chweha ebeensow elltaughtbyF ourier
yofHeat andinthiswayit was
mulas1to15 ; theso lutionbeing( 15
.. . .. . . .1. . .. . . ( 1 ;
ancef romridgetoridge. Thus reproducing( 15 PartIII. withthenotationmodifiedtoshortenit in
mericalcomputation w eha e
(
e- i *
abo emeanle e lo f thew ater
o into eroneof the
lareaofoneof the
( 3 ) .
f P ar t I I I. ( 6 t o ( 1 8 o r
eane pressionforthesurface-effectofan
uidistantridgesonthebottom.W eshall
uccessionofridgesisfinite theresult
l lnotbeappro imatedtoby increasingthe
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A R Y W A V E S I N F L O W I N G W A T E R- I V
renceinthe effectofamillione uidistantridgesfromthatofa millionandonee uidistantridges
tionsonthesurfaceof thef luido erany
beasgreatas thedifferencebetweenthe
f athousandandone orbetweenthe
en: andtheabso luteef fecto f four orsi ,
bly thesameas ormaybegreaterthan or
ef fecto famillion inrespecttothecondit ion
pacebetweenthetwomiddle ridges.The
nsiderationofinfinityforourpresent case
w ith a f terthemannerofF ourier by
nitecana l" an" endless* canal " oracana l
t:acircular canalaswemayimagine
htbecur ed o fany form pro idedonly
cularornotcircular theradiusofcur ature
tcomparedwiththebreadthof thecanal.
necessarytoallowthe motionofthe
ecanal tobesonearlytwo-dimensional
imensionalmotioninastraight
pplicableto thewaterinthecur ed
gralnumbernofe uidistantridges
etabethedistancef romridgetoridge.
dditionofsolutionsofthe formula( 2
eef fect
)
e i - e -i
w ord" endless shouldincommonusage andespecia lly
odif ferentameaningf rom" inf inite . Thuse eryone
tbyan" endlesscord. A n" inf initecord means
ninfinitelylongcord-acordwhich hasnolimitto the
lusagein mathematicallanguage accordingto
e isca lleda" closedcur e " musthenceforthbeabso lutely
d toendlesstroubleinelectrical nomenclature according
nguage ane lectriccircuit issa idtobeclosedw hena
andtobeopenwhena currentcannotpassthroughit.
ta ll Englishw ritersone lectrica lsub ectsha ebeenguilty
whetheranyoneofthemwouldsaya roadroundapar
isclosed andisclosedw hene erygateonit isopen.
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R
ofdifferentv aluesofn e enorodd
tionsbothof mathematicalprinciples
dynamics butforthepresentI confine
I fo rw hich( 4 becomesidentica l
t i f M ( e i - e - i / e + e - i [ p r ac t ic a ll y c on s ta n t
saninteger thedenominatoro f (2 v anishes
othisinteger. Thisisthecase inw hich
thecanalis anintegralnumberof
o f f reew a esinw aterofdepthD. The
s andisinterestingbothinitselfandin
gproblemsin manybranchesofphysical
that w henthev a lueof
e- i
toany integerj , thechie f termof ( 2 is
andalltheothertermsarerelati e lyvery
ctisforcedstationaryw a esofw a elengtha/ . Thus ifw econsiderdifferentv e locit iesof f low
renearly totheve locityw hichma es
e-i aninteger themagnitudeof theforced
terand greaterforthesamemagnitude
nisstillperfectlydeterminate.Suppose
dgesmallerandsmaller sothatthew a eheightof thestationaryw a emayha eanymoderateva lue as
moreandmorenearly tothatwhichma es
e - i a n i nt e ge r t h e ma g ni t ud e o f th e r id g e mu s t
andinthe lim itmustbez ero . Thus
mayha estationaryw a esofanygi en
helim itingcase - thatinwhichtheve locity
e l o ci t y of a w a e o f wa e - le n gt h a / .
r th e c as e o f M( e i - e -i / e i + e - i a s f ar
ninteger thatistosay
- i j + . . .. .. .. .. .. .. . 5 ,
ora llv a luesof ilessthanj +1 the
sclearlynegati e w ithdecreasingabso lute
andfora llva luesof igreaterthanj it is
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A R Y W A V E S I N F L O W I N G W A T E R- I V
sngv a luesf romi= j + 1to i= o . Thus
of thecoef f icientsofcosi rinthesuccessi etermsof theseriesf romthebeginningarenegati e w ith
esupto i= j ; andaf terthatposit i e
con ergingultimatelyaccordingtothe
hate= 2rD/ a w eseethatthecon ergence issluggishw hena thedistancef romridgetoridge( or
thecaseof anendlesscanalwithone
y large incomparisonw iththedepth butthat
epth ornotmorethanf i eortentimes
dingly interestingclassofcases , thecon ergence isv ery rapid.
how e er anotherso lutionstil lmore
morecon ergentindeedforthegreaterparto f
te erbetheratioofDtoa aso lution
entine erycasee ceptforv a luesofx
thedepth.Thecalculationforthese
ecessary togi etheshapeof thew ater-surface
ofthev erticalthroughtheridgesmall
th:for thispurpose andforthispurpose
2 indispensable . F orin estigatinga ll
tionthe newsolutionismuchmore
o l e s o n t he w h ol e v e r y mu c h le s s of
ndby summationfromthesolution
m gi e n i n Pa r t II I . ( 4 0 , ( 4 1 , a s
i dg e s be j + j ' + 1 a n d le t i t be
apeof thesurfacebetw eenthev ertica ls
+ 1andj + 2. Ta etheoriginof the
rticalthroughnumberj + 1 ridge andlet
heposit i esideof it . Theso lutionw illbe
lution( 40 PartIII. j so lutionsdiffering
i n g r es pe ct i e ly x + a x + 2 a . .. x + j a
j ' so lutionseachthesameas( 40 PartIII.
- x + 2 a . .. - x + j ' a s ub st it ut ed f or x . T h us
heef fectsof the j + j ' + 1single - ridges
1 -f + l f ix + ( 1 _f i ' ) f i i- l a.
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R
~ i
c o sa i
a
0 orthenumericbetw een
isfiesthee uation
n a i -D /b = 0
7 .
yof thef low
mridge toridge
onalareaofone ofthe
ontalcoordinatex , the
ethemeanle e lo f
herupstream or
es
gt D.Inthiscase asweha ealready
a , a c 2. .. a s a re a l l re a l a n d th e re f or e
achrea landlessthanunity . Hence inthiscase
s e ri e s o f w hi c h th e s um s a pp e ar i n ( 6 , a r e
d i f we t a e j = c m a nd j ' = o o ( 6 b ec om es
) . f i
ee pressionforSw hiche erof theridges
fx ; andtheva lueforx = a ise ua lto
Thew ater-disturbance isthereforee ua land
omridgetoridge andthesolution( 8 ,
e pressesw ithintheperiodtheheightof the
e e l n o t n ow a s i n ( 2 , t h e m ea n l e e l
buta le e latahe ight S. d / aabo e
b y i nt e gr a ti o n of ( 8 w e fi n d
. .. . .. . .. . .. ( 9 .
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A R Y W A V E S I N F L O W I N G W A T E R- I V
rmingthesecondmemberof this
t ha t by ( 7 a bo e a nd ( 3 4 P ar t II I. w e ha e
' C os a C A l a. N i
/ D. s i n2 a i 1 - D /b )
a r t II I . ( 2 9 a n d ( 2 4 , w e fi n d
. .. . . .. . . .. . . 11 .
, ( 9 b ec om es
.. . .. . . .. . 12 .
e by Itheheightabo emeanle e l
f indf rom( 8 ,
.. . .. . . .. ( 1 ) .
thisand( 2 abo e twodifferente pressionsforthesameq uantity( with forsimplicity D= 1 , leads
abletheoremofpureanalysis
e -i
ai6- io+ e-0( a- ) 11
' 1 - e -~ a
enumeric
gt 1
etw eenzeroandv r/ 2
tion. . 15
t a n a- 1 /b = 0
7 r - a i
t i enumeric& lt a
asilyv erifiedbyta ing d c.cos --
memberofthe resultisob iously
- e - ) T h e se c on d m em be r m o di f ie d b y
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S P R O D U C E D B Y A S I N GL E I MP U L S E I N
P TH O R I N A D I S PE R SI V E M E D I U M .
ca l M ag a i n e V o l . x x I I i .M a rc h 1 88 7 p p . 25 2 -2 5 5 h a i n g
ya lSociety 3 rdF ebruary 1887 P roceedings
mplicity consideronly thecaseof tw odimensionalmotion.
nowofthemediumisthe relation
locityandthewa e- lengthofanendless
es. Theresulto fourwor w il lshow
progressofaz ero orma imum or
fav ary inggroupofw a esise ua lto
sofperiodicwa esofw a e- lengthe ua l
chmaybedef inedasthewa e- lengthin
particularpointloo edtoin thegroup
rallybe intermediatebetweenthe
nsideredtoits ne t-neighbourcorrespondingpointsontheprecedingandfollowingwa es .
e locityofpropagationcorresponding
TheF ourier-Cauchy-Poissonsynthesis
- t f m ] . . .. . .. . .. . . 1
dtime( x , t o faninf inite ly intense
me( 0 0 . Theprincipleof interference assetforthbyProf. Sto esandLordRay leighinthe ir
andw a e- e locity suggeststhefollow ing
l:Whenx - tf m isv ery large thepartso f the integra l( 1
ofasmallrange p- ato / + a v anish
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E S O N W A T ER [ 3 0
, be ingava lue ortheva lue o fm
] } = . . .. .. .. .. .. .. .. .. 2 ;
f ( A = V t . .. .. .. .. .. .. .. .. . 3 ) ,
f / L + , L f ' ( / * . .. .. .. .. .. .. .. .. .. . 4 ;
stheorem form-A v erysmall
/ x : -tf ) ] - t[ ~ f / ( ) + 2f ( ) ] I ( m - ) . .. 5 ;
,
= t { /2f ( A / + [ - Lf ( / - 2f ( A ] ( m -. 2 .. ( 6 .
1 -A d â € ” l â € ” . .. .. .. e
2f ( tA ] 2
1 , wef ind
A + o - 2
f ( p ]
einghere-ootooo becausethe
sso inf inite lygreatthat though + a the
are infinitely small amultipliedby it
2= . . . . . .. . . 9 .
- si n[ t L 2f / ( L ] 1 /2 co s[ t a 2f /( L + T r r
f ( / . ] 2 2 rr 2t f [ ( ) - 2f ( 4 ] ~ . .. .. .. .. 1 0 .
locityaccordingtoLordR ay leigh sgenera li a tionof
aresult. [ F orfurthere tensiononthe linesof thepresent
or e L ec t ur e s ' A p p. C p p . 52 8 -5 1 a n d pa p er o n ' D e ep - Se a
e d in f ra ~ ~ 8 0 s e . a l so H . L am b H y dr o dy n am i cs 3 r d e d.
. H a e l oc , P r o c . Ro y. S o c. A u g. 1 9 08 p p . 3 9 8 -4 0 a n d
E. v o l . x x i , J u ly 1 90 9.
at thisconditionofv erygreatdenominatorisnot
ssof tsuff icingby itse lf to j ustify the inf inite lim its: cf .
a .
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E S P RO D U C E D B Y A S I NG L E IM P U L S E 3 0 5
e-lengthandwa e- elocityforany
ar t h at b y ( 3 ) ,
- tf ( ) ] ,
toro f (10 ise ua lto / 2cos0 w here
] + I r .. . .. . .. . ( . . . .. . .. . ( 1 0
) ,
) ] } = 0
dO / d t = - l f p . .. .. .. .. 1 0 ) ,
sition.
f ir s t e a m pl e t a e d e ep - se a w a e s w e
.. . .. . . .. . . .. . 1
( 3 ) , a nd ( 1 0 t o
12 ,
t. . .. . . .. . . .. . . .. . ( 1 ) ,
gt27r
n = a os - ( 1 4 ;
2~ ^ , 4 4
son sresultforplacesw herex isv ery
hew a e- length27r// thatistosay
atgt2/ 4 isv ery large.
e s i n wa t er o f d ep t h D
- 2m .- .. .. . .. .. .. . 1 5 .
gh t i n a di s pe r si e m e di u m.
la ry g r a i t at i on a l wa e s
. .. .. .. .. .. .. ( 1 6 .
la ry w a e s
. .. . .. . .. . .. . .. . .. . .. . .. . .. . 1 7 .
20
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R
e s o f fl e u r e ru n ni n g al o ng a u n if o rm
. . .. . . .. . . .. . . .. 18 ,
e ura lrigidityandw themassperunito f
esha ebeenta enbyLordRayleigh
ra li a tionof thetheoryofgroup elocity andhehaspointedout inhis" StandingWa esin
ndonMathematica lSociety December1 ,
t an t p ec u li a ri t y of e a m pl e ( 4 i n r es p ec t t o
w hichgi esminimumw a e- e locity
ocitye ua ltow a e- e locity . The
entproblemforthiscase oranycase
minimumsorma imums orboth
ms o fw a e- e locity isparticularly
esnotpermit itsbeingincludedinthe
a n d ( 6 t h e de n om i na t or o f ( 1 0 i s i ma g in a ry
on f rom( 7 forwards gi esfor these
o f ( 1 0 , t he f ol lo wi ng :c os [ t 2f ( ) / ] + s i n [ t p U f ( a ] 1 9 .
t + 2f ( p ]
downforeachofthetwo last
a nd ( 6 ] .
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T A N D R EA R O F A F R E E PR O C E SS I O N O F
TER.
a n . 7 1 8 87 P h il . M ag . V o l . x x i i i.
11 - 1 20 .
perondif ferentlinesinPhil. Mag. O ct. 1904:
inaPaperinPhil. Mag. J an. 1907 ~~ 127-158:
ause inmyR . S. E. paperofF eb. 1 anditssuccessor
substanceof it w ithpromisede tensions isgi enin
e ea s il y r ea d f o rm . K . ( M e nt o ne M a rc h 3 0 1 9 04 .
S .
onw iththe InstitutionofMechanica lEngineers
A ug. 3, 1887.
Mech.Eng.1887 inPopularLecturesand
pp. 450-500 someof the il lustrationsbe ingomitted.
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[ 3 3
G A TI O N O F L A M I NA R M O T I O N T H RO U G H
V I N G IN V I S C ID L I Q U I D .
Re p or t 1 8 87 p p . 48 6 -4 9 5 P h il . M ag . V o l . x x I .
5 .
estigateturbulentmotionofwater
nes forapromisedcommunicationtoSection
tionatitsMeetinginManchester Iha e
lytowardsasolution( manytimestried
years o f theproblemtoconstruct by
ontoanincompressible in iscidf luid amedium
esoflaminarmotionasthe luminiferous
f l ight* .
dedona llsides andletu v , w
en t s a n d p th e p re s su r e at ( x , y z , t . W e
+ w ~ + 7 ~ ' x . . .. .. .. .. .. 2 ,
\ d dy d d ) ( 2
d dp
+ w ~ + ~ . .. .. .. .. . )
d y .. . .. . '
dp
d . .. ..
, ( 4 w ef in d t a i ng ( 1 i nt oa cc ou nt
d w 2 ( d d w d w dd w d ud
+ 2 - € ” + - ~ - + - --d d y \ d \ d d y d d dy d . ... ... .. ( 5 .
G e ra l d N a tu r e M a y9 1 8 89 P r oc . R oy . D ub . S oc . 1 89 9 a n d
o r i n Sc i en t if c P ap e rs 1 9 02 p p . 25 4 4 7 2 4 8 4. S e e al s o
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M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 0 9
o ne n ts u v w m ay h a e a n yv a l ue s
ac e s u b e c t on l y to ( 1 . H e nc e o n
w e h a e a s a p er f ec t ly c o mp r eh e ns i e
onatany instant
m + e o s ( n y+ f c os ( q z + ) . 6 ,
s ( m + e s il n( n y + f c O s ( q z + g .. . 7
os m) e o s ( n ym f s in ( q z + g ... 8 ,
f g ( e f a
q ) ' ( , q ) ' f e ) a r e an y th re e v e lo ci ti es s at is fy in g
g ( e f g ............ 9 ;
q ) + ' l ( m n q ) + q y m n q ) ............ 9 ;
on(orintegration fordifferentv a luesof
g . Th e su mm at io ns f or e f g m ay w it ho ut l os s of
f inedtotw ova lues: e= O , ande= 7r
= 0 a n d g = ~ 7 r . W e s h al l a dm i t la r ge v a l u e s
- l n- l q -1 undercerta inconditions[ ~ 4
( 1 2 , a nd ~ 1 5 be lo w , b ut o th er wi se w es ha ll s up po se
chof themtobeofsomemoderate or
linearmagnitude. Thisisanessentia lo f the
now proceed.
, x y a d en ot e sp ac e- a e ra ge s l in ea r s ur fa ce
telygreatspaces definedandillustrated
wo r e d ou t fr om ( 6 , ( 7 , ( 8 , a s f ol lo ws L
atlength orav erygreatmultipleof
n - , q - l m ay b e c on c er n ed : I1 ( L 7 c f g g
a .2 aA o ' , q ) c os ( n y+ f c os ( q z + g
.. .. . 1 0 ,
d d u = 2 a f ) c os ( ny + f . .. 11 ,
p r 0 0
- ) L j d dy d u = , ) .. ... ... ... 1 2 ,
e , ) ] 2 os ny + f cos2 ( q + g . . 1 ) ;
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R
sthat
e = 0 w e t a e 0 i n p l a c eo f
e = - 7 1 , ,
2 ( n . . . .. . .. . .. . .. . .. ( 1 4 ,
( r f g ( 0 , )
m n q )
f g
m n q ) ] c os ( n y+ f si n( n y f . .... ... . 5 ;
1 4 t h at
e= 0
e ad o f -
0andg= w ir
e= Tr 1 1
a n d g = 0 " 2 " 4
= T 7r n = O , f = - r , , , 1
nsfor( 15 .
e 2
( m n q ) ) * . . .. .. .. .. .. .. .. . 1 6 ,
rosofmandq, ana logoustothoseof ( 14 .
a eragingsforthepresent ta e
T hu s we f in d
t( e fg ( e f g ( e f g 2
8-2 { m a m n2 + q n ( m , q ) - + q y m , q )
7 .
ious.
ra lpropertyof thisk indofa eraging
.. . .. . . .. . . .. 18 ,
w hichisf inite forinf inite lygreatva lues
ntobe homogeneouslydistributed
iesthat thecentresofinertiaof all
idha ee ua lparalle lmotions if anymotions
herefore w eta eourreference linesOX ,
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M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 1 1
e d r el a ti e l y to t h e ce n tr e s of i n er t ia o f t hr e e ( a n d
sof inertiao f largevo lumes inother
anslatorymotionofthefluid asawhole.
erylargea erageofmandof v andofw
mayremar , w ithreferencetoournotationof
s a sw es ee by ( 1 0 , ( 1 1 , ( 1 2 ,
= a n 0 q ) = a m n 0 = / 0 , q ) ... = ( m , , 0 . * ( 19 .
how e er encumberingourse l esw ith
nandnotationof~ 3 , w emayw rite asthe
null ityo f translationalmo ementinlarge
= a e w .. .. .. .. .. .. .. ( 2 0 ;
a eragethroughanygreatlengthofstraight
aofplaneorcur edsurface orthroughany
i ednotationofa erages homogeneousnessimplies
e v 2 = V 2 a e W 2= W 2.. . 2 1 ,
a ewu = CA a e u = AB . .. 22 ;
A B , C a r e si v e l oc i ti e s in d ep e nd e nt o f t he
whichthea eragesareta en. These
er inf inite ly shorto f imply ing though
ousness.
utionofmotionto beisotropic.
telymorethan isimpliedby the
ermsof thenotationof~ 8 w ithfurther
ew hatw esha llca llTHEA V ER A GEV ELOC ITY
R 2. . .. . .. . .. . .. .. . . ( 2 ) ,
. .. . . .. . . .. . . .. . . .. . . .. . . . (24 .
wpresentthemsel esastotransformationswhichthedistribution ofturbulentmotionwill
e li u idle f tto itse lfw ithanydistribution
initialdistributionbe homogeneous
esofspace e ceptacerta inlargef inite
hichthere isinit ia llye ithernomotion or
neousornot butnothomogeneouswith
roundingspace willthefluidwhich
uiremoreandmorenearlyastime
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R
ogeneousdistributionofmotionasthatof
tillultimatelythemotionishomogeneous
s coulditbethatthise ua li a tion
hsmallerand smallerspacesastime
words w ouldanygi endistribution homogeneousona largeenoughsca le becomemoreandmoref ine-gra ined
obablyyesforsomeinit ia ldistributions
obablyyesforv orte motiongi en
onelargeportion ofthefluid while
itia lmotiongi enintheshapeof
ho lt rings o fproportionssuitable for
andeachofo era lldiameterconsiderably
edistancefromnearestneighbours.
ghtheringsbeofv erydif ferentv o lumes
obablyyes if thediametersof therings
enotsmallincomparisonwithdistances
he indi idua lrings eachanendlessslender
ornearlyentangledamongoneanother.
: If the init ialdistributionbehomogeneousandceo lotropic w il l itbecomemoreandmorenearly isotropicastimead ances andult imatelyq uite isotropic P robably
tialdistribution whetherofcontinuous
orofseparatef initev orte rings.
metricalinitialdistributionof v orte
table .
nbehomogeneousandisotropic
andominrespecttodirection w ill it
yes. Iproceedto in estigateamathematica lformula deducible f romtheansw er w hichw illbeofuse
y ( 2 2 a nd ( 2 4 w e ha e
f or a ll v a l u es o f t. .. .. .. .. .. . ( 2 5 .
) w e fi nd
d ( u ) d( u ) dp dp
- z a u - -+ v - + w + v - + u --
d d dy . .. .. .. .. 2 6 .
e th e rh a ll A u g. 1 0 1 8 89 . S e e p. 2 0 2 su p ra .
nfactsuchfeo lotropyasthatof~ 20ismerely translationa l
rt ices. W. T .
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M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 1
d ( z v ) d ( u ) d p dp..
v + M s -+ V mu * * ( 2 7 .
c d y
sfor e eryrandom caseofmotion
lo w b e ca u se p o si t i e a n d ne g at i e v a l ue s o f
ua llyprobable andthereforetheva lueof the
isdoubledbyaddingto itselfw hat. it
, w w e su b st i tu t e -u - , - w w h ic h
a n d v e r if i ed b y l oo i n g at ( 5 , d o es n o t
etheinitial motiontoconsistof
, O , 0 s u pe r im p os e d on a h o mo g en e ou s
u 0 v o w ) ; s o t h a t we h a e
+ u 0 v = v o w = wo .. .. .. 2 8 ;
t o f in d s uc h a f un c ti o n f y t , t h at a t
componentsshallbe
w .. .. .. .. .. .. .. .. .. .. .. . 2 9 ,
uantit iesofeachofw hiche ery largeenough
o t ha t p ar t ic u la r ly f o r e a m pl e
z a v = x z a w .. .. .. .. .. .. .. .( 3 0 .
o r u v , w i n ( 2 w e fi n d
y t d u df
~ d y
3 1 .
. .. 3 1 .
fbothmembers.Thesecondtermofthe
ondtermof thesecondmemberdisappear eachinv irtueof ( 3 0 . Thef irstandlasttermsof the
r eachinv irtueof (18 a lone anda lso
0 . T h er e r em a in s
z a U d + v -+ w .... .. 3 2
econdmember[ by( 1 ]
d w) - + u + . ( 3 3 ) ;
- + U - .. .. .. .. . 3 3 ) ;
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R
pairoftermsofthe thus-modifiedsecond
1 8 f i nd
( u )
.. . .. . .. 3 4 .
t t hi s r es u lt i n o l e s b e si d es ( 1 , n o
( u v , w t h an ( 3 0 ; n o i so t ro p y n o
ct toy andonlyhomogeneousnessof
andz, w ithnomeantranslationa l
ncomponentofthemotioniswholly
, a n d s o f ar a s o ur e s ta b li s hm e nt o f ( 3 4
ofanymagnitude greatorsmallre lati e ly
oftheturbulent motion.Itisa fundamentalformulainthetheoryoftheturbulentmotionof water
ndIhadfoundit inendea ouringtotreat
erProf.J amesThomson stheoryofthe
niformR egimeinR i ersandotherOpen
a ouringtoad anceasteptow ardsthe law
armotionat differentdepths Iwas
seemingpossibilityofa lawofpropagation
inanelasticso lid w hichconstitutesthe
ommunication onthesupposition
tionU0 v 0 w 0isisotropic andthat
i de d by t he g re at es t v a lu e of f y t , i s in fi ni te ly
hesmallestva luesofm n q , inthe
, ( 7 , ( 8 f o r th e t ur b ul e nt m o ti o n.
eethat if theturbulenttmotionremained
casatthebeginning f y t w ouldremain
a luef y . Tof indw hetherthe
mainisotropict and if itdoesnot to
ow of itsde iationf romisotropy le tus
/ dt b y ( 2 a nd ( 3 ) , a s f ol lo ws :- i rs t b y m ul ti pl yi ng ( 3 1 b y v , a n d ( 3 ) b y u a nd a dd in g w e f in d
) d ( U v ) d f y t
W - f y t + ) } 3
y
15 1878.
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M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 1 5
s andremar ingthatthefirst termofthe
by ( 3 0 , andthef irsttermof thesecond
w e fi nd w it h V 2 a s in ~ ~ 8 9 t o de no te t he
e locityof theturbulentmotion
) } - 2 t . .. .. .. .. .. .( 3 6 ,
( d ( v ) dp d ( 3 7
~ + w- + V d + y ( 3 7 .
dy
. 3 8 ,
w ouldbe if fw erezero . We f ind
d ( 3 9 ,
3 7 ,
d + u . . . .. . .. . .. . .. . .. . .( 4 0 .
de itherthesupposit ionof init ia l
motion oroftheinfinitesimalnessof
duceanduseboth suppositions.
ationof ( 3 9 , w enow useour
y t , di idedby thegreatestv a lueof
t el y s ma l l in c o mp a ri s on w i th m n q , w h ic h a s
e s
(
es
d d - d
+ U d -2 .. .. .. 4 2 .
i so tr op y w eh a e
0 A
d V * . 4 )
+ U o -+ W o - 7 V 02 V . 4 )
d dy
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R
ypartsforthelast twotermsofthe
us i ng ( 1 , w e fi n d
d u o dw d -
} ~ 0 = -x z a ( u + d - V o2
d d d y
2 o
) a n d ( 4 2 ,
d2 \ d d
o ( + + y V . -2 .. ( 4 4 .
u ri e r e p a ns i on ( 7 f o r v o w e f in d
f g c os ( a + e si n( n y + f c os ( q z + g
q ) 7 m2 + 2 + q 2 . .. .. .. .. 4 5 .
uf i e s & a m p c . d ro p pe d ,
- - m2 n2 + 2 .. . 4 6 * ,
n 2 + q 2
nSY M2+ q 2 / 2
-2 â € ” q . ... .. .
o = 8M 2+ 2 2 * 2 .( 4 7 .
a erageuniformityof theconstituentterms
omogeneousness( ~ ~ 7 8 9 , thesecond
u al t o â € ” 8 E E2 S / 2 a nd t he re fo re ( ~ 9
im ilarlyweseethatthesecondmemberof
2 R2 . He nc e f in al ly b y ( 4 4 ,
;
w i th ~ R 2 f o r V 2 o n a c co u nt o f i so t ro p y
} = - 2Rs dy t . 9
d( U V ) O - 9. t= 0 .. .. .. .. . 4 9 .
o
opy whichthise uationshowst is
o f thesmallnessofdf / dy and( 27 doesnot
ds i n v i r t u e of ( 3 0 . H e nc e ( 4 9 i s n ot
ues( v a luesfort= 0 o f thetw omembers
nitesimalde iationfrom2R2inthe
hana eragingthroughy -spacessosmallastoco erno
y t , butinf inite ly large inproportionton-1 isimplied.
W . T . A u g. 1 0 1 8 8 9 .
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M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 1 7
member consideringthesmallnessof
fora llva luesof t unlesssofarasthe
referredtoattheendof~ 1 maybelost
rt icesv it ia ting( 27 ,
2R2 df y t ( 5 0 .
emberf romthise uation by ( 3 4 ,
51 .
yremar ableresultthatlaminardisturbance
othew ell- now nmodeofw a esof
uselasticsolid andthatthev elocity
/ , o rabout' 47of thea eragev elocityof
fluid. Thismightseemtogofar
ity tothevorte theoryof the luminiferous
edoubtfulpro isoattheendof~ 20.
tionof themediumbeastable
fv orte - ringsthesuggestedv it ia tion
annotoccur. F ore ample le titbesuch
w herethesmallw hiteandblac circles
g
" ' ; : - /
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ P 3 , H
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R
therings: thewhitewheretherotation
blac w here it isinthesamedirectionas
ofawatchplacedon thediagramfacing
aginef irsteachv orte - ringtobe ina
nedw ithinarigidrectangularbo of
edby thefinelinescrossingone
ughoutthediagram andtheother
per atanydistanceasunderw eli eto
olumeoftherotationallymo ing
utingtheringtobegi en there is
ape anddiametralmagnitude inwhich
erthatthemotionmaybesteady . Let
llspacew ithsuchrectangularbo esof
oneanotheroppositelyinthe manner
ulnowthe rigidityofthesidesof
ontinuesunchangedlysteady.B utis
igidpartit ionsaredoneaw aywith No
nthatit is. If it is laminarw a es such
uldbepropagatedthroughit andtheve locity
R/ V 2/ if thesidesof the idealbo es
dplanesof theringsares uare( w hich
e w 2 , a n d if t h e di s ta n ce b e tw e en t h e s u a re
stheproperratio tothesideof thes uare
a e U 2 = a e w2.
r e a m pl e p l an e w a e s o r l am i na r
erpendiculartothe undisturbedplanesof
nfigurationof thev orticesinthe
odofaharmonicstanding v ibration
sincy ( w hichismoreeasily il lustrateddiagrammatica lly thanaw a eorsuccessionofw a es , isi l lustratedin
hef luidoneachsideofy= 0. Theupper
entsthe stateofaffairswhent= 0
2 t o . B u t i t mu s t no t b e o e r l o o e d t h at
dependsontheunpro edassumptionthatthe
tisstable.
btful so farasIcanj udgeafter
tionfromtimetotimeduring theselast
theconfigurationrepresentedinfig.1 or
angement isstablewhentherigidity
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M O T I O N T H RO U G H A T U R B U L E NT L I Q U I D 3 1 9
passim.
a n s an a e r ag e o f th e k i n d d e sc r ib e d in t h e fo o tn o te o n ( 4 6 ;
ebeinge panded
harebeingcontracted.
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R
singeach ringseparatelyisannulled
siblethat therigidityoftwo three
maybeannulledwithoutv itiatingthe
metricmotion butthatifit be
eofspace fora llthepartit ions the
able andtheringsshuf fle themsel es
re lati eposit ions w itha eragehomogeneousness l i e theult imatemoleculesofahomogeneousli uid.
erthesecondit ions the" v it ia tingrearrangement re ferredtoattheendof~ 20canbee pectednot
eperiodofaw a eorv ibration. To
meterofeachringtobev erysmallin
gedistancesf romneighbours sothatthe
srathertothe moleculesofagasthan
ouldnothe lpustoescapethev it ia ting
uldbeanalogoustothatin estigatedby
ek inetictheoryof thev iscosityofgases.
mt inconclusion thatthemostfa ourable
hepropagationof laminarw a esthrougha
iscidli uidistheScottishverdicto fnot
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2 1 )
O N O F V I S C O U S F L U I D B E T W E E N TW O
ca l M lf a ga i n e V o l . x x I . A u gu s t 1 8 87 p p . 18 8 -1 9 6
etheR oya lSocietyofEdinburgh J uly15 1887.
tionofthefirst ofthisseriesof
yof EdinburghinApril andits
phica lMaga ine inMayandJ une , the
steadymotionofav iscousfluidhas
ctfortheA damsPri eof theUni ersity
epresentcommunication( ~ 27-40
etwocasesspeciallyreferredtoby the
ouncement andpreparesthew ay forthe
simpleby apreliminarylayingdown
a t i on s ( 7 t o ( 1 2 b e lo w o f t he f u nd a me n ta l
av iscousf luidk eptmo ingbygra ity
boundariesinclinedtothehori onat
nw ithanymotionde iatinginf inite ly little
dymotionwhichwouldbetheuni ue
tionifthe v iscosityweresufficiently
almostcertainindeed thatanalysis
8and3 9w illdemonstratethatthesteady
iscosity how e ersmall andthatthe
ntedoutbySto esforty-fouryearsago
tigatede perimentally f i eorsi years
ds istobee pla inedby lim itso f
erandnarrowerthesmalleristhe
neof theboundingplanes para lle lto
al motion andO Yperpendicularto
1 8 87 p . 1 42 .
p. 166se . Thenumberingof thesectionsiscontinuous
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R E C TI L I NE A L MO T I O N O F V I S CO U S F L U I D 3 2
t o ( 1 a n d ( 2 , a n d in t h em s u pp o se
h in f in i te l y sm al l : ( 1 i s u nc h an g ed ( 2 , w i th U
b e co me
( d =
v C Y. .. 7 ,
) - ~ - d
dp
] V 2
d p
] = V 2 w- _ " ' ( 9 ;
sin I/ L. . .. . . .. . . .. . . .. . . .. . . .. 10 ,
denotes insteadofasbeforethepressuresimply
sI. y .
beafunctionofyandtdeterminedby
us ( 1 a nd ( 7 , ( 8 , ( 9 a re fo ur e u at io ns wh ic h
daryconditions determinethefour
v , w p i nt er ms of x , y z , t .
im inateuandw by ta ingd/ d ,
( 8 , ( 9 , a n d ad di ng . Th us w ef in d i n v i r t ue
. .. . .. . . ( 1 1 .
v /
uationsforthedeterminationofv andp.
m wefind
d2 _
c b 2- y2 ] = V 4 ... 12 ,
h w ithproperinit ia landboundarycondit ions determinestheoneun now n v . Whenv isthusfound
9 de ter mi ne p u an dw.
acticallyimportantcase ispresented
f theboundingplanestobek ept
thatis F and~ of ( 6 tobeperiodic
m pl e t a e
. . .. . .. . .. . .. . .. . .. . . 1 ) .
csolutionof( 4 is
b - y / o / 2 / /
/ - _ e -b -2 - co s ( o t - y 1 4 .
2~ 2 1- 2
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R E C TI L I NE A L MO T I O N O F V I S CO U S F L U I D 3 2 5
becomesreducedto
V 2
, ( 1 6 , ( 1 7 b ec ome
,
f l * d . ( 2 ) ,
2 4 ,
( 2 5 .
- d -. . .. . .. . .. . .. . .. . . 2 5 .
at e u at io ns ( 2 2 - 2 5 i mp ly ( 1 a nd
determinethefourquantit iesu v , w p.
noccasiona lly touse( 1 . We proceed
nof theproblembeforeus consisting
, w p s a ti s fy i ng ( 2 2 - 2 5 f o r a ll v a l ue s o f
n d th e f ol l ow i ng i n it i al a n d bo u nd a ry c o nd i ti o ns : wh e n t = 0 : u v , w t o b ea r bi t ra r y fu n ct i on s ( 2 6 ;
e ct on ly to ( 1 . .
= 0 f or y = 0 an da ll v a lu es of x , z , t .. . 2 7 .
w= 0 for y= b , ( )
a p a rt i cu l ar s o lu t io n u v , w p w h ic h
ditions( 26 , irrespecti e lyof the
7 , e c e pt a s f ol l ow s : = 0 w h en t = a n d = 0
andy= b. . .. . . .. . . .. )
a rt i cu l ar s o lu t io n u b b p s a ti s fy i ng t h e
darye uations: u= 0 b=0 to=O , w hent=0. . . .. . . .. . . . 29 ;
0 t u+ w = 0 w he ny = 0. .
olutionwillthenbe
v , w = t o+ w . .. .. .. .. .. . 3 1 .
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R
w r e ma r t h at i f p w er e z e r o t h e co m pl e te
dbe
y t ;
trialfo ratype-so lutionw ith/ notz ero
n -n ml t y + q z ] . .. .. .. .. .. .. .. .. . 3 2 ;
andtdenotes/ -1. Substituting
w e f in d
m/ t 2 + q 2 T .. .. .. .. .. .. 3 3 ) ;
n2+q + q 2-nZmt+ ( m2/ ) s2t2 ........... 3 4 .
, a nd ( 3 2 , w ef in d
mPt y + q z ]
_ M8 t 2 + q 2 . .. . .. . .. . .. . .
q 2 ( 3 5 ;
y+ q z ]
+ ] . .. .. .. .. 3 6 .
q 2 2
a n d pu t ti n g
( n -m n t y + q z ] . .. .. .. .. .. .. .. .. .( 3 7 ,
2/ m T
- 2 + q 2 q -
+ ( n -t + q ] - 2 + ( n _t 2+ q 2 2 3 8
e s W .
nd w w e fi nd u b y( 1 , a s fo ll ow s: n - m ft v + q w 3 9 .
. . .. . .. . .. . .. . .. . . 3 9 .
ddingtype-so lutionsfor+ b and+ n
w earri eatacompleterea ltypeso lutionwith forv , thefo llow ing- inw hichK denotesan
- t- t m 2 + n 2+ q ' 2 - t m p t+ ( m 2 /m ) 3 2 t 2 CO S
t 2 + q 2 s in [ m + ( n m ) q z ]
+ n mp t+ ( m 2/ ) 3 2 t ] C O S. .
2 si n- -7 -r [ m - n + mR t y + q q z ] J . .. 4 0 .
: s in I
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R E C TI L I NE A L MO T I O N O F V I S CO U S F L U I D 3 2 7
0
nn y ( m + q z ) . .. .. .. .. 4 1 ,
e ma e
42 ;
ersummationfora llv a luesof if rom1
rintegrationwithreferenceto mand
rminedva luesofK , a f terthemannerof
n y ar b it r ar i ly a s si g ne d v a l u e to V t = o f o r e e r y
o y = o , y = b. .. .. .. .. .. . 4 ) . Z = - o z = + C - C
ntegrationappliedto( 40 gi es
x , y z ; a nd th en by ( 3 8 , ( 3 7 , ( 3 9 w ef in d
tev aluesofwandu.
r bi t ra r y in i ti a l v a l u e w 0 t o t he z c o mp o ne n t of v e l oc i ty f o r e e r y v a l ue o f x , y z , a d d to t h e
, w h ic h w e ha e n o wf o un d a p a rt i cu l ar s o lu t io n
) f ul fi ll in g th e fo ll ow in g co nd it io ns : = 0 f or a ll v a l u es o f t x , y z
0 a nd a ll v a l u es o f x , y z
5 a n d ( 1 , b y r e ma r i n g th a t v ' = 0
p = O , a nd th er ef or e( 2 ) a nd ( 2 5 b ec om e
W ' . .. .. .. .. .. . ( 4 6 .
us ta swe sol e d( 2 1 , by ( 3 2 , ( 3 3 ) , ( 3 4 ; an dt hen
osatisfythe arbitraryinitialcondition
, ( 4 1 , ( 4 2 , w e ac hi e e t he d et er mi na ti on
w edeterminethecorrespondingu , ipsofacto
y puttingtogetherourtw osolutions w e
, w = w + w . .. .. .. .. .. . 4 7
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R
w ithout( 27 , inansw ertothef irstre uisit ion
to f i nd u b [ 0 i n a ns w er t o t he s e co n d
.
f irstf indingarea l( simpleharmonic
, ( 2 2 , ( 2 ) , ( 2 5 , f ul fi ll in g th e co nd it io n
n wt
ncotw heny=
n co t
snt ( 4 8 ,
s in c ot w he n y= b
ot
D E F , X 3 , 3 , S ) E , a re tw el e ar bi tr ar y
) . T h en b y t a i n g j d w f o o f e a c h of t h es e
rier w eso l etheproblemofdetermining
ughoutthef luid bygi ingtoe ery
imatelyplaneboundariesaninfinitesimal
hofthethree componentsisanarbitrary
t . L as t ly b y t a i n g th e se f u nc t io n s ea c h = 0
andeache ua ltominusthev a lueof
y po i nt o f e ac h b ou n da r y w e f in d t he u b t o o f
o fourproblemof~ 3 2isthencompletedby
Todoa llthisisamereroutineaf teranimaginary
dasfollows.
a s su me
q z ) J
) { H ey / m 2+ q 2 + K e -y / m 2+ q 2
[ y V ( W ' 2+ q 2 dye V ( , 2 + q 2 [ L f y M ( ]
d y/ .2 + q 2 [ f ( y + y M ( ] ] } . .. 49 ,
M a r e a r bi t ra r y co n st a nt s a nd f F a n y tw o
d - ( m 2+ q 2 . .. .. .. .. 5 0 .
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R E C T1 L IN E AL M O T I O N O F V I S CO U S F L U I D 3 2 9
ut 3 1/ LU = y andm2+ q 2+ /= ........( 51 ,
( X + V r y - . .. .. . .. . .. . .. . .. . . ( 5 2 ;
sc e nd i ng p o we r s of ( X + t y y , g i e s t wo
ichwemaycon eniently ta eforourf
= - 2( x + y y 3 7 -4 ( X + t y 6 7 -6 ( + v y 9 + & a mp
. 2 + 6. 5. . 2 9. 8. 6. 5. . 2
( X y y 7 7 6 ( X + t y) l + & a mp c .
+ 7 . 6. 4 . 1 0 .9 . 7. 6 .4 . . . .. . .. . . 5 ) .
sentia lly con ergentfora llv a luesofy .
easo lutioncontinuousf romy= 0toy= b
nstantswecangi eanyprescribed
/ d y f o r y = 0 a nd y = b . T hi s d on e f i nd p
; andthenintegrate( 25 forw inan
eriesofascendingpow ersofX + tyy w hich
butneednotbew rittendownatpresent
ll o ws : w = 0 t W t + m + q z ) . . .. . .. . .. . .. . .. . .. . ( 5 4 ;
+ K 2 ( X + t yy + L ( X + t yY
+ Pey ( n 2+ q 2 + Q e -y m2+ q 2 " ( 55
wof reshconstants duetothe integration
gi etoW anyprescribedva luesfor
y b y ( 1 , w i t h( 4 9 , w e h a e
m + q z ) I
- . â € ” ( 5 6 .
q . . . . .. . .. . .. . .. . 5 6 .
s an ts H K , L M P Q , c le ar ly a ll ow
e d v a l ue s t o ea c h of 6 V i W , f o r y= O
ompletionof therea li edproblemw ith
tions asdescribedin~ 3 7 becomesa
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R
e ( u v , w s o lu t io n o f ~ 3 4 c o m e s
asymptotica llyastimead ances asw e
3 4 , a nd ( 3 8 . H enc et he ( u b b O ) o f ~ 3 7 whi ch
att= 0 comesasymptotically toz ero
We concludethatthesteadymotion
W I N G D O W N A N I N CL I NE D P LA N E B E D .
ca l M ag a i n e V o l . x x i . S e pt e mb e r 1 8 87 p p . 27 2 -2 7 8.
ndofthe twocasesreferredtoin
ecaseofwateronaninclinedplane
d p ar a ll e l pl a ne c o e r ( i c e f o r e a mp l e , .
directionsandgra itye erywhere
e asasub-case the icyco ermo ing
w ithit w hichisparticularly interesting .
ngentia lforceattheuppersurface it is
hesamecaseasthatofabroadopenri er
perfectlysmoothinclinedplanebed. It
eptw henthemotionissteadily laminar the
surface isk eptrigorouslyplane butnot
cf . LordRay leigh " O ntheQuestionof theStabil ityo f the
P h il . M ag . x x x i . 1 8 92 p p . 59 - 70 : S ci e nt i fi c P ap e rs i . p . 5 82
forcedmotiondeterminedin~ 40 thef luidiscapableofa
hofwhichthev elocityattheboundaryisnull. Inthe
, w ( o r wh at i s th e sa me W , V a nd d V / dy i n ~ 3 9 , a t ea ch
y theratiosof thesi constantsare in o l ed toaperiod
oducingforeachf reeperiodnormaltypesofmotionw hosesca le
ined: imaginaryva luesof thesef reeperiodsmightin o l e
mis appliedbyLordRayleighhimselftothe argument
owing. Thee perimenta lin estigationofO sborneR eyno lds
ppeartoshow howe erthatw ithincerta inlimitso f theve locity
w ispractica lly stable . O nesuggestion mentionedbyLord
that asthere isnocontinuoustransit ionfromsteadymotion
motionofperfectf luidw ithnov iscosity theactua lmotion
stymay in o l e instabil it iesinav ery thinlayera long
. Inanycase the in estigationsinthete tw ouldperhaps .
nggeneralremar s stillretainanapplicationas determininghowfarsteadylaminarmotion ifsomehowestablished issusceptibleto
foutside forces.
LordR ay le ighnow referstoapaperbyProf . W. MC . O rr
v ii. No . 3, 1907 e tendinghisow npre iouscriticism
w ithw hichheisdisposedtoagree. Inthatpaper how e er
pp. 72 74 99 w hichareheldtoma eitprobablethatthe
eawaye ponentia lly andthattheforcedoscil lat iondeterminedinthete tistheactualso lution it isurgedthatitdoesinfactsatisfy
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E R F L O W I N G D O W N A N I N CL I NE D B E D
byarigidco er w hile theopensurface
uiterigorouslyplanebygra ity and
entia lforce . B ut pro idedthebottom
ssofthedimplesand littleroundhollows
ce producedby turbulence( w henthe
seemstopro ethatthemotionmustbe
t wouldbeifthe uppersurfacewere
andfreefromtangentialforce.
bedin~ 3 1ha ingbeendisposedof
ow ta ethe includingcase describedinthe
1 forw hichw eha e assteadyso lution
.. . .. . . ( 57 ,
b ot to m up wa rd s. T hu s ( 7 , ( 8 , ( 9 , ( 1 1 ,
dp
( 3 - y U d .. .. .. ( 5 8 ,
d .
dp
= 2 V d .. .. .. 5 9 ,
-.dy.
d p ( 6
,
d .
V 2 p. .. 6 1 ,
2 -
) = V v . . .. .. .. .. .. ( 6 2 .
ow anysuchsimplepartia lsolutionas
5 3 6 f o r th e s ub - ca s e th e re d e al t w it h a n d we
irtually inclusi e in estigationspecif ied
in ~ 3 8 a ss um e
+ q z ) V . . .. .. .. .. .. .. .. .. .. . ( 6 ) .
tacit lyassumed. Therea lsodiscussionsofproblemsof thistypeby
ethodaregi eninpp. 122-1 8 w ithanaccountofpre ious
a p re i ou s pa rt ( l oc . ci t. N o. 2 ~ ~ 3 A 5 8 P ro f. O r r g i e s
Rayleigh sconclusionthatin theabsenceofv iscositythe
table inthesensethatthisstabilityw oulde istonly
nces cf . supra ~ 27.
Liou il leanaly sis( F ourier Theoriede laC haleur Sturm
i l le s J o u rn a l fo r t he y ea r 1 8 6 a n d Lo r d Ra y le i gh s T h eo r y
o l . I. s h ow s h ow t o e p r es s a n ar b it r ar y f un c ti o n of x , y z b y
lu ti on s of ~ ~ 3 7 3 9 a b o e a nd ~ 4 ( 6 ) , ( 6 7 , ( 7 0 h er e
etherforourpresentcaseor formersub-case thefulfilment
, ( 2 7 , w i t h ou t u si n g th e m et h od o f ~ ~ 3 4 3 5 3 6 .
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tm and V - -m2-q 2...... 64 ;
herefore
d 2~ ? )
+ t [ a + m ( M yC â € ” y2 ] } 2 + tI a( M 2 + q 2 2
Y / ) ~ ~ ~ ~ dy 2
~ y _ C y2 ] ( M + q 2 - to ml 61 0 .. .~ ( 6 5 ,
2 + ( h + I c y+ l ys V = 0 .. . 6 6 .
me
2 + C o + C 4 o4 + & a mp C .. . .. . .. . . 6 7 ;
e r o th e c oe f fi c ie n t of y i i n ( 6 6 w e f in d
( i + 2 ( i + 1 , U C i+ 4 + ( i + 2 ( i + 1 e cif2
l [ ] i ( i - .1 g + f h C i+ k c C i- 1+ l Ci -2 = 0 .. . 6 8 .
e l y i = 0 i = 1 i = 2 . .. a n d re m em b er i ng
s u ff i i s z e r o w e fi n d
e c 2 + h o 0 = 0
2 . ec + 2 . 1 .f c2 ~ h c + k 0 = 0
e o4 ~ 3 . 2 .f c + ( 2 .1 .g + h c + k c + 1 00 = 0
5 . 4 .e c5 + 4 . 3 . f C4 + ( 3 . 2 . g + h C + 1 0 02 ~ l C I = 0
a m p c . & a m p c .. .. .. .. .. 6 9 .
eninorder gi esuccessi ely04 05 06 ... each
c ti o n of c I c 1 0 2 C ; a n d by u s in g i n ( 6 7
ined w ef ind
Ci Ai y + C 2& a mp ( y + c A ( Y .. ... . 7 0 ,
c . a re f ou r a rb i tr a ry c o ns t an t s a n d S , , ~ ,
ho llydeterminate e pressedinaseriesof
hchby ( 68 w eseetobecon ergentfor
spbez ero . Theessentia lcon ergencyof
a s i n ~ 3 9 f o r th e c as e o f no g r a i t y t h at t h e
v = 0 w = 0 i s s ta b le h o we e r s ma l l be E
o.
essthecon ergence. When/ w is
ergenceformany terms butult imate
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E R F L O W I N G DO W N A N I NC L IN E D B E D 3 3 3
t h e di f fe r en t ia l e u a ti o n ( 6 5 , o r
ducedf romthe4thtothe2ndorder andmaybe
m 7
+ m ( y - c . * . . .. .. .. 7 1 .
o-dimensionalmotion( q = 0 , agreeswith
e pressedinthe laste uationofhis
orInstabilityofcertain F luidMotions
. F eb. 12 1880 . The integra l butnow
nstants( C 0 cl , isst i l lg i eninascending
n d ( 6 8 , w h ic h w i th L = 0 a n d th e t hu s si m pl i fi e d v a l ue s o f e f g p u t in p l ac e o f th e se l e tt e rs b e co m es
1 oc i+ 2 + ( i + 1 i m/ ci + ]
h C i + k i - + c i -2 = 0 .. .. .. 7 2 .
es o f i t hi s g i e s
m c ci = 0 . . . .. . .. . .. . .. . 7 ) ,
ly e ceptinthecaseofoneparticular
. .. . . .. 74 ,
mallerrooto f thee uation
.. . . .. . . .. . . .. . . .. . 75 .
otcon ergenceforva luesofye ceeding
andthustheproofof stability islost.
e uation simplif iedin( 71 forthe
maynodoubtbetreatedmoreappropriately
onofstabil ityorinstabil ity byw ritingit
e no t in g t he t wo r o ot s o f ( 7 5 ] ,
( I - ) } 2 .. .. .. 7 6 ,
- y. _ - y.
alconsiderationoftheinfinities at
. O n e w ay o f do i ng t h is w h ic h I m er e ly
ddonotfo llow outforw antof t ime isto
' -y 2 + o ( ~ - y + & a mp c . ,
- y 2 + C / ( ' - y 3 + & a mp c. ... 77 ,
o ar b it r ar y c on s ta n ts a n d C2 c , . . . c 2 c / . .
edsoas tosatisfythedifferential
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R
easilydone andw hendoneshow sthat
fora llv a luesofy lessthan4 ande ceeding
f thisindeta ilw ouldbevery interesting
llmathematicaltreatmentofthe
sstream- lines( cur esofsines throughout
w o" cat s-eye borders( correspondingto
w hichIproposedinashortcommunicationto
A ssociationatSw ansea in1880 , " O na
R ay le igh sso lutionforWa esina
. It istoberemar edthatthisdisturbing
mingproofofstabilitycontainedin Lord
( 5 6 , ( 5 7 , ( 5 8 t .
a n d in t er p re t in g t he r e su l t in c o nn e i o n
t h at
hw eha efoundconsistso faw a edisturbancetra e ll inginany ( x , z ) direction o fw hichthe
nthex -directionis-o/m.
4 ) o f ( 7 5 a re v a l ue s of y at p la ce s wh er e
sturbedlaminarf low ise ua ltothexv e locityof thew a e-disturbance.
ounding-planestobeplastic andforce
othofthem soastoproducean
orrugation accordingtotheformula
z ) , thissurface-actionw illcausethroughoutthe
finitesimalwa e-motionifo/misnot
foranyplaneof thefluidbetw eenits
nitycorrespondingtoy= 4 ory= 4 w ill
o /m ise ua ltotheva lueofU forsomeone
oplanesof thef luid andthetrue
he" cat s-eyepattern o f stream- lines and
atthisplaneortheseplanes.
ctispublishedinNatureforNo ember11 1880 andin
olumeReportfortheyear.In thisabstractcancelthe
w ithreferencetoacertainsteadymotiondescribedinit
t se e n e t f o o t no t e .
sinreply ( loc. cit. supra thatw henX ciscomple there
sothattheargumentdoesnotfa il regardedasonefor
a luesofw thoughitmaynotcompletelyensurestabil ity .
ectinProc. Lond. Math. Soc. x x v II. 1895 Scientif icPapers
erence inthete t andsupra p. 186 isthattherecannot
ases unlessthisbandofv orticeshasbeenestablished.
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E R F L O W I N G D O W N A N I N CL I NE D B E D
enmo ingw iththesteady
parallelboundaryplanes e pressed
h wo u ld b e a c on d it i on o f k i n e t ic e u i li b ri u m ( p r o e d
erthe influenceofgra ityandv iscosity and
scositybesuddenlyannulled. Thef luid
brium butisthee uil ibriumstable
letoneorboth bounding-surfacesbe
nyplaceandleft freetobecomeplane
esisofthissurface-operationis
) c os c ot co sm c os q z . .. .. . 7 8 ,
m q ) { c os ( o t- m )
c os q z . .. .. . 7 9 ,
safunctionofmandq w hichimplies
tionstra ellinginoppositex -directions
o ~ o o f ( c o /m t h e wa e - e l oc i ty . He n ce
disturbanceessentia lly in o l ese ll ipticw hirls.
ensteadylaminarmotionisthoroughly
tobrea upintoeddiesine eryplace on
t shoc orbumponeitherplastic plane
egreeofv iscosity asw eha eseen
onstable butthesmallerthev iscosity
fgsinI o rthegreatertheva lueofgsinI
thenarrow erarethe lim itso f this
beenledby purelymathematicalin estigationtoastateof motionagreeingperfectlywiththe
escriptionsofobser edresultsbyOsborne
March15 188 , pp. 955 956 : Thefactthatthesteadymotionbrea sdow nsuddenly
stateof instabilityfordisturbances
ause ittobrea dow n. B utthefact
willbrea downforalargedisturbance
erdisturbance showsthatthereisa
so longasthedisturbancesdonote ceed
fsurprisetometoseethesudden
ssprangintoe istence showinga
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E R F L O W I N G D O W N A N I N CL I NE D B E D 3 3 7
g ma esitcertainthatifw aterbegi en
eplanesboth atrest andifoneof the
nottoogradually setinmotion andk ept
emotionof thewaterw illbeatf irstturbulent
ofuniformshearingwill beapproached
timateannulmentofthe turbulence.
municationonthissub ecttoSectionA of
nManchester andtoha eitpublished
f thePhilosophicalMaga ine. C orrespondingquestionsmustbee aminedw ithreferencetothe
blem ofaninfinitelylong straight
ginw aterw ithinaninf inite ly longf i edtube.
1888A damsPri ew illbringout
nsonthissub ect.
artero fapoundpers uarefoot( ) istheresistancedueto
aterbetw eentw opara lle lplanes-o facentimetre( 9 ofa foot asunder w henoneof theplanesismo ingre lati e ly totheotherat
tres persecond if thew aterbeatthetemperature0~ C ent.
ca lculatedf romPoiseuil le sobser ationsonthef low of
is1- 4x 10-5ofagrammew eightpers uarecentimetre .
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T W O - D I M EN S IO N A L W A V E S P RO D U C E D
TINGDISTUR B A NC E .
. E di n . V o l . x x v . r e ad F e b . 1 1 9 04 p p . 18 5 -1 9 6
une 1904 pp. 609-620.
swaterinastraight canal infinitely
w ithvertica lsides. Letitbedisturbed
pressureonthesurface uniformin
atotheplanesides andle ftto itse lf
re . It isre uiredto f indthedisplacementandv elocityofe eryparticleof thewateratany future
onw illbefully specif iedbyagi en
city andagi ennormalcomponentdisplacement ate erypo into f thesurface.
tatadistancehabo etheundisturbed
para lle lto the lengthof thecana l andO Z
etf bethedisplacement-components
componentsofanyparticleof thew ater
onis( x , z ) . Wesupposethedisturbance
wemeanthatthe changeofdistance
ofwateris infinitelysmallincomparison
tance andthe line j o iningthem
directionwhichareinfinitelysmalli n
n.W aterbeingassumedincompressible
ion startedprimarilyfromrestbypressure
isessentiallyirrotational.Hence
d
; = d ( Q , , , 0 t ; = ; = . .. 1 ;
t o r h a s w e ma y wr i te i t f or b r e i t y wh e n co n e n ie n t i s a f un c ti o n of t h e v a r i ab l es w hi c h ma y b ec a ll e d th e
andb( x , z , t isw hatiscommonlyca lled
blemstreatedinthefo llowinggroupofpapersandthe irhistory
ss' O nDeep-WaterWa es byProf. H. Lamb Proc.
. ( 1904 , pp. 3 71-400.
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R
C o sg 2 + ( P - z ) s i n 4 p2 .. . ( 7
p2
. -- -. 8 ,
e 4p2. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . ( 8 ,
2 a n d 0 = t a n- l / +
changeswhenx passesthroughz ero.
a n d de n ot i ng b y { R D t h e di f fe r en c e
idedby2t w eha eanotherso lutionofour
entf rom( 6 asfo llow s
{ R D 4 + x 4 + ) . .. .. .. .. .. .( 9 ,
+ z ) c si os e -/ - 4t 2( l o
2...........................11 .
m f ig. 1 representsfort= 0theso lutions
, w ithz = 1forcon enience inthe
ichw ef indby ta ingt= 0in( 7 x 2/ 2
t h e mi n us s i gn i n ( 1 0 b e in g o mi t te d f or
x 2 + 2 + ] , [ / x 2 + 2 - _ Z ]
2 4 2 + ) . x ( 1 2
cticalinterpretationof oursolutions
containfullspecif icationsof tw odistinct
neachofw hichq bmaybeta enasa
orasav e locity -potentia l o rasahori ontal
orv elocity orasav erticaldisplacementcomponentorv elocity.Thusweha ereallypreparationforsi
o fw hichw esha llchooseone -= V / 2x ( 7 ,
n.
1 f o r th e wa t er - su r fa c e l e t th e t wo c u r e s o f
splacements ( 12 of thew ater-surface
i enf irst inP roc. R . S. E. J an. 1887 andPhil. M lag.
p. 3 07: ithasnotbeenreprintedhere .
s( loc. cit. thatthe init ia ldisturbance isnotentire ly
sf d doesnotcon erge. Seea lso infra ~ 101.
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A T E R TW O - D IM E NS I O N A L W A V E S 3 4 1
-̂ ^ - ^ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ I
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . - q
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R
re erywhereatrest.Thedisplacements
t a r e e p r es s ed i n r ea l s ym b ol s b y ( 7 ( 1 0
andby ( 8 , ( 11 w ithafactor/ 2introduced e itherofw hichmaybechosenaccordingtocon enience
asthusbeencalculatedf rom(8 , w ith
f or si v a lu es of t ' 5 1 1 5 2 2 5 an d5 an df or
rofv aluesofx torepresenttheresults
igs. 2and3. E ceptforthetimet= 5
entlyallthemost interestingcharacteristics
atthecorrespondingtime.Thecur e
ptibly lea ethez ero lineatdistances
t if w e c ou l d se e i t i t w ou l d sh o wu s t wo a n d a ha l f
ery interestingcharacterist ics shownin
7be low byw hichw eseethatse era l
lesofordinatesmagnifiedfromone to
nemill ion andtotenthousandmill ion
ibitthemgraphically.
e s f or t = 0 a n d t = ; w e se e t ha t a t
istancesfromthemiddleof the
= 19 andfa llsatlessdistances. A nd
= 0 r e ma i ns a c r es t ( o r p o s it i e m a i m um
foret= 1 w henitbeginstobeaho llow .
istencebeside itandbeginstotra e l
u r e t = 1 w e s ee t h is c r es t t r a e l le d
7 f romthemiddlew here itcameintobeing
s i t h s e e nt h cu r e s ( f i gs . 1 2 w e
9 4 8 6 5 2 2 a t t he t im es 1 X , 2 2 ,
ghtwardsonourdiagramshasits
dua ldowntotheundisturbedle e lat
pe ismuchsteeper andendsatthe
middleofthedisturbance attimes
sometime w hichmustbeverysoon
o llow beginstotra e lrightw ardsfromthe
eshcrest shedofffromthemiddle.At
g ot a s f ar a s x = ' 9 a t t = 2 ~ , a n d 5 r e sp e ct i e l y i t h a s re a ch e d x = 1 7 5 a n d x = 6 7 . Lo o i n g in
nsionofourcur esle f tw ardsfromthe
w ef indane actcounterparto fw hatw e
ontheright. Thusw eseeaninit ia l
calonthetw osidesofacon e crest o f
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A T E R TW O - D IM E NS I O N A L W A V E S 3 4
eundisturbedle e l sin inginthemiddle
s. Thecrestbecomeslessandless
w n t o he i gh t 1 1 w he n i t be c om e s co n ca e
arw a e-crestsareshedof fonthetwo
y f romitrightwardsandlef twardsw ith
eachremainingfore ercon e . Thusw e
oendlessprocessionsofwa estra e ll ing
ons originatingasinfinitesimalwa eletsshedoffonthe twosidesofthemiddleline. Eachcrestand
increasingve locity . Eachw a e- length
romhollow tohollow becomeslongerand
tw ards a llthisaccordingto law fully
f ~ 3 a bo e .
fnumberspromisedin~ 5abo e
msand magnitudesofthetwoanda
e n x = 0 a n d x = 2 w h ic h t he s p ac e -c u r e f o r
f a il s t o sh o w.
l g= 4 t = 5 - = , s in s( + 0 e P2 .
. 4 C o l. 5 C o l. 6 C o l. 7
o 0
e X e p 2 m '
|
42 1 0000 10-10' 1 57+11-0 196
4 14 0 ' 3 4 4 ' 1 47 8 , , 0 71 7
0
09 - 7541 , 1778-10-10 1891
- ' 8 9 97 , - 0 66 , , , - 8 82
9 - ' 0 0 2 , 3 6 2 , , , 0 016
1 3 7 0 - 8 99 7 , 1 -0 94 + 1 0 -1 0 1- 6 2
0
3 3 8 - ' 5 45 1 , 4 6 6 - 10 -1 0 3 - 24
8 1 26 2 - 2 4 1 , 1 0 - 9 , 3 1 -84
9 ' 7 5 9 1 0 -5 * 0 2 9 6 + 1 0 - 5 02 2 7
1- 09 9 - 89 62 , 2 95 8 , , , 3 1 5 2
00 7 - 68 1 , 5 -7 9 , , , , 4 42 4
92 87 - 49 2 , , 4 5 6 , , 2 6 7
2 0
16 - -68 2 , 212-5 10-144-6
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R
.
- = 2 si n( + 2 +
4 C o l. C o l. 6 C o l. 7
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A T E R TW O - D IM E NS I O N A L W A V E S 3 4 5
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E S O N W A T ER [ 3 5
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A T E R TW O - D IM E NS I O N A L W A V E S
show ninthepre ioustable forthe
tute- ' ; -w eseethat thef irstfactor
w ly f romx= 0tox = oo thesecondfactor
tw een+ 1and-1w ithincreasingdistances
f romz erotozeroasx increases. Thethird
sesgradually f rome- t / hatx =0 to1 at
t h e th i rd f a ct o r is ' 9 9 w hi c h is s o n ea r ly
ofamplitude is fora llgreaterva lues
enby thef irstfactora lone w hichdiminishes
t o 0 at x = o .
gi e n f i gs . 1 2 3 , m a yb e c al l ed
neachof themabscissasrepresentdistancef rom
ance. F ig. 4isatime-cur e( abscissas
= 2h. Itrepresentsav erygradua lrise
f o ll o we d b ya f a ll t o a m in i mu m at t = 2 8 a n d
ns w ithsmallerandsmallerma imum
ions andshorterandshortertimesf rom
t= o . Thesamew ordsw itha lteredf igures
waterle e latany f i edposit ionfarther
ncethan x = 2.Thefollowingtable
100h a llthetimesofz ero lessthan71h
epressionsattheintermediatetimeswhen
5of~ 7 hasitsma imumandminimum
eseele ationsanddepressionsareveryappro imately thegreatestinthe inter a lsbetweenthezeros because
~ 7 v ariesbutslow ly asshow ninthef irst
e.
forthef reesurface in o l esno
aparticularcaseofthe general
ssumptiong= 4 merelymeansthatourunit
acefallenthroughinourunit oftime.
sof~ 3 maybegeometricallye plained
i n g 0 o u r or i gi n o f co - or d in a te s a t a h ei g ht h
anddefiningpasthedistanceofany
t. When asin~ ~ 5-9 w eareonly
the freesurface( thatistosay when
ha t i f x i s a l a rg e m ul t ip l e of z , p x . S e e fo r
f thetableof~ 9. A ndifw eareconcerned
urface w estil lha ep x , if x isa
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R
p = 1 00 -0 05 .h O = t ar l 1 = 4 51 8 .
TimesofZ ero
andof A ppro imate
i m um A p p ro i m at e Ma i m um
a t io n s an d F P M a i m um E l e a t io n s an d
ons Ele ationand Depressions
Depression
+ ~ 1 4 0 - 7 7 18 5 0 -9 0 + 1 09 1
2 42 0
60 -7478 5 90 - -1058
5- 4 0
~ 1 1 7 * 7 2 4 7 56 7 4 + ~ 1 0 2 5
0 0
1 2 77 6 7 02 5 9 4 5 - ~ 0 9 9
0 75 0
12 7 6 80 6 6 2 0 + 09 6 2
29 0. 8480 40-61 - .1199 * 6595 6451 - -09 3
0. 8219 44- 1 - -1162 * 6 92 6690 + 0904
6807 0
157 -6195 69-21 -0876
3 4 0
sw e
o imation
_gt__
X Z ) CO S 4- + V / x - Z ) S iy : 4 X 2.. 1
resent- f ( insteadof -5 asin% 5-9 ;
1 41 ,
m(1 w ithoutfartherrestrict i esupposit ions. B utifw esupposethatz isnegligibly smallin
andfartherthat
. t 2
. .. ..
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A T E R TW O - D IM E NS I O N A L W A V E S 3 4 9
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E S O N W A TE R [ 3 5
insteadof+ , isCauchy ssolution ; of
thetimehasad ancedsomuchasto
u i a l en t t o ( 1 5 , " l e m ou e m en t c ha n ge
ppro imation. TheremainderofhisNote
ges ischie f lyde otedtoverye laborateef forts
orthe largerva luesof t. Thisob ect
thee ponentia lfactorin( 8 o f~ 3
ripplingrestrict ionz l . 0w hichv it ia tes( 16
l. I . no te x v i . p. 1 9 .
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T A N D R EA R O F A F R E E PR O C E SS I O N O F
TER.
. Edin. J une20 1904 Phil. Mag. V o l. v III. O ct. 1904
cationissubstitutedforanother
whichwasreadbeforetheRoyalSociety
y7th 1887 becausetheresulto f that
fectandunsatisfactorybyomissionof
e ferredto in~ 10ofmypaperofF ebruary
nceforthto thelast-mentionedpaperas
e e s u pr a p 3 0 7 .
processionsproducedbysuperpositionofstaticinitiatingdisturbances ofthetypee pressedin
e graphica lly representedby f ig. 1 andleading
n~ 1- , 5-10. Theparticulartypeof
choose isthatchosenattheendof~ 4
butusefulmodif ication maynow w rite
gtX 2
c o / - e . .. .6 ) .
2 2. .. .. .. 1 7 .
z 2 + x 2 , a nd X = ta n- l( x / )
rdv erticalcomponentofthedisplacementofthefluid attimetfrom itsundisturbedpositionatpoint
chmaybeeitherinthefreesurfaceoranyw herebelow
7 , w e h a e f o r th e i ni t ia l h ei g ht o f t he
undisturbedle e l
( ) = .. ... .( 1 8 .
fo rir- tan- / - - sa esconsiderable labourand
ecia llywhen asinourca lculations z =1.
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R
e asinit ia tingdisturbance arow
ooofsuperposit ionsof ( 18 ; a lternate ly
andplacedate ua lsuccessi edistances
e
o -l i + i , o ...... 19 ,
x + iX , o . ... ... .. 1 9 ) ,
= ( x , 0 - ( x + , 0 . .. .. .. .. .. . 2 0 .
aspace-periodicfunction w ithX foritsperiod.
stitutedfor0 represents-t beingthe
abo eundisturbedle e latt imet in
ncerepresentedby ( 19 .
hate erfunctionberepresentedby
19 impliesthat
P ( x , 0 . .. .. .. .. .. .. .. .. . 2 1 ,
ace-periodicfunctionwithX forperiod.
sthat
- P , 0 . .. .. .. .. .. .. .. ( 2 2 ;
A ndw iththeactualfunction ( 18 , w hich
x , 0 , t he fa ct th at & l t b ( x , 0 = q ( - x , 0
- 0 . .. .. .. .. .. .. .. .. .. . ( 2 ) .
of thecharacterfig.5 symmetricaloneach
andminimumordinate . TheF ourier
, 0 , w he n su b e ct t o( 2 2 a nd ( 2 ) , g i e s
7r
c os -- + A c os 3 - + A c os 5 + .. .( 2 4 .
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N T A N D RE A R O F A F R EE P R O C E SS I O N 3 5
unctionsgeneratedbyadditionof
ore uidifferentarguments. Letf x ) be
periodicornon-periodic andlet
i X ) . .. .. .. .. .. .. .. .. . 2 5 ;
x ) = P ( x + X ) . . .. . .. . .. . .. .. . .. . .. 2 6 .
c e p a ns i on o f P ( x b e e p r es s ed a s
c os a + A c os 2 a + A c o s 3 a + w h er e ac = 2
s i n 2c a+ B 3 s in x + . .. X . .. .. . 2 7 .
er w eha ebyF ourier sana ly sis
.2 .r . . 2
c l O . 2 7w - C 00 . 27 r X
+ iX ) c os d f x ) cos
- 0 .2 7 rr
x + iX ) sin d f x ) sin2
- o X . . .. .. 2 9 .
, a sb y( 1 9 ) , ( 2 0 ,
0 -P X + X O ) . ......... 3 0 .
stoz ero reducestheA stoz erofore en
or o dd v a l ue s of j g i e s i n v i r tu e of ( 2 2 ,
, 0 cos l . ( 3 1 .
4 ( 6 , ( 1 2 , a bo e a nd a cc or di ng to th e
e
SI 2 ( P / x ) . ... p. . . ( 3 2 .
c e p an si on ( 2 4 o f P ( x , 0 , w e ha e
. 2w r 4 ~ S RS + C X V 2 co .2 wr
cos - X R d x - cs
Z + t X ) . . .. .. 3 3 ) .
2
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E S O N W A TE R [ 3 6
ast memberofthise uationfacilitates
negral. Insteadofcos inthe last
. 2 w ~ ~ ~ ~ ~ ~ 2 7 r . 27 -r : 2 77 - V 2 n
( 3 4 .
snodifferenceinthe summationf__ d ,
ppearsforthe samereasonthatthe
sa p pe a r be c au s e of ( 3 0 . T h us ( 3 3 ) b e co me s
n L
3 5 ;
+ t )
+ t X ) = t o
- an dt = â € ” z . ( 3 6 .
w emayomitthe instruction{R S because
sintheformula:thuswefind
2i r z 2 r 8V 2 _ 2
fd ae 2 2 C A
ismadeinv irtueofLaplace scelebrated
W
allowsusreadily toseehow neartoa
a ph o f P - x 0 f o r an y p ar t ic u la r v a l u e of
4 r
1 A 5/A = V 3 .e.
= 4 ; w e ha e
14 A /A = 0 24 95 A / A = - 0 3 4 7. .. 3 9 .
ut_ ofA ; andA about-LofA .
nusoidality butnotq uitenearenough
Tr y n e t X = 2 ; w e ha e
" = 0 0 18 67 A / A = - 00 10 78 .. . 4 0 . \ /
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N T A ND R E AR O F A F R EE P R O C E SS I O N 3 5 5
ndthofA1 andA about1~ x 10-6of
enoughappro imationforourpresent
bleinany ofourcalculations:A is
eptibleifincludedin ourcalculations
ofoursignif icantf igures : butitw ould
ourdiagrams.Henceforthweshall
yw iththef reesurface andta ez = h the
fcoordinatesabo etheundisturbedle e l
ceatany timetafterbeingleft
cedaccordingtoanyperiodicfunctionP( x )
s e as i n ( 2 7 ; t a e f i rs t f o r th e i ni t ia l
cement a simplesinusoidalform
. . .. . .. . .. . .. . .. . . 4 .
( 3 ) , a n d ( 4 a b o e l e t w ( z , x , t b e t he d o wn w ar d s v e r t ic a l co m po n en t o f di s pl a ce m en t . W e t hu s h a e a s t he
themotion
.. 42 ,
4 ) .
- c c o s t /g n . .. . .. . .. . .. 4 4 ,
l- nownlawoftwo-dimensionalperiodic
w ater. A ndformula( 44 w ithC e-m= A
h( 41 . Hencetheadditionofso lutions(44 ,
h A s uc c es s i e l y pu t e u a l to A , A 2 . . .
i t hc = 0 fo r th e A s a nd = - r f or t he B ' s g i e s
ertica lcomponent-displacementatdepthz -h
timet= 0thew aterw asatrestw ithits
g t o ( 2 7 . T h us w i th ( 3 8 , a n d ( 4 4 , w e
a n d ( 2 7 , a n d p u tt i ng m = 2 7 r/ X , w e se e
nduetoanyoneof theA sorB ' sinthe
ndlessinfinite rowofstandingwa es
e u a l to X / a n d ti m e- p er i od s e p r es s ed b y
4 5 .
..........
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R
riodic becausetheperiodsofthe
e ingin erse lyas/ , a renotcommensurable .
2 h a s pr o po s ed i n ~ 1 7 w h ic h a c co r di n g to
s f o r th e f re e s ur f ac e o n ly a l i tt l e mo r e th a n 1/ 1 00 0
aranapproachtosinuso ida lity thatinour
dthemotionas beingperiodic with
1 . Th is m a e s r = V / 7 wh en a s in ~ 5 w e
~ 10 simplif yournumerica lstatements
d h = 1 w hi c h ma e s t he w a e - le n gt h = 2 .
o f " f rontandrear " remar now
parallel straightstandingsinusoidal
startede erywhereo eraninf initeplaneof
er maybeidea lly reso l edintotwo
w a esofha lf the irhe ighttra e ll ingin
ua lv e locit ies2/ /7r.
gthew holesurfacewithstandingw a es
ati esideof the line( notshownin
, thatisthe le f tsideof0theoriginof
ea ethew aterplaneandmotionlessontheright
distancesonthe le f tsideof0 there
standingw a ese ui a lenttotw otra ins
o f w a e - l e ng t h 2 t r a e l li n g ri g ht w ar d s an d
//7r.Thesmoothwaterontheright
dedby therightwardprocession.
pro esthatthee tremeperceptible
ession(mar edR inf ig. 10below does
R onthe le ftsideof0 broadeningwith
therighto f0 perceptiblydisturbthe
lsopro esthatthesurfaceat0has
roughalltime.It farthershowsthat
sveryappro imately sinuso ida l w ith
throughaspaceOF ( f ig. 9 to theright
me andthat atanyparticulardistance
appro imationbecomesmoreandmore
ances. WhatIca llthef ronto f the
isthew a edisturbancebeyondthepo intF ,
stancerightwardsfrom0 wherethe
idalityo f shape andsimpleharmonic
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N T A ND R E AR O F A F R EE P R O C E SS I O N 3 5 7
nly j ustperceptiblyatfault. Wesha llf ind
esare asshowninf ig. 9 lessandlesshigh
atgreaterandgreaterdistancesfrom 0
butthatthew a e-he ightdoesnotat
bruptlytonothing.Thepropagational
ofthe disturbanceisinrealityinfinite
terasinfinitely incompressible.
efrontoftherightwardprocession
o llow ingitf rom0 issimplygi enby the
ev a luesofx , o f themotionduetoaninit ia l
ofsinusoidalfurrowsandridgesonthe
esentsastatic initialconfiguration
x , 0 , appro imately rea lisingthecondit ion
epresentsonthe samescaleofordinates
atthetime25r inthesubse uent
conf iguration w hich forany timet w e
d ef in ed as fo ll ow s: Q ( x , t = ( ( x , t - ( x + 1 t + ( x + 2 t - .. .a di nf .. .. 4 6 ,
d ef i ne d b y ( 1 7 , w i th z = 1 a n d g = 4 .
ata lldistancessofarle f tw ardf rom0
earoftheleftwardprocessionhasnot
particulart ime t a f terthebeginning
t o f ~ 1 c a lc u la t ed a c co r di n g to ~ 1 8 1 7
il lmere ly standingwa es idea lly
dandleftwardprocessions. LetI
offig.10 bethepointof theideally
otprecise lydef ined w herethe le f tward
rtime t becomessensiblyinfluenced
nIandRthe wholemotionistransitional
egularsinusoida lmotionP( x , t o f the
toregularsinuso idalmotionofw a eheight2P( x , t , f romR to0 andontoF o f f ig. 9 thebeginning
anceintherightwardprocession.Hence
wardprocessionfromthewholedisturbancedue totheinitialconfiguration weha eonlytosubtract
m Q ( x , t c al c ul at ed f or n eg at i e v a l ue s of x . T h us
holeofthe leftwardprocessionis
, t f or ne ga ti e v a lu es of x . .. .. . 4 7 .
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R
esurface thusfoundfortheleftward
25r.
t , w h ic h a pp e ar s i n ~ 1 a s a n it e m
m mi n g sh o wn f o r P ( x , 0 i n ( 1 9 ) , a n d
t a t t he e n d of ~ 1 , a n d wh i ch h a s be e n us e d
nsforQ ( x , t ; isrepresentedinf igs. 6
n d t = 2 5 r re s pe c ti e l y.
hepo intsof f ig. 6 representing
ca lculationhasbeenperformedso lely forintegra l
atf irstscarce ly tobee pectedthatafa ir
uldbedrawnfromsofew calculatedpoints
ctua llybeendraw nbyMrWitheringtonw ith
anthesepoints e ceptinformationastoa ll
ngthe lineofabscissas , throughthew holerange
ulatedpo intsaremar edoneachcur e:
w iththek now ledgeof thezeros the
ryclose ineachcasetothatdraw nby
x , t , f o r po s it i e i n te g ra l v a l u es
sedby thefo llow ingarrangementsfora o iding
mmationofa sluggishlycon ergent
4 6 , b y u se o f o ur k n o wl e dg e o f P ( x , t .
a nd ( 1 9 ,
0 t - ( 1 t + ( ( 2 t -... ad. inf.... 48 ,
( - ) i ( i t . .. .. .. .. .. .. .. ( 4 9 .
- i t = b ( i t ,
0 t . .. .. .. .. .. .. .. .. .. .. 5 0 .
4 6 , w e ha e
x , t - ( + l t + ( + 2 t - ( c + 3 , t + ...
( + 1 t - ( + 2 t + ( + 3 , t -...
Q ( x , t = g[ ( , t - ( x + 1 t ] = = D( x , t ... 5 1 .
t ionsof thise uation w ef ind
( - i2Q ( x , t - -1 iD , t + ... + D x + i - t ...... ( 52 .
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N T A ND R E AR O F A F R EE P R O C E SS I O N 3 5 9
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N W A TE R [ 3 6
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0 , andaportionof thecur eofsineswhichveryappro imatelyagreesw ithitatgreatle f twarddistances.
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o f r / G o f 1 f
nto f rightwardprocession. Graphof ( 46 fort =25T r.
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N T A ND R E AR O F A F R EE P R O C E SS I O N 3 6
_ _
-4 -I I
~ ~ ~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
i
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R
0 a n d us i ng ( 5 0 , w e fi n d fi n al l y
) i p( 0 t - ( -y1 D ( 0 t + ... + D( i -1 t ( 5 ) .
nienttocalculateQ ( 1 t , Q ( 2 t ...
tt ingthepointsshow ninf ig. 9.
dofassumingasin( 47 theca lculation
ne g at i e v a l ue s o f x , a v e r y tr o ub l es o me a f fa i r w e
us . W e h a e b y ( 4 6
t- 4( x + 1 t + ( - + 2 t -...
t - ( - + l t + ( -+ 2 t -....
( - t = ( x , t - ( x + 1 t + ( x + 2 t -...
- + 2 t -. ... ... .. ( 5 4 .
usedinthef irsttermof ( 54 , thatits
s it i e a n d ne g at i e v a l ue s o f x , w e h a e
( ( x - i t . He nc e( 5 4 m ay be wr it te n
( - x , t = 2 ( -1 i & gt ( x + i t = P ( x , t ( 55 .
t = P ( , t - Q ( x , t . .. .. .. .. .. .. .. 5 6 .
w e fi n d
t . .. .. .. .. .. .. .. .. .. .. 5 7 ,
aterdueto theleftwardprocession
ancex f rom0onthe le f tside x be ing
integra lorf ractional. Ha ingpre iously
t f o r po s it i e i n te g ra l v a l u e s of x , w e h a e f o u n d
atedpo intsof f ig. 10forthe le f twardprocession.
o ingplansdescribedin~ 11-28
ymeansforunderstandingandw or ingoutin
mt= 0tot= co o fagi enf initeprocession
hsuchdisplacementof thesurface andsuch
w thesurface astoproduce att= 0 a
rmorew a esad ancingintostillw ater
sti l lwaterintherear. Toshow thedesired
tendf ig. 10le f tw ardstoasmanyw a e- lengths
epoint I describedin~ 24. In ertthe
e ly torightandle f t andf it itontothe
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N T A ND R E AR O F A F R EE P R O C E SS I O N 3 6 5
tendedrightwardssofarastoshow noperceptible
00 or3 00 o foursca le . Thediagramthus
hewatersurfaceattime25rafter acommencementcorrespondinglycompoundedfromfig.8 andanother
esentther earofthefinite( two-ended
owconsidering.
heproblemthusindirectly so l ed
f1000w a e-crestsinthebeginning the
on
- i t . .. .. .. .. .. .. .. 5 8 ,
undaccordingtotheprinciplesindicated
pressthesamesurface-displacementasour
andtheproperve locit iesbe low thesurface
arightw ardprocessionofwa es. Ourpresent
ytheinitialsinusoidalityof thehead
nfiniteprocession tra ellingrightwards
thehydro ineticcircumstancesofaprocessionin adingstil lw ater. Ourso lution andthe itemtow ards
nd7 andinf ig. 2of~ 6abo e show how
med.Thewholein estigationshows
nganydef inite" group- e locity w eare
oupof tw o three four oranynumber
a es. Ihopeinsomefuturecommunication
inburghtoreturntothissub ectin
yprinciplesetforth byO sborne
heinterferentia ltheoryofSto estandR ay le igh+
f initegroup- e locity inthe ircaseofan
ysupportinggroups.B utmyfirst
eperformanceofw hichIhopemaynotbe
f imypromisesregardingship-wa es and
ingina lldirectionsf romaplaceofdisturbance inw ater.
showsomeofthe mostimportant
enca lculated andw hichmaybeuseful
esub ectofthepresentpaper.
i . 18 77 p p. 3 4 - 4.
r C a rm b .U n i . C al e nd a r 1 8 76 .
o l . i. 1 8 77 p p . 24 6 -7 .
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A V E S O N W A TE R 3
= , , / p+ Z D x , 0 = ob x , ) - o X + 1 0 .
D( x , 0 [ x ' o X , O 0 I D( x , O )
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N T A ND R E AR O F A F R EE P R O C E SS I O N 3 6 7
~ ~ o E~ 2 2 25 D1 2T
' - 00 02 * 0 00 0 + 00 01
4 1 * 0 00 5 - 0 0 01 - ' 0 00 2
1 - 0 01 1 + 0 00 1 - - 0 0 0 2
2 0 02 4 ~ 0 0 0 + 00 06
' 0 04 4 - - 0 00 + 00 20
0 0 75 - - 0 02 - . 00 1 8
' - 0 11 8 - - 0 0 05 - - 0 05 5
' 0 1 74 ~ 0 0 5 0 + 0 1 17
0 0 24 6 - - 0 06 7 - ' 0 1 6
- 0 3 3 + - 0 06 9 + ' 0 14 6
- 04 4 - ' 0 07 7 - ' 0 18 8
- 0 55 0 + - 0 11 1 + 0 28 1
- 06 79 - ' 0 1 70 - ' 0 8 6
- 0 82 0 + 0 2 16 + 0 7 7
-0917 - -0161 - . 0101
7 - 11 1 - - 0 06 0 - ' 0 7 2
P 1 2 99 + 0 1 2 + 0 55 8
6 - 1 47 2 - - 0 24 6 - - 0 0 2
5 0 - 16 51 - - 02 14 - ' 0 62 6
4 9 - 1 8 2 + 0 4 12 + 02 6 7
9 - 20 16 ~ 0 14 5 ~ 06 7
' - 22 01 - - 04 92 - ' 0 26 6
- 2 8 5 - - 0 22 6 - ' 0 7 1
4 - 2 56 9 ~ 0 4 8 7 + 0 0 21
7 * 2 75 2 + 0 46 6 ~ 0 7 28
* 2 9 4 - 0 2 62 + , 0 4 25
8 - 1 12 - - 06 87 - ' 0 41 0
* 3 2 8 7 - - 0 27 7 - - 0 7 51
- 4 59 + 0 47 4 - ' 0 29 0
* 3 6 29 + ' 0 76 4 + 0 4 4
- 7 94 + -0 3 0 + ' 0 7 41
* 3 9 5 6 - 0 41 1 + 0 42 9
- 4 11 2 - 0 8 4 0 - 1 0 1 9 0
* 4 2 6 7 - - 06 50 - ' 0 64 2
- 4 41 6 - - 0 0 08 - - 0 6 5 7
- 4 56 0 + 06 4 9 - ' 0 2 82
- 47 02 - + 0 9 1 + 0 22 4
4 8 40 + 0 70 7 + 0 5 82
- 49 7 + 01 25 + 06 4
- 5 10 1 - - 0 5 18 ~ , 0 4 17
5 22 6 - - 09 5 + ' 0 0 5
8 * 5 4 8 - ' 0 97 0 - -0 3 2
- 54 64 - - 06 8 - - 05 56
5 5 80 - 0 0 8 2 - ' 0 5 78
- 5 69 0 + 0 4 96 - ' 0 4 21
- 5 79 7 + 0 9 17 - ' 0 1 52
- 5 90 0 + 1 0 6 9 + 0 1 41
- 60 01 + 0 92 8 ~ ' 0 7
4 ' - 60 98 + -0 55 5 + ' 0 50 1
- 61 9 ~ 0 05 4 + 0 5 06
* 6 28 4 - ' 0 45 2 + ' 0 40
5 * 6 7 2 - ' 0 8 5 5 + 0 22 6
- 6 45 9 - - 1 08 1 + 0 0 2 2
6 54 0 - ' 1 10
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-WA V ES.
gsof theR oyalSocietyofEdinburgh J une20 1904
u n e 1 90 5 p p . 7 3 - 7 57 .
p- a es.
ew hatcumbroustitle" Tw o-dimensiona l "
" C a n al W a e s t o d en o te w a e s i n
abottom andv erticalsides w hich if
irsource becomemoreandmore
ensionalatgreaterandgreaterdistances
presentcommunicationthesourceis
ontwo-dimensionalthroughout the
pecti elyperpendiculartothebottom
ofthecanal:thecanal beingstraight.
p inthepresentcommunicationand
1- 1 isusedforbre ity tomean
epthatthe motiondoesnotdiffer
dbe ifthewater beingincompressible
onditionispracticallyfulfilled in
edistancebetw eene erycrest( po int
n , andneighbouringcrestone itherside is
hirdof itsdistancefromthebottom.
e s I m e an a n y wa e s p ro d uc e d in o p en
inggenerator andforsimplicity I
generatorto berectilinealanduniform.
p floatingonthewater orasubmarine
uniformspeedbelow thesurface or
notincludean interestingclassofcanalwa esofwhich
firstgi enbyK ellandintheTrans.Roy.Soc.Edin. for
chthewa elengthisvery longincomparisonw iththedepth
andthetrans ersesectioniso fanyshapeotherthan
nta lbottomandv erticalsides.
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WA V ES
anelectrif iedbodymo ingabo ethe
wa es if themotionof thew aterclose
dimensional theshiporsubmarine
ngitssides( orasubmergedbarha ing
ingtothesidesof thecanal w ithf reedom
Thesubmergedsurfacemustbecy lindric
endiculartothesides.
rcylindricbarof diametersmall
below thesurface mo inghori ontally
amathematicalproblemwhichpresents
worthyofseriouswor foranyonewho
t. Thecaseofaf loatingpontoonismuch
eofthediscontinuitybetweenfreesurfaceof
ressedbyarigidbody ofgi enshape
sierproblemthaneitherof those I
aoraforci econsistingofagi encontinuous
thesurface tra e ll ingo erthesurface
understandthere lationof thistothe
inetherigidsurfaceofthe pontoonto
imagineappliedto it agi endistributionII
ereperpendicularto it. Ta e0 anypo intat
undisturbedw ater- le e l draw OX para lle l
andOZ v erticallydownw ards. Let
nt-componentsofanyparticle ofthewater
onis( x , z ) . Wesupposethedisturbance
wemeanthatthe changeofdistance
ofwateris infinitelysmallincomparison
tance andthatthe line j o iningthem
directionwhichareinfinitelysmalli n
n. F orliberalinterpretationofthis
w . Waterbe ingassumedfrict ionless its
yfromrestbypressureapplied tothe
llyirrotational.B utweneednotassume
mediate ly thatit ispro edbyour
w heninthemw esupposethemotiontobe
av eryusefulw ordintroduced a f tercarefulconsultationw ith
mybrotherthe lateProf. J amesThomson todenoteany
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R
ionsofmotion w henthedensityof the
ty a r e
d . ( . .. .. .. .. .. .. .. ( 5 9 ,
e o f gr a i t y an d p t he p r es s ur e a t ( x z t .
dtobe incompressible w eha e
.. . .. . . .. . . .. 60 .
sumedtobe infinitesimal thesecond
tmembersof ( 59 arenegligible and
become
ferenceoftwodifferentiations gi es
timethemotionis z eroorirrotational
e er.
srotationalmotionin anypartof
stingtok now w hatbecomesof it . Lea ing
trestrictiontocanalw a es imagine
moothsea inaship k eptmo inguniformly
-ropeabo ethew ater. Loo ingo erthe
erofdisturbedmotion showingbydimples
littlewhirlpools.Thethic nessof
othingperceptiblenearthe bowto
arthestern moreorlessaccordingto
moothnessof theship.Ifnowthe
scosityandbecomesaperfectfluid the
otionte llsusthattherotationallymo ing
e ship andspreadsoutinthe more
andbecomeslost ; w ithout how e er
mecerta inthatifanymotionbegi enw ithinaf initeportion
ble li uidorigina llyatrest itsfate isnecessarilydissi
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WA V ES
whichbecomesreducedtoinfinitely
f inite ly largeportionof li u id. Theship
lm seawithoutproducinganymore
dstern butlea ingw ithinanacuteangle
smoothship-w a esw ithnoeddiesor
The idea lannulmentof thew ater s
siderablythetensionofthetow-rope but
thasstillw or todoonane erincreasing
ese tendingfartherandfartherright
r ea o f 1 9~ 2 8 ( t a n- / - o n e ac h s id e o f
alsee inabout~ 80below . R eturningnow
oandcanalw a es: w e inv irtueof
. .. .. . .. . .. . 6 )
t iscommonlyca lledthe" v e locity -potentia l ;
ie n t w e s h al l w ri t e in f u ll ( ( x , z , t . W i t h
esby integrationw ithrespecttox andz,
.. . .. . .. . .. . .. ( 6 4 .
d + = 0. . .. . . .. . . .. . . .. . . .. . . .. . . 65 .
m e th o d t a e n o w
= - k e -n s in m ( x - v t . .. .. .. .. .. . 6 6 ,
nde pressesasinusoida lw a e-disturbance
tra e ll ingx -w ardsw ithv e locityv .
y -pressure I w hichmustactonthe
motionrepresentedby ( 66 , w henmn v , k
ap p ly ( 6 4 t o t he b o un d ar y . Le t z = 0 b e t he
n d le t d d en o te i t s de p re s si o n a t ( x , o t ,
thatistosay
= d f( x , z , t z = = m sin m( x - v t ... 6 7 ,
sw ithinf inite ly smallv e locit iese erywhere w hilethe
ainsconstant.Aftermanyyearsoffailure topro ethat
Helmholt circularringis stable Icametothe conclusionthatitis essentiallyunstable andthatitsfatemust betobecomedissipated
thisconclusionbye tensionsnothithertopublished
ribedina shortpaperentitled:" O nthestabilityof
dmotion inthePhil. Mag. forMay1887. [ R eprinted
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R
hrespecttot
. .. . .. . .. . .. . .. . .. . . ( 6 8 .
urface wemust ing , putz = d andin
becaused k , are infinitely smallquantit ies
eirproductisneglectedin ourproblemof
ts. Hencew ith( 66 and( 68 , andw ith
ace-pressure ( 64 becomes
t = g k c os m( x - v t - I + gC... 69 ;
raryconstantC ta en= 0
m x - t . .. .. .. .. .. .( 7 0 ;
( 6 8 , w e h a e f i n a ll y
.. . .. . .. . .. . .. . .. . 7 1 .
/gm weha eI= 0 andthereforewe
usoida lw a esha ingw a e- lengthe ua lto
nownlawofrelationbetweenv elocity
eaw a es. B utif v isnote ua lto / g/m
sw ithasurface-pressure( g-nm 2 dw hich
thedisplacementaccordingas
g /m .
be:-gi enI asumofsinuso idal
s in g le o n e a s i n ( 7 0 ; - r e u i re d d t he
f thew ater-surface. Weha eby ( 71
operlyalterednotation
- v t+ ) . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . ( 72 ,
v t + ) + A co sf ( - t + y ( 7 ) ,
2
r e gi e n c on s ta n ts h a i n g di f fe r en t v a l u es i n t he
s andv isagi enconstantve locity .
e presses withtwoarbitraryconstants
of f reew a esw hichwemaysuperimposeonany
andinstructi e inrespecttothe
s toapply ( 72 toaparticularcaseof
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WA V ES. 7
nofperiodicarbitrary functionssuchasa
onstantpressures andz eros one ua l
a e ll ingwithve locityv . B utthismustbe
t to letusgetonwithship-w a es and
a e a s a c as e o f ( 7 2 , ( 7 ) ,
s 0+ e 2 c os 2 0 + e tc . = g c 2 2 c + e
e co s 0 + e 2 . .. . .. . .. 7 4 ,
o - c os + - 2 c o s0 + e t c. .. . .. . .. . 7 5 ;
- v t + / ) . .. . .. . .. . .. . .. . .. . .. 7 6 ;
g a . . _ a. ( 7 7 ;
.. . .. . . .. . 7
c & l t 1 . R em a r t h at w h en v = 0 J = o o
a n d ( 7 4 , d = l / g w h i ch e p l ai n s ou r u ni t o f
amicalconditionsthusprescribed
remar f irstthat( 74 , w ith( 76 ,
icdistributionofpressureonthe surface
yv ; and( 75 representsthedisplacement
resultingmotion whenspace-periodic
sthe surface-pressure. Anymotion
uentonany init ialdisturbanceandnosubse uentapplicationofsurface-pressure maybesuperimposedonthe
75 toconstitutethecompleteso lution
e motioninwhichthesurface-pressure
.
roughlytheconstitutionofthe
f o r 1 i t i s he l pf u l to k n o wt h at n d e no t in g
e i n te g er w e h a e
e 2 co s 20 + e tc . = S b7 + ( x n a ( 7 8 ,
/ e .
9 .
oo
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R
15abo etotheperiodic function
dmemberof ( 78 .
membersof( 78 isillustratedbyfig.11
9 ( C
e = 5 an d c on s e u e n t ly b y ( 7 9 , b / a = 1 10 ;
esentsthefirstmember andthetwolight
rmsof thesecondmember w hichareas
agramallowstobeseen onit.There
mentbetweeneachofthe lightcur es
ycur ebetw eenama imumandthe
t. Thusw eseethate enw itheso
e a n o t v e r y ro u gh a p pr o i m at i on t o e u a li t y
fperiodsof thefirstmemberof ( 78 anda
member. If e is& lt 1byaninf inite ly
imationisinfinitelynearlyperfect.
9thatfig.12 cannotshowany
asca leofordinatesone-tenthof thato f f ig. 11.
ntbetweenthefirstmemberof( 78 and
memberwithv aluesofeapproaching
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WA V ES
e followingmodificationofthelast
e 2 y - ( 1 - e2
- C 1 - e 2 + 4 e s in 2 0"
* 8 9 O 9
~ ~ F
IIisv erygreatwhen0is v erysmall
ess0isv erysmall( orv erynearly= 2i7r .
e
. .. . .. . .. . .. . . * ( 8 1
gIIappro imate lybyasingletermof the
.
a lso lution( 75 ; andremar that
e te r m of ( 7 5 i s i nf i ni t e o f wh i ch t h e
arin( 70 . Hencetoha ee ery term
st h a e J = j + 8 w h er e j i s a n i n t eg e r an d
m a y co n e n ie n tl y w ri t e ( 7 5 a s f ol l ow s :
e c o s j 0
1 + j l + +-2 +
0 e + 2 cos ( j + 2 0 _ a di nf. . . 8 2 ;
. . . . . .. . . .. . . .. . . .. . . .. 8 ) ,
initeandinf initeseriesshow nin( 82 .
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R
e 8 = - a n d in t h is c a se J c a n be
s asfo llow s. F irstmultiplyeachtermby
nd we fi nd
e + [ cos ( j + 1 0 + 2 cos ( j + 2 0 + etc.
d e e- 8c os ( j + 1 + e l- co s( j + 2 0 + e tc .
+ ' de e- { RS q j + l ( 1 + e + e 2 2 + etc. ;
a n d a s i n ~ 3 a b o e { R S d e no t es
lf sumfor+ t. Summingthe infinite
fde forthecase8= a w ef ind
+ ( { R S q + 2 l og .. .. .. .. .. .. .. . 8 4 ,
+ l o g1 + V e c o s 0 + e s in 1 0
2 & g t 2 = 2 t a n2 1 mu -2 10 . 8 5 ,
In - t/ esin0
o g
+ l g 1 - 2 e c os 1 0 e
+ t a n 2 - e8 i n ( 8 6
o f t- ( 8 2 g i e s
1 - e 0 e c o s.
8 ) g stanhesoinofo
c os ( j 0 lo g - 2 ec os e
0 ta n- 2 es in .( 8 6 .
o f 8= ~ , ( 8 2 g i e s
C os
+ . .. +
p r es s ed ( 8 ) g i e s t he s o lu t io n o f ou r
o f ~ ~ 4 6 -6 1 I h a e t a e n e = ' 9
ineticil lustrationsinLectureX . o fmy
p. 11 , 114 f romw hichf ig. 12 andpartof
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WA V ES
e n . Re su lt s ca lc ul at ed f ro m ( 8 ) , ( 8 6 , ( 8 7 ,
- 1 6 a l l fo r t he s a me f o rc i e ( 7 4 w i th
rdifferentv e locit iesof itstra e l w hich
s 2 0 9 4 0 o f j . T h e wa e - le n gt h s
th e se v e l oc i ti e s ar e [ ( 7 7 a b o e 2 a /4 1
. Theve locit iesare in erse lyproportiona l
Eachdiagramshow stheforci ebyone
f fig. 12 andshow sbyanothercur ethe
ew ater-surfaceproducedby it w hentra e ll ing
speeds.
be ingthehighest o f thosespeeds
forci etra e ll ingatthatspeedproduces
tupwardswherethedownwardpressureis
umdownwarddisplacementwherethepressure
ard isleast. J udgingdynamica lly it iseasy
eaterspeedsoftheforci ewouldstill
e themeanle e lw herethedownw ard
sgreatest andbelow themeanle e lw here
inishingmagnitudesdow ntoz erofor
e f or a ll p os it i e v a l ue s of J & l t 1 a s er ie s
houghsluggishlyw hene 1 , byw hichthe
actly calculatedfore eryva lueof0.
f o r wh i ch J = 4 4 a n d th e re f or e b y
a /9 7r a nd X = a /4 5 . Re ma r t ha t th e sc al e of
only1/2 5of thesca le inf ig. 16 andseehow
ter-disturbancenowincomparisonwith
meforci e butthreetimesgreaterspeed
a e - le n gt h ( v = \ / g a/ 7 r X = 2 a . W i t h in
w eseefourcompletew a es v ery
al betweenM M tw oma imumsof
oste actly (butv eryslightly lessthan
sbetw eenC andC . Imaginethecur etobe
ghout andcontinuedsinusoidallytocut
nC C atra inof4-sinusoida lw a es
edthroughouttheinfiniteprocession...CC...weha eadiscontinuousperiodiccur emadeupof
44periodsofsinusoidalc ur ebeginning
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S O N W A T ER
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WA V ES
~ ~ b
~ ~ ~ ~ ~
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P W A T ER S HI P -W A V E S 3 8 1
D
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R
hechangeateachpoint ofdiscontinuity
hangeofphase.Aslightalteration
ew ithin60~ oneachsideofeachC
tinuouswa ycur eof f ig. 15 w hich
aceduetomotionofspeedV ga/9wrofthe
entedby theothercontinuouscur eof
8isapplicableto f igs. 14and1 e cept
forci e w hichisV / a / 197rforf ig. 14and
andotherstatementsre uiringmodification
" w a e s " i n r es p ec t t o fi g . 15 s u bs t it u te
nd20~ inrespectto f ig. 1 .
def iningMMinrespectto f igs. 15 14
thecaseof f ig. 1 .
at assa idin~ 48 theformula
, ( 8 7 } g i e s fo r a wi de r an ge o f ab ou t 12 0~ o n e ac h
1 80 . si n( j + ) 0 .. .. .. .. . 8 8 ,
8 49insymbols itbe ingunderstoodthatj
4 andthate is99 oranynunericbetw een
uldgi eashortansw ertothisq uestion
neticideas Here istheonlyanswerI
andseehow intheforci edef ined
re isa lmostw holly confinedtothespaces
sideofeachof itsma imums andisv erynearly
3 00~ . It isob iousthatif thepressure
theselast-mentionedspaces whilein
neachsideofeachma imumthepressure
74 , theresult ingmotionw ouldbesensibly
ewerethroughoutthewholespace
6 0 ~ ) , e a ct ly t ha t gi e n by ( 7 4 . H en ce w e
oughnearly thew holespaceof240 f rom
lmoste actly sinusoida ldisplacementofwatersurface ha ingthew a e- length3 60~ / j + 2 duetothetranslationa l
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WA V ES
te pectsosmalladif ferencefrom
wh o le 2 4 0~ , a s c al c ul a ti o n by { ( 8 ) ( 8 6 ,
d a n d a s i s sh o wn i n f ig s . 18 1 9 2 0 b y t he
ndsideofC w hichrepresentsineach
- - d ( 180 .sin j + ) . ..... 89 ,
0 f romonecontinuoussinuso ida lcur e .
ssofthisdifferencefordistancesfrom
0~ , andthereforethrougharangebetw een
0 ~ , i s v e r y r e ma r a b le i n e ac h c as e .
p re t at i on o f ( 8 8 a n d fi g s. 1 8 1 9 2 0
e so l ut i on t 8 ) , ( 8 6 , ( 8 7 } a " f r ee
d in g to ( 7 ) , t a e n as
. s in ( j + ) . .. .. .. .. .. .. ( 9 0 .
ulstheappro imatelysinusoidalportion
nf igs. 1 , 14 15 andappro imately
elysinusoidaldisplacementinthecorrespondingportionsofthespaces CC andCConthetwo sidesof
stingso lutionofourproblem~ 3 6 and
cial itleadsdirectand shorttothe
efollowinggeneralproblemofcanal
e the iso lateddistributionofpressure
e ll ingatagi enconstantspeed re uired
splacementofthewater intheplace
before itandbehindit w hichbecomesestablishedafterthemotionof theforci ehasbeenk eptsteady for
resynthesisof thespecialsolution
e so l esnotonly theproblemnow proposed
onfromthe instantoftheapplication
. Thissynthesis thougheasilyputinto
or edouttoanypractica lconclusion. O n
mypresentshort butcompletesolutionof
dymotionfor whichweha ebeen
ngoutil lustrationsin~~ 32-5 .
e finitely asa cur eofsines theD-cur e
2 0 l e a i n g th e f or c i e c u r e F , i s o l at e d
iagrams.O r analyticallystated:
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R
P u b l i c D o m a i n
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WA V ES
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R
e u a l v a l ue s o f d ( 0 f o r e u a l po s it i e a n d
f ro m 0~ t o 40 ~ or 5 0~ b y t 8 ) , ( 8 6 , ( 8 7 1
of0ta e
1 8 0~ ) s in ( j + ) 0 .. .. .. .. .. .. 9 1 ,
ca lc ul at ed b y { ( 8 ) , ( 8 6 , ( 8 7 } . T h is u se d in
0 0 f o r al l p os i ti e v a l u e s of 0 g r ea t er t h an 4 0 ~
s i t th e d ou b le o f ( 9 1 f o r al l n eg a ti e v a l ue s o f
.
orPontoons introducedtoapply thegi en
thewater-surface .
amsshowingawater-surface
bef i ed f itt ingclosetothew holew atersurface. Now loo attheforci ecur e F , onthesamediagram
osensiblepressureremo etheco er.
ninsomeparts ofthewholewaterremains
e a m pl e i n f ig s . 1 , 1 4 1 5 1 6 l e t th e
if f co ersf itt ingitto60~ oneachsideof
facebef reef rom60~ to3 00~ ineachof
co ers.Themotionremainsunchanged
dunderthef reeportionsof thesurface. The
egi enforci e andrepresentedby the
isnow automatica llyappliedby theco ers.
8 19 20w ithreferencetothe
heyshow . Thusw eha ethreedifferent
idco er w hichwemayconstructas
ontoon k eptmo ingatastatedv e locity
terbefore it lea esatra inofsinuso ida l
Dcur erepresentsthebottomof the
earrowshowsthedirectionof the
heF cur eshow sthepressureonthe
g.20 thispressureisso smallat-2
supposedtoendthere anditw il l lea e
almoste actlysinusoidaltoan
t( infinitedistanceifthemotionhas
etime . TheF cur eshow sthatin
uidanceasfarbac as-3 q , andinf ig. 18
eepitsinuso idalw henleftf ree q be ingin
a e- length.
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WA V ES
elessPontoons andthe irF orci es.
chasthoserepresentedinf igs. 18 19
thatifany twoe ualandsim ilarforci es
ance\ X betweencorrespondingpoints and
itutediscausedtotra e latspeede ualto
c co r di n g to ( 7 7 a b o e t h e v e l oc i ty o f f re e wa e s
rw illbe le f tw a eless(atrest behindthe
pletheforci esandspeedsof f igs. 18 19
chforci e inthemannerdef inedin~ 57 w e
sof tw onumbers ta enf romourtables
f igs. 18 19 20 thenumbersw hichgi e
erin thethreecorrespondingwa eless
showngraphicallyinfig.21 onscales
e locity . Thef reew a e- lengthforthis
inthediagram.
andthethreew a elesswater-shapes
show ninf igs. 22 2 , 24ondif ferentscales
pressure chosenforthecon enienceofeach
f thethreecasesta ethatderi ed
na lin estigation. B y loo ingatf ig. 2 w e
ngitsbottomshapedaccordingto the
o + 3 q , 1 f r e e wa e - le n gt h s w i l l le a e t h e
estif itmo esa longthecana latthe
reewa e- lengthis4 . A ndthepressure
mofthepontoonisthat represented
cur e .
scissas ineachofthe fourdiagrams
gedtenfold.Thegreateststeepnessesof
arerenderedsufficientlymoderateto
arealwater-surfaceunderthegi en
b e s ai d o f fi g s. 1 5 1 6 1 8 1 9 2 0 a n d of
habscissasenlargedtw enty fold. Inrespectto
eticsgenerally itisinterestingto remar
retationoftheconditionof infinitesimality
spractica llyallow able . Inclinationstothehori on
( 5 ~ " 7 o r s a y 6 ~ ) , i n a ny r e al c a se o f
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N W A T ER [ 3 7
~ ~ ~ ~ ~ ~ ~ J ~ ~ ~ ~ ~ ~ ~ r
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ d
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WA V ES
ances w illnotseriouslyv it ia tethemathematica lresult.
heca lculationsofd 0~ ) and
f o r tw e nt y -n i ne i n te g ra l v a l u e s of j ; 0 1 2 3 , . . . 19 2 0 3 0 4 0 . . . 90 1 0 0 f r o m th e f ol l ow i ng f o rm u la s f o u n d
180~ ; andw ithe= 9ineachcase
+ 1 e [ - e log + 1 + - + + ..
"
1 j ( 2 + 1 e [ l tan- - -- + 1â € ” + +
( - 1 J 2 + 1. .. 9 )
) showninthediagramise plainedby
isinf inite lygreat thetra e ll ingve locityof
small andtherefore byendof~ 41 the
aticallyduetothe forci epressure.
e ualto
f thecur esof f ig. 17forpo ints
dingto integra lva luesof j ise ceedingly
edby it intoanin estigationof the
themotionofasingle forci e e pressed
(94 ;
uturecommunication whenitwillbe
ary toseaship-w a es.
yaidof periodicfunctionsthetwodimensionalship-wa eproblemforinfinitelydeepwater adopted
tion w asgi eninPartIII. ofaseries
a esinF low ingWater publishedin
ine O ctober1886to J anuary1887 w ith
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S O N W A T ER [ 3 7
forwateroffinitedepths.The annulmentofsinusoidalwa esinfrontof thesourceofdisturbance( a
thecana l by thesuperposit ionofatra in
sw hichdoublethesinuso ida lw a esinthe
December1886 byadiagram[ p. 295supra
wtheresidualdisturbanceof thewaterin
5 a b o e a n d re p re s en t ed i n f ig s . 18 1 9 2 0 .
tothan MrJ . deGraaffHunterfor
ousco-operationw ithmeina llthew or o f
on andforthegreatlabourhehasgi en
andtheirrepresentationbydiagrams.
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WA V ES
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S O N W A T ER [ 3 7
~ ~ ~ F
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eofabscissasisquarter-wa e- lengths.
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A V ES.
gsof theR oyalSocietyofEdinburgh J uly17 1905
V o l . x i . J a n ua r y 1 9 0 6 p p . 1 â € ” 2 5 .
w emust forthepresent astime
ledinterpretationof thecur esof f ig. 17:
accordingto~ 44 if8= 0( w hichmeans
thedisturbance d isinf inite lygreat o f
ningisclearin( 70 o f~ 3 9.
pressionofthewaterat distancex
thedisturbance isduetoasingle forci e
la
.. . .. . .. . .. . .. . .. . . 9 5 ,
tanyv e locityv . If thisforci ew ere
aceof wateratrestit wouldproducea
) , asw eareta ingthedensityof the
forci e II( x ) w ouldshapethew atertoan
fcross-sectionshowninfig.25 representing
2 o n t he s c al e o f k = 1 0 c m. a n d b = 1 c m.
95 wefindtan- ( x /b .b .Hencethearea
, o r 6 6 . b rb , a n d th e t ot a l ar e a of t h e
f inityoneachside is7rb . Hencethearea
of thetota larea . Thistota larea W rb ,
e f o rc i e a r ea a n d v r b I c a ll t h e me a n
rea. Thebreadthof theforci ew here
n by t h e do t te d l in e B B i n t he d i ag r am i s b .
i n t hi s a nd f o ll o wi n g e p r es s io n s i s t he ( x - t o f
iginofco-ordinatesbe ingnow f i edre lati e ly tothetra e ll ing
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P S EA S H IP - W A V E S 3 9 5
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R
besuddenlysetinmotion andk ept
anyve locityv intherightw arddirection
oduce agreatcommotion settling
orenearlysteadymotionthrough
cesfrom 0. Thein estigationof
b. 1904 , andparticularly theresultsdescribed
llustratedinf igs. 2 3 , show thatinourpresentcase
e r v i o l e nt e e n i f in c lu d in g s pl a sh e s , d i i d es
tra elawayinthetwo directionsfrom
-speedincreasinginproportiontos uare
ingtothe law of fa ll ingbodies and
throughe erbroadeningspaces w hatwould
absolutequiescence if theforci ew ere
ingactedforany time longorshort.
ontinuesacting andtra e ll ingrightwardsw ithconstantspeed v , accordingto~ 67 thetra e ll ing
heinitialcommotioninthe two
fmere lyapointo f re ference mo ing
lea esthew ater asshow nby f ig. 26 ina
arlyq uitesteadymotionthroughan
ontherearsideof0 andthroughasmall
pro idedcerta inmoderatingcondit ions
, b v .
e~ 68 f irstsupposev inf initely
nitelylittle disturbedfromthestatic
infig. 25 anddescribedin~ 66. Small
a ev erysmalldisturbancewithany f inite
there tremeandletv bev erygreat.
a lprinciplesw ithoutca lculation thatv
ma ebutv ery littledisturbanceof the
ersteepbethestaticforci ecur e . A
ndaricochettingcannonshot i l lustratethe
namicalprinciplein three-dimensional
hematica lca lculation( ~ 79below w eshall
re a t en o ug h w e ha e
97 ,
dgreatthecommotionis themotionof the li uidis and
onalthroughout.
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S EA S H IP - W A V E S 3 9
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S O N W A T ER [ 3 8
htofcrestsabo emeanw ater- le e lin
w a esle f t intherearof thetra e ll ing
e ar e a of t h e fo r ci e c u r e ( f i g. 2 5 ; b e in g
e u a ti o n
. .. . . .. . . 98 :
3 9 ( 7 1 ] by
.. . . .. . . .. 99 ,
hof f reew a estra e ll ingw ithve locityv .
heoreminrespecttoship-wa esis
W i t h ou t c al c ul a ti o n we s e e th a t i f X i s v e r y
w rb( the" meanbreadth o f theforci e
66 , hmustbesimplyproportiona ltoA for
e ll ingatthesamespeed. Thisw esee
a lueofb h/ isthesame andbecause
orci eswithinanybreadthsmallin
esforhthesumof thev a lueswhichthey
artherw ithoutca lculation w ecansee
ca leofourdiagrams thathX / A must
tcalculationIdo notseehowwecould
, asin~ 79below .
tionprescribedin~ 71isillustrated
eringcasesinwhichi tisnotfulfilled.
o forci esbesuperposedw iththeirm iddlesat
g i e h = 0 t h at i s t o sa y n o tr a in o f w a e s .
ceforthis caseisrepresentedinfig.27.
X or-X ; thetw ow illgi ethesamevalue
eonly . Orletthetw obeatdistanceX ;
asgreatasoneforci ema esit.
29 3 0 representingresultso f theca lculationsof~~ 78 79below theabscissasarea llmar edaccording
leofordinatescorresponds ineachof
o k = 2 4 - 89 a nd b rb = 1 02 51 .1 0- . X . T hi s
( 9 7 A = I X , a nd h = t . F i g . 3 0 r ep re se nt s
hema imum intheneighbourhood o f0
e:about1720 timesfortheabscissas
dinates.
theright-handside thew aterslightly
ra e ll ingforci e w hichisadistribution
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V ES
hosemiddle isatO. O nthe le f tsideof
surfacenotdif feringperceptibly f romacur e
e-lengthrearwardsfrom0.Asmall
hofatruecur eofsinesinthediagram
e ssurfacedif fersf romthecur eofsines
ncef rom0asaq uarter-w a e- length.
atin realitythewatersurfaceis
r ly l e e l a n d in c o ns i de r in g a s w e sh a ll h a e
doneby theforci e w emustinterpret
aggerationofslopesshowninthe
o remar thatthestaticdepression
e ifatrestw ouldproduce isabout87times
roducedabo e0by theforci e tra e ll ing
ew a es o f thew a e- lengthshow ninthe
sinterestinga lsotoremar thatthe lim itationtoverysmallslopesisnotbindingonthestaticforci ecur e .
istributionofstaticpressure e erywhere
urface producingstaticdepression
ig. 25 w ould if causedtotra e lata
w a e- lengthisv ery large incomparison
rbance representedby f ig. 26w ithw a es
assa idin~ 69abo e w ouldproduceno
eedoftra ellingwereinfinitelygreat.
gasshow ingthew a elessdisturbance
andsim ilarforci esw iththeirm iddles
lthew a e- length. Thisdisturbance is
frontandrearof themiddlebetween
namicalconsiderationsof thee uil ibrium
w eseethattheareaof f ig. 27( portion
beingrec onedasnegati e mustbee actly
o f theareasof thetwoforci es representing
wnwardpressure.Thisarea being
er i ca l d at a o f ~ 7 , i s n um e ri c al l y ~ X ; t h at i s
lengthisI andbreadththeunito f
o imatemensuration w ithavery rough
dtherange ofthediagram continued
s v erif iesthisconclusion.
nthesameplanasf ig. 27butw ith
thsasthedistancebetweenthetwoforci es
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N W A T ER [ 3 8
~ ~ ~ ~ F ;
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V ES
e- length. Li e f ig. 27 it issymmetrica lon
eof thediagram but insteadofbe ing
27 i t s ho ws f o ur a n d a ha l f wa e s a l l v e r y
al w ithtwodepressiona lha l esofw a es
e le ationscomingasymptotica lly tozero
diagram.Thecur erepresentedby
y theright-hande tremeof f ig. 28: and
rightto le ft isthe le f t-hande tremeof
ththe waterwhollyatrest andstart
erspeed w ithforcegradua lly (orsomew hat
ptotheprescribedamount themotion
resentedby f ig. 28 w ith superimposed
uic lydisappearingine erlengthening
mplitude tra e ll ingaw ay inbothdirections
withtheregularregimerepresentedby
asetoapply theforci es w eha elef ta
a halfv eryappro imatelysinusoidal
ontandarearde iatingfromsinusdidalityas
omtheinstantofbe ingle f tf ree the
itsrearwillrapidlybecomemodified:
central partoftheprocessionwill ha e
lengths w ithvery litt lede iationf romsinuso ida lity. B ut a f terfourorf i eperiodsf romtheinstantofbe ing
processionwillha egotintoconfusion. A f ter
eriods thewaterwillbesensibly
roughthespacewherethe processionwas
partof thespaceo erwhichitwould
ontandrearhadbeenk eptguardedby the
wotra e ll ingforci es. A tnotimeaf ter
escanw ereasonablyorcon eniently
city " to thegrouporprocessionofwa esw ith
. A pre a lentidea is Ibe lie e thatsuch
escouldberegardedastra e ll ingw ithha lf
" o f wa e s o f th e l en g th g i e n i n th e o ri g in a l
e reasonsaregi enforacceptingthetheory
only inthecaseofmutua lly supportinggroups
s S mi t h s P r i e e a m in a ti o n pa p er p u bl i sh e d
ersityC a lendarfor1876: andforre ecting
egroupofw a es. Inreality thef ront
ctseereferencesonp. 304supra .
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S O N W A TE R [ 3 8
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V ES
f tra e lsw ithacce leratedve locitye ceeding
w a esof thegi enw a e- length insteadof
teadymotion symmetricalinf rontand
ngforci e w hichisasolutionofour
nstablesolution( asprobablyarethe
of ~ 4 5 a bo e s h ow n i n fi g s. 1 , 1 4 1 5 .
fthewaterisgi eninmotionaccording
a m pl e 5 0 w a e - l e ng t hs p r ec e di n g 0 ( t h e
e- lengthsfo llow ing0 thef ronto f thew hole
of0 w il lbecomedissipatedintononperiodicw a estra e ll ingrightwardsandlef tw ardsw ithincreasing
asingve locit ies andtheappro imately
itw il lshrin bac w ardsrelati e ly to
etheforci ehastra e lledf if tyw a elengths theperiodicw a esinf rontof itarea llgone: butthere
both beforeandbehindit.Afterthe
ahundredwa e- lengths thew holemotionin
emotionforperhaps3 0w a e- lengthsormore
ettledtonearly thecondit ionrepresentedby
sasmallregulare le ationinad anceof
gulartra inofappro imately sinuso ida lwa es
esbeingofdoublethew a e-heightgi en
a s s ai d a bo e i n ~ 6 8 w i ll g o o n l e a i n g
nofsteadyperiodicw a es increasingin
eseanirregulartrainofw a es shorter
dlesshigh thefartherrearwardweloo
0of~ 26 27abo e . It isaninteresting
lem to in estigatethee tremerear
lessw aterbehindit o f thetrainofw a es
etra e ll inguniformly fore er. Ihopeto
henw ecometoconsiderthew or doneby
stigationof theformulasby the
2 6 2 7 2 8 2 9 3 0 h a e b e en d r aw n a n d
spro ed. Gobac totheproblem of
ea d o f ta i n g e= 9 a s i n ~ ~ 4 6 -6 1 t a e
1 / 2 + 1 . B y ( 8 6 a nd ( 8 7 o f~ 4 5 we ha e
100 ,
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R
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S H IP - W A V E S 4 05 /
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i
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R
~ 2 V e s in 10
( j + - 0 ta n- l 2 -e sn ~
e co s 0+ e
g1 - 2 e c. .. 1 01
2 + * e c os j . .. 102 .
2 -3
tedbyputting0= - . andta ing
a ti o n is t h at a s w e sh a ll s e e by ( 7 8 o f ~ 4
( 1 01 , ( 1 02 , e p re ss t he w at er d is tu rb an ce d ue t o an
tconsecuti edistanceseache ua lto
p r es s io n f or e a ch f o rc i e b e in g
n a 2 ~ ~ . .. 1 0 ) ,
. .. . . .. . . .. . . . (1 ) ,
posit i eornegati e integer andby ( 79
. . .. . . .. . . ( 104 .
sureat0dueto eachoftheforci es
o si d es i s 1 / 1 + ( 2 7 r. 1 0 4 2 o f t he p r es s ur e
secentre is0. Thusw eseethatthe
orci es e ceptthe lastmentioned may
era lw a e- lengthsoneachsideof0: and
, ( 1 0 1 , ( 1 0 2 e p r es s t o a v e r y hi g h de g re e
edisturbanceproducedinthewaterby the
ew hosecentre isat0.
a e = 18 0~ i n ( 1 00 , ( 1 01 , ( 1 02 ; w e
e i { V e t an -1 -e 1- e +
. o .
. .( 1 05 .
1-10-4 asw etoo inourca lculations
ow t a e e = 1 . Th i s re d uc e s ( 1 0 5 t o
1 -. .. + 1 j + . .. 1 06 .
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V ES
te lygreatoddore eninteger andw ef ind
) 7 r .. . .. . .. . .. . .. . .. . 1 0 7 .
a eseen foundbysuperimposingonthe
29aninfinite trainofperiodicwa es
2 7 r / X ) a n d th e re f or e h = r w hi c h
two-dimensionalproblemofcanalship-wa estothethree-dimensionalproblemofsea-ship-wa es we
hodgi enbyR ayle ighattheendofhis
tandingw a esonthesurfaceof running
edtotheLondonMathematicalSocietyin
ninf initeplanee panseofw ater consider
s u ch a s t ha t r ep r es e nt e d by ( 9 5 ' o f ~ 6 6 w i th
eneratinglinesindifferentdirections
a e ll ingw ithuniformvelocity v , inany
onoftheseforci es andofthedisturbancesofthe waterwhichtheyproduce eachcalculatedbyan
( 101 , ( 102 , gi esustheso lutionofathreedimensiona lw a eproblem w hichbecomestheship-w a e-problem
ntsinfinitelysmallandinfinitelynumerous.
nstituentforci easconf inedtoaninfinitely
batedtheconse uenttroublesomeinfinityby
be annulledininterpretationofresults
rto 0.Iescapefromthe troublein
m of w a e s b y t a i n g ( 9 5 t o e p r es s
e intheforci e andma ingbassmall
s i nd i ca t ed i n ~ ~ 7 9 7 , 7 6 b y t a i n g
w e c al c ul a te d a f in i te v a l u e f or d ( 0 . B u t f o r
erablygreaterthanhalfaw a e- length w ew ere
lationsby ta ingb= 0.
nsionalsystemlet inf ig. 31 J be
of therearw ardw a e-normalo foneof
wa es. Thisisalsothe inclination
of thetra e ll ingforci etow hichthat
enow fortheforci eobta inedby the
c. 188 : republishedinRay leigh sScientif icPapers
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R
numberofconstituents asdescribed
b 2
( x c o s+ y si n) 2 + b 2 * ( 1 08
ionof r andbisthesamefora llva lues
arforci esystemw e
w he re r = 2 y2 .. .. .. 1 09 .
w hethercircularornot bek ept
nofx negati e w ithve locityv : and
pondingf reew a e- lengthgi enby the
thew a e- lengthof theconstituenttra in
gto r= 0. F orthe* -constituent the
rpendiculartothef rontisv cosk , andthe
2. Loo ingnow tof ig. 26 w ithX cos2
othedirectionof themotionof theforci e inf ig. 26.
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V ES
fg. 3 1 a nd t oe u at io ns ( 9 7 , ( 9 8 ; w e
edepressionat( x , y duetotheconstituentof forci eshow nunderthe integralin( 108 is
x c os r + y s in r . ( 1 1 0
. .
y sinsisconsiderablygreaterthan-X cos2.
t( x , y duetothew holetra e ll ing
s in d 27 r( x c os + + ys in ~ )
, ' k
.. .. .. .. 1 11 .
singthe lim its- - r7r-0 to ris
i egi esatra inofsinuso ida lw a esin
bledisturbanceinits frontatdistances
aw a e- length. Loo now tof ig. 31 and
erofmediallines oftheforci es
108 , ( 111 ; a llaslinespassingthrough
P Y Y L , X X ' o f th es e li ne s ar e sh ow n
dingrespecti e ly to= = - ( 7r-0 ,
e a c ut e a ng l e r = - 7 r. O n e a c h o f th e f ir s t
dicatestherear. Thefourth X X ' , is
on andhasneitherfrontnor rear.
ustincludeall andonlyall themediallines
dsP. HenceQP isone lim ito f rin
ssesthroughP X X ' istheotherlim itbecause
r.Thusallthe linesincludedinthe
seanglePOX ' . Thusthe integral( 111
onatP ( x , y duetothe j o intactionofa ll
becausenonee ceptthosew hosemedial
contributeany thingtothedisturbance
dappro imatelye a luatingthedef inite
enientlyput
= c .. .. .. . 1 1 2 ,
llow s:
rr a
2b 2- si n. .. 1 1 ) .
' X
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R
erygreat therewillbee ceedingly
ne ua lposit i eandnegati ev a luesof
hichwillcausecance ll ingofa llportionsof the
ifany thereare forw hichdu/ d rv anishes.
ha t t he r e ar e t wo s u ch v a l ue s i * 2 b o th
u b e in g a m a i m u m ( u ) f o r on e o f th e m
fortheother andthat w hen0hasany
a n d 27 r - ta n -~ ^ / t h e v a l u es o f ~ , , 5 2
derationofthislast-mentionedcase
eareaofsea inad anceof two lines
tra e ll ingforci e inclinedate ua l
1 9~ 2 8 ) , o n e ac h s id e o f th e m id - wa e t h er e
ceat distancesofmuchmorethana
hecentreof theforci e . Themain
es therefore l iesintherearw ardangular
ines. It isi llustratedby f ig. 3 2 asw e
y theproperinterpretationof ( 11 ) .
ntof thesinin( 11 ) byTaylor stheorem
ngf rom 1 bysmallf ractionsofaradian
, , . ~ / li /.
- 1 2 = a- ... 114 ,
d 2 u
( ( - ) d- .. . 1 15 .
1 15 w e fi nd d = d l / , 3 i V 7 r , w he re
cttoq2andv a luesof tdifferingbut
+ q ~ 2 i n s t ea d o f th e - q 2 o f ( 1 1 4 , a n d
ad o f th e -( d 2u /d q 2 o f ( 1 1 5 ; b ec au se u i s th e
i ni m um . Ca l li n g k , k t h e v a l ue s o f k
#2 andusingthesee pressionsproperly in
f o r th e d ep r es s io n o f th e wa t er a t ( x , y ,
co s1 d , s i n( a l - q 2
1 17 .
a + q ) . 117 .
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PSEA SHIP -WA V ES 411
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ c
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
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R
assignedtotheintegrationsrelati ely
ca u se t h e gr e at n es s o f r/ X i n ( 1 1 5 a n d th e c or r es p on d in g f or m ul a r el a ti e t o # 2 m a e s q , a n d q 2 e a c h v e r y gr e at
e , fo rmoderateproperly smallposit i eor
- J r a n d - # . N o wa s d is c o e r ed b y
re g or y s E a m pl e s p . 4 79 , w e h a e
2 / 2
w ef ind
- c os a l k , ( s in ~ l + C O S U 2 I
#/ 2c os 2 2 j . .. .. 1 18 .
a luesby (1-15 w ef ind
N
n r ul -
8
l l 8 .
/
ntit iesdenotedby8 , / 2in( 116
e ( 1 1 2 a s fo ll ow s: ru = ( x + y t V i + t 2 w he re t = t a n* . . .. .. .. . 1 1 9 .
ononthesupposit ionofx , y rconstant w e
2l 6. .. .. 1 20 .
y t 5 + t 6 .. .. .( 1 21 .
r t in w h ic h m a e s i t a m a i m um o r m in i mu m
0 .. .. .. .. .. .. .. ( 1 22 ;
wh i ch w h en ( y / X ) 2 & l t - h a s re a l ro o ts a s
x 2
2 4 ~ y + 4 y ~ y I .. . .. . . 1 2 ) .
therof these fort in( 121 , w ef ind
t 2 + 2 yt j \ V 1 + t 1 2. .. .. . 1 2 4 ,
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V ES
119 ,
x ( 1 + t m 2 3 2 . .. .. .. .. .. . ( 1 2 4 .
rstfactoro f ( 124 by ( 122 w ef ind
( 2 y + 2 t .t mV . .. 1 24 " ,
a nd m = 2 g i e s / a n d / b y ( 1 1 6 .
e s e e th a t ( d 2 u /d 2 M v a n i s he s w he n x = y V 8
o r t a n d po s it i e f or t 2 w he n x & g t y V 8 .
/ dr2negati e . Thereforeu isthema imum
e. Thereforeu2istheminimum and( 119
mumandminimumv alues
V / l+ t 2 r u2 = ( x + y t2 V I + t 2 2. .. ( 1 25 .
2 ) w es ee th at wh en y/ = 0 w eh a e - t = + o o
easey f rom 0to+ x / V 8 - t1fa lls
/ ~ , and- t2risescontinuously f rom
b e co m e e a ch o f t he m V / ~ ; w h ic h
1 6 .
asystem ofautotomic monoparametricco-ordinates . ~~ 87-90.
ru = a . . . . .. . . .. . . .. . . .. . . .. . . .. . 126 ,
ameterO W of thecur eO C C f ig. 3 2
scribe be ingthecur egi enintrinsica lly
w i th s u ff i ' m o m it t ed f r om t . I n th e p re s en t
ybecalledisophasals becausetheargument
be low isthesamefora llpo intsonanyoneof
1 22 f or x a nd y w e fi nd t
t . .. .. .. .. .. . 1 2 7 .
-ordinatesinaplane w eha eaw ell- now ncase inthe
tingofconfocalellipsesandhyperbolas.
p -W a e s ( s u pr a N o . 3 2 p . 3 0 7 s e e Po p . Le c t. I I .
e c h e lo n c ur e s " i s g i e n l i e f i g. 3 2 w i th t h e li n e of c u sp s
meinclination19~ 28 ; butthee uationsaredif ferentf rom
esentingtheef fecto fadif ferenttra e ll ingdistributionofpressure .
i. M ag . V o l . X L V . 1 8 98 p p . 10 6 -1 2 ; L a mb H y dr o dy n am i cs 3 r d e d. 1 9 06 ~ 2 5 , a l so n e wm a tt e r in t h e Ge r ma n t ra n sl a ti o n a n d
c.Roy.Soc. Aug. 1908 assupra p.3 04 whostatesthathis
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R
r esshowninfig.3 2hasbeen
a luesofx yca lculatedf romthesetw o
n g to - t v a l ue s t an 0 ~ , t a n 10 ~ , t a n 20 ~ , . . .t a n 90 ~ .
alspartiallyshowninfig. 3 2 allsimilar
eendraw ntocorrespondtose ene uidifferentsmallerv a lues 19X , 18X . . . 1 X , o f theparametera ifw e
a l to 2 0 X .
amthate ery tw oof these isophasals
nts ate ua ldistancesonthetw osides
thesystemdow ntoparameter0 e ery
Cistheintersectionof twoandonly
e n b y( 1 2 7 w i th t w o di f fe r en t v a l u es o f t he
completeeachcur ea lgebra ically w e
mbyan e ualandsimilarpatternon
ledpattern thusobta ined w ouldshow
ualandsim ilarinthef rontandrear w hich
p os s ib l e bu t i ns t ab l e. W e a r e h o we e r a t p re s en t
ableship-wa escontainedinthe angle
w o si d es o f t he m i d- w a e a n d we l e a e t h e
honly theremar thata llpointsinthe
m andtheoppositeangle lef tw ardof0
aluesoftheparametera:whileimaginary
real pointsinthetwoobtuseangles.
127 , w ef ind
. .. . . .. . ( 128 ;
istheanglemeasuredanti-cloc w ise
t o th e c ur e a t a ny p o in t ( x , y , i n t he
Eliminationoft betweenthetwo
g i e s a s t he c a rt e si a n e u a ti o n of o u r cu r e
a 2 ( 8 y4 - 2 0y 2 -4 + 6 a 4y 2 = 0. .. 1 29 .
t ions( 127 aremuchmorecon enient
estingtoverif y ( 129 forthecase
2 7 , c o rr e sp o nd i ng t o e it h er o f t he t w o cu s ps
86andthecontinuousvaria tions
ethat- t and- t2arerespecti e ly the
ui a lenttoLordK el in s. Theorigina lreporto f the
pressingthe law ofamplitudeof thew a es w hichare
nt.
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V ES
ns rec onedf romO Ycloc w ise o f
C andof theshortarcWC intheupper
ifw ecarryapointf rom0toC inthe
W intheshortarc w eha ethechangeof
entedcontinuouslybythedecreaseof
t o 3 5 ~ 1 6 , w hi l e y in c re a se s f ro m 0 t o x / 8
of t a n- l ( - t ) f r om 3 5 ~ 1 6 t o 0 ~ , w hi l e y
0aga in. The inclinationtoO Yof the
t he c u sp C i s 3 5 ~ 1 6 ( o r t an - l V I / .
ortarcC WC of thecur euor
isaminimum. Ineachof the longarcsuisa
po i nt o f t he c u r e t h e v a l ue o f u w h et h er
isa / r. Hencefordifferentpo intsof the
lyproportiona ltotheradiusv ectorf rom0.
18 ' w enow seethatfora llpo intson
r u a n d ru 2 h a e b o th t h e sa m ev a l ue b e in g
thecur e . Thef irstparto f ( 118 ' isone
onatanypoint oneitherofthe long
parto f ( 118 ' isoneconstituentof the
theshortarc. Ta ingfore ample
sshowninf ig. 32 w enow seethatforany
rcs thesecondconstituentofthe
o becalculatedfromthesecondpart of
ranypo into f itsshortarc thesecondconstituent
ca lculatedf romthef irstparto f ( 118 ' .
m ilarly thedeterminationofd( x , y
o f thesmallercur esw hichw eseeinthe
rarcsof the largestcur e w earri eat
sthecompletesolutionof ourproblem.
ingwa esinthewa eofthe
enby thesuperposit ionofconstituents
127 w ithgreaterandsmallerv a luesof
telysmallsuccessi edifferences.
n lo o i n g at t h e wa e s f ro m ab o e i s e a c tl y
andv a lleys w ithridgesandbedsof
cordingtothe isophasa lcur esshownin
yoneof theshortarc- ridgesandfo llowing
w ef inditbecomingthemiddle lineofa
garcsof thecur e . A ndfo llow inga
oughthecusps w ef ind inthecontinuation
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E S O N W A TE R [ 3 8
g ri d ge s . E e r y ri d ge l o ng o r s ho r t i s
lthecur edridgesandv a lleysareparts
o fcur es i l lustratedby f ig. 32and
ra i c e u a ti o n ( 1 2 9 .
nsw emayw rite( 118 ' asfo llow s:
ec si ll ( r u + ) . .. .. 1 0
( + d ) . .. . .. . .. . .. . .. . ( 1 1 .
rhapsthemostimportant featureof the
actua lly seeonthetw osidesof themidw a eofasteamertra e ll ingthroughsmoothw ateratsea oro fa
asfastasitcaninapond isthesteepness
esw hichw ek now tobeinclinedat19~ 28
etheoryof thisfeature ise pressedby
in( 1 0 , andisw elli l lustratedby the
ore le enpo intsofanyoneof the
heresultso fw hichareshowninco lumn6of
ye pressthedepressionbelow and
le e l duetooneconstituentof thesystemof
C o l. 4 C o l. 5 C o l. 6
d 2 a / a s ec 2 t
d \ / ' X
00 100000 1100000 10-000
6 85 ' 9 7 2 9 ' 9 7 8 2 1- 0 64 7
2 0 1 -9 1 58 7 - 7 4 9 7 1- 2 1 0
5 0 ' 8 72 90 3 3 3 3 3 2 - 0 94
3 8 4 9 8 6 6 0 2 0 0 00 0 0 o c
7 7 8 7 22 5 - ' 4 0 8 0 2 - 66 6 0
6 6 - 9 6 2 4 - 1- 8 40 7 0 1 -7 8 9
16 5 1 1 0 94 1 - - 5 00 0 0 1 7 8 88
00 1-5 041 -14-0987 2-279
9 7 2 - 91 2 22 -6 - 3 4 1 4 - 16 7 2
0 0 00 oo - o o
thehighestspeedattainedbyaduc ling thisangle is
terthan19~ 28 becauseof thedynamicef fectof the
ofw ater. SeeB alt imoreLectures p. 59 ( le tterto
2 rdA ug. 1871 andpp. 600 601( lettertoWill iamF roude
6thOct. 1871 .
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V ES
ysdescribedin~ 92. C o lumn1is-* .
caandy / a ca lculatedf rom( 127 . C o lumn4
2 6 f r om c o lu m ns 2 3 . C o l u mn 5 i s
f rom( 124 ' andco lumns2 4. C o lumn6
f r om ( 1 1 a n d co l um n 1 .u b e in g a s
nm um f o r v a l ue s o f - r fr o m 0 to 3 5 ~ 1 6 , a n d
esf romthisto90~ , w eseethattheproper
forthef irstfourlinesofeachcolumnis2
sis1.
s g e n er a ll y a f un c ti o n of * ; b u t if t h e fo r ci e
e , k i s a c on s ta n t a n d fo r p oi n ts o n o ne o f
a= constant theonlyv ariablecoeff icientsof
d3- l. B utfordif ferentisophasa lcur es
0 e pressingthemagnitudeof therange
e e l v a r ie s i n e r se l y as V / a .F o r m id w a e ( r = 0 a i s s im p ly t h e di s ta n ce f r om t h e fo r ci e : a nd w e
orourpoint- forci e butforagreatship
ery largenumberofw a e- lengthsright
e ightin erse lyasthes uarerooto f the
eorf romthemiddleof theship.
3 5~ 16 representsafeatureana logous
eisin naturenoinfinityforeither
niteanddistributed notinf inite ly intense
ysmall space. Accordingtothe
1-80abo e w eha eine erycaseaf inite
o f fo r ci e e c e pt i n ~ 8 0 wh e re w e h a e
ll incomparisonw ithX , andw ea oid
mn6: andcan bygreatlabour ca lculate
bers risingtoav ery largema imumat
b u t no t t o in f in i ty a n d so a r ri e m at h em a ti c al l y
eryhighw a esseenonthetw obounding
bance inclinedat19~ 28 tothemid-w a e.
memberthatweseein realityaconsiderablenumberofwhite-cappedwa es( would-beinfinities
argeglassyw a esw hichformso interesting
sturbances.
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S O N W A T E R [ 3 8
tpaperaremerelyaw or ingoutof the
ra itationa lw a esw ithnosurfacetensionontheprinciplegi enbyR ayle igh in188 forthemuch
ofcapil la rywa esinf ront inw hich
efconstituentof theforci e andw a es
hchie fconstituentof theforci e is
ical a lgebra ic graphicof~ 3 2-95
muchva luableassistancef romMrJ . deGraaff
stnow beenappointedtoapostinthe
ory.
c V o l . x v . p p . 69 - 78 1 8 8 ; r e pr i nt e d in L o rd
apers V o l. II. pp. 258-267.
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DEEP-SEA WA V ESO F THREEC LA SSES:
G LE D IS PL AC EM EN T ( 2 F R O M A G R O U P O F
D I SP L AC EM E NT S ( 3 B Y A P E R I O D I CA L LY
E-PRESSUR E.
. E di n . J a n . 22 1 9 06 P h il . M fa g . V o l . x I I I .
.
toanInitia tiona lF ormmorecon enientthan
r i o us P a pe r s on W a e s . ~ 96 - 11 .
f~ 5- 1 includingthe" f rontand
reeprocessionsofw a esindeepwater
aldisturbances accordingtothe
describedin~ 3 , 4. Inthisform
erywheree le ationore erywhere
mount atgreatdistancesf romtheorigin
es uarerooto f thedistancep f roma
bo ethewater-surfaceinthemiddle
presentpaperanewformof typedisturbanceisderi edindifferentlyfrom eitherthefirstor
sof~~ 3 , 4: f romthef irst bydouble
ncetotime t f romthesecond by
referencetospace x .
f~ ~ 1 2 slightlymodif iedwithrespect
f rictionlessincompressible li uid( ca lled
stra ightcana l inf inite ly longandinfinitely
des. Letitbedisturbedf romitsle e lby
nthesurface uniformine ery line
sides andletitbe lefttoitselfunder
re uiredtofindthe displacement
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R
particleofwateratany futuretime. Our
yspecif iedbyagi ennormalcomponentofv e locity andagi ennormalcomponentofdisplacement
rface.
tadistance habo etheundisturbed
para lle lto the lengthof thecana l andO Z
etI ~ bethedisplacementcomponents
omponents o fanyparticleof thew ater
onis( x , z ) . Wesupposethedisturbance
wemeanthatthe changeofdistance
ofwateris infinitelysmallincomparisonwiththeirundisturbeddistance andthelinej oining
gesofdirectionwhichareinfinitelysmall
an.W aterbeingassumedincompressibleandfrictionless itsmotion startedprimarilyfromrest
freesurface isessentiallyirrotational.
d
t ; C = d = ; = d... ... ... 1 2 ,
t , o rF asw emayw rite itforbre ityw hencon enient isa functionw hichmaybeca lledthedisplacementpotentia l andF ( x , z , t isw hatiscommonlyca lledthev e locitypotential. Thusak now ledgeof thefunctionF , fo ra llva luesof
pete lydef inesthedisplacementandthev e locityof
edeterminationofF w eha e inv irtue
hefluid
3 ) .
t ion thew ell- now nprimary theoryof
hat if F isgi enfore erypo into f the
andisz eroate erypo intinf inite ly
ofF isdeterminatethroughoutthefluid.
mal andthedensitybeingta enas
fundamenta lhydro ineticsgi es
.
- g + ( z - h~ - € ” ) + . .. 1 4
)
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A N C E IN I TI A TE D F R O M A G I V E N F O R M 4 2 1
y I1theuniformatmosphericpressureon
p th e p re s su r e at t h e po i nt ( x , z + ' ) w i th i n
x , z , t i s a f un c ti o n wh i ch b e si d es
s a ti s fi e s al s o th e e u a ti o n
5 ;
atthecorrespondingf luidmotionofw hichF
ntia l( 1 2 , hasconstantpressureo er
g ; t h at i s t o sa y e e r y su r fa c e wh i ch w as
sundisturbed. Thusourproblemof
esimalirrotationalmotionofthefluid
sunderanyconstantpressure isso l ed
1 3 ) a nd ( 1 5 .
nw everif y that asfoundin~ 3
- 2 e4 Z + ) . .. .. .. .. .. .. .. .. .. 1 6
nd ( 1 5 . B y c ha ng in g ei nt o - a nd b y
tionsperformedon(1 6 accordingto
d , w h e re i j , k a r e a n y in t eg e rs p o si t i e o r
r e f r om ( 1 6 a n y nu mb e r of i m ag i na r y
ofthese withconstantcoefficients
f realisedsolutions. If asin~ 97 w e
mulasthusobta inedasadisplacementpotential thenby ta ingd/ d o f itw ef ind' thevertica l
t w hichwesha llta easthemost
nineachcaseforthesolutionsw ithw hich
emay ifw eplease ta eanyso lutionof
t ing notadisplacement-potentia l butav e locitypotentia l o rahori ontalcomponentofdisplacementorve locity
tofdisplacementorv elocity.
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R
12w etoo
+ t ) - 2e4 z ' x )
- os - e ( 1 7 ,
2 , a nd x = t an -l ( x / )
thisnotationband- wasconsistently
w henposit i e upw arddisplacementof
byupw ardordinatesinthedraw ings .
4 f i g. 1 t h at w h ic h h as i t s ma i m um
7 , f o r t= 0 . T h e ot h er c u r e o f f ig . 1
eordinatesonthetwosidesof0
w i th - R D i n st e ad o f { R S . T h e s ym b ol s
e r e i nt r od u ce d i n ~ 3 a b o e { R S t o d en o te
ha lf thesumofw hatisw rittenaf terit
odenotearea li a tionby ta ing1/ 2to f the
us1/2tof thesameformulawith+ t
ur e inw hichtheordinatesare
-o f theordinatesof thesecondof \ / 2LC
isnow gi enintheaccompany ingdiagram
bo eitthef irsto f theo ldcur esof f ig. 1is
atesreducedintheratio2/ 2to1 for
w iththenew cur e. Thisnew cur e
enientinitiationalformreferredtoin
per.
yt a i ng t = 0 i n ( 1 9 o r in ( 1 44 [ m os t
formof ( 1 9 ] , isasfo llow s:
( ' , ' ) 2 = 2 p ( 2 -p ......... 1 8 .
tionofthenewparticularsolution
f r om t h e pr i ma r y ( 1 6 a s i nd i ca t ed i n
efo llow ingformula:
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A N C E IN I TI A TE D F R O M A G I V E N F O R M 4 2
+ 1 )
{ R D d â € ” ( Z + ) )
g c os _ 4 p2
\ 4p 2 j
2 , a nd X = t a n- ~ ( x / ) . . .. .. .. .( 1 9 .
a forthesamederi ation whichw illbe
n~ ~ 1 5-157below isasfo llow s:
4 z I t ) - ~ ~ __ 0 ( . . .. t
= { R S d t2 â € ” 2 + ( , z t dt .... ... .. ( 1 40 .
1 9 and( 140 iseasilypro edbyremar ing
nd ( 1 5 ,
-gt
= { X ) RS -g z + i - C , + L ... ( 142 .
9 dt \ I z + tX )
3 , andseew ithinhow narrow aspace
+ 2 i n t he n e wc u r e t h e ma i n in i ti a l
w hile intheo ldcur e itspreadssofar
20itamountstoabout-16of thema imum
andaccordingtothe law of in erse
otofdistance w hichholdsforlargev a lues
atx = 80itw ouldstil lbeasmuchas-1of
mparati enarrownessof the init ialdisturbancerepresentedby thenew cur e andtheult imate law of
( insteadofx - 2fortheo ldcur e are
enew cur e intheapplicationsandil lustrationsof thetheory tobegi enin~ ~ 1 5-157below .
tthetotalareaof theo ldcur ef rom
great w hile it iszero forthenew cur e.
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/ -Tl a. Z d Z - a n oJ ad dy * - c * 1
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R
entialenergyofthe initialdisturbance
2 .. .. .. .. .. .. .. .. ( 1 4 ) ,
odcur e w hile forthenew itisf inite .
maybew ritteninthefo llowingmodif ied
n enientforsomeofourinterpretations
~ 4 p 2
” 2t - CO S c os A. .. .. . 1 4 4 ,
- tan- gt2sinX
X - t an - s . .. .. 1 45 .
- 2 p
w hichforbre ityw eshallca llw atercur esintheaccompany ingsi diagramsof f ig. 3 4 representthe
cordingtoournew solutionr( x , z , t f o r
sp ec ti e ly 0 2 V , , V / wr V 7 r 4 V r 8 V 7 r.
dbyta ingg= 4.Thisismerely
gasourunito f lengthha lf thespacedescended
byabody fall ingf romrestunderthe
orsimplificationinthewriting offormulas
eundisturbedle e lo f thew ater-surface. The
pla inedin~ 107below areca lledargumentcur es asthey representtheargumentof thecosine in( 144 .
ycuriousandv ery interestingfeatureof
reasingnumberofv a luesofx forw hichthe
stimead ances andthe largef igures
u w h ic h i t r ea c he s a t th e t im e s 4 \ / V r a n d
w odiagrams. Thesezeros foranyva lueof t
a ti o n
.. .. . .. . .. . .. .. . .. . 1 4 6 .
highlycomplicatedcharacterofthe
145 thez erosareeasily foundby tracing
w ithA asordinate andx asabscissa( asshown
esof thesi diagramsontwodif ferentscales
tion notformeasurement anddraw ing
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A N C E IN I TI A TE D F R O M A G I V E N F O R M 4 2 7
e atdistancesfromitrepresenting
T e t c. A p a ra l le l a t di s ta n ce - 1 7 i s an
argument-cur es andisshownin
ononescaleofordinates. Thepara llelcorrespondingtodistance-1 - risshowninthef if thandsi th
allersca leofordinatesusedinthe irargumentcur es.
w sz erosatx = + / , o fw hichthat
d1. Intheseconddiagramtheargumentcur e indicatesz erosforthe- rrand- - rparalle ls w hichare
er-cur e . Thezerocorrespondingto
dattheoriginat thetimewhen~ gt2
at i s w he n t w as 1 / V 2 o r ' 7 0 7. I t i s a
sforx -posit i eandx -negati e .
at shortlybefore itst ime ama imum
eintheargument-cur e w hichstil l indicates
aremar edbycrosses.
t inthe inter a lbetweenitstimeand
oz e r os o f t he w a te r -c u r e f o r x - p os i ti e h a e
Theseandthecorrespondingzerosfor
distinctlyonthew ater-cur e andthe ir
earemar edby fourcrossesonthe
betweenitstimeandthatof No.4
a e c o me i n to e i s te n ce o n ea c h si d e of O Z ,
tedfore ampleontheargument-cur e
Nineonlyoutofa llthesi teenzeroson
eonthewater-cur e . These en
oneachside a ll l iebetw eenx= 0and
betweenitstimeandthatof No.5
orx -posit i eha ecomeintoe istence one
by thepara lle l- rr. F ourteenonlyout
rosoneachsideareperceptibleonthe
ofthefiftyimperceptiblez erosoneach
a n d x = + 1 .
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R
thez erosoriginate inpa irsonthe
x - p os i ti e a n d x - n eg a ti e : t h os e o n th e
intersectionsofoneofthe parallels
1 r/ 2w iththeargument-cur e . The
ment-cur etra e lsslowly intheoutward
1astimead ancesto inf inity . A ttimes
d ia g ra m s 5 an d 6 i t h as r e ac h ed s o c lo s e to
nthasbeenregardedastheactua lposit ionof
orthepurposeofdraw ingthecur e andfor
talnumberof z eros.
ginatesaccordingtoan intersection
eargument-cur etra e lsoutwardsw ith
nfinity astimead ances. Eachof the
ros thatistosay eachzerooriginating
ononthe inw ardsideof theargumentcur e tra e lsveryslow ly inwardsw ithv e locitydim inishingto
estoinfinity.Thusthemotionof the
enx = - 1andx = + 1becomesmoreand
numberof inw ardtra e ll ingw a es
shingtoz ero and asw eseeby the
144 , w ithamplitudesandw ithslopesa lso
ro : astimead ancesto inf inity .
neof theseq uasistandingwa esis
a p pr o i m at e ly e u a l to - 2 p wh e n th e t im e
# gt2isv erygreatincomparisonw ithp.
odisinfiniteat theorigin.Thisagrees
olemotionattheorigin w hich asw e
n ( 1 9 , w it h z = 1 a nd g = 4 i s e p r e ss ed
.. .. . .. . .. . .. . 1 4 7 .
thespacebetweenx = - 1andx = + 1
nterestingcharacter.Towardsafull
ument-cur etothesideof theoriginforx -negati e w e
ev a luesof iin( 146 : butforsimplicityw eha econf ined
positi ev a luesofx .
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A N C E IN I TI A TE D F R O M A G I V E N F O R M 4 2 9
maybecon enienttostudy thesimplif ied
€ ” t 2
- ) ep2. .. . . .. . . . (148 ,
f ( 1 9 gi esw henIgt2isv ery large in
lingz erosonthetw osides beyond
heorigin di idethew aterintoconsecuti eparts ineachofw hichit isw hollye le atedor
w emayca llha lf -w a es. They tra e l
easinglengthandpropagationalv elocity.
d e e l op e d af t er t = / r a s i t tr a e l s
irsttoama imumele ationorma imum
hatdiminishestoz eroastimead ances
tracetheprogressofeachof thez eros
nthetimesofoursi diagrams. Thisis
smar edonse era lo f thez erosinthe
confiningourattentiontotheleft-hand
eindiagram 1asinglez eronumbered1.
e numberedintheorderoftheir coming
, 3 ; 4 4 ... 10 1 0 . .. 3 3 , 3 3 ; . ..a ll in
sdiagram2show sz ero1considerably
thatis outwards ; andz ero2beginningits
m3 showsz eros1and2eachad anced
therthan2. Diagram4show sa llthezeros
istenceattime3 / 7r. Thesearez eros1
w ardsthanattimew / 7r andapa ir 3 , 3 ,
istenceshortlybeforethetime- / Tr.
elsoutwardsandtheinnerinwards.
meintoe istencebetween3 and3 : la ter
istencebetween4and4.
haspassedoutof rangele f tw ards: but
wa r d z e r o s 2 3 , 4 5 6 7 8 9 a n d in d ic a ti o ns o f t he i n wa r d z e r o s 9 8 . T he w ho l e tr a in o f z e r o s f or
eallycontinuedtothe middlebynumbers
5 6 7 8 9 9 8 7 6 5 4 3 ; s i t een in all.
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R
tof therangeofdiagram6 butw eseein
eros4 5 . . . 12 andanindicationof the
w h ic h h as c o me i n to e i s te n ce b e fo r e th e t im e 8 W / a r.
sfortime8 V /T indicatedbynumbers is
, 3 3 , 3 2 .. .4 3 ; s i t y- fo ur in al l.
the Indef initeE tensionandMultiplicationof
a l Deep-SeaWa esInit ially F inite
7.
standfree afterbeinginitially
nofa finitenumberofsinusoidal
f i emounta insandfourva lleys inthe
eSociety.Theinitialgroup ofwa es
f ig. 35 isformedbyplacingsidebyside
( t a e n a s un i ty , n i ne o f t he c u r e s o f
a lternatelypositi eandnegati e . Diagrams
aremadebycorrespondingsuperposit ionsof
5and6 o f f ig. 3 4. Thusw hat according
ep - se a p er i od i c wa e s ( ~ 1 9 a bo e w o ul d
y thewa e- length if thenumbersof
f inite lygreat w ouldbe2 andasw e
eperiodw ouldbe/ 7r andthepropagationa l
7r.
ewaterisleft free thedisturbance
otwogroupsofw a es seentra e ll ing
themiddle lineofthediagram.The
twogroupse tendrightwardsandleftwardsfromtheendof theinitialsinglestatic group farbeyond
s " supposedtotra e latha lf thew a e elocity w hich(accordingtothedynamicsofOsborneR eyno lds
mportantandinterestingconsiderationof
feedauniformprocessionofw ater-w a es
f thefreegroupsremaineduniform.
grealisedis illustratedbythe
w hichshow agreate tensionoutwardsin eachdirectionfarbeyonddistancestra e lledathalf the" w a e elocity . Whilethere isthisgreate tensionof thef ronts
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/ M ..
m n . i
J 1 3 5 1 0 1 0 5S
f f i ee le ationsandfourdepressionsemergingastw ogroupstra e ll inginoppositedirections.
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R
w eseethatthetw ogroups a f ter
stence inthemiddle tra e lw iththeirrears
ebetweenthemofwater notperceptibly
ryminutew a eletsine er-augmenting
ndslowerintherear ofeachgroup.
ereartra elsataspeed closelycorrespondingtothe" halfwa e- elocity " foundbySto esase actly
suniformsuccessionofgroups produced
o-e istentinfiniteprocessionsof
ingslightlydifferentw a e- lengths.
arv elocityisillustratedindiagrams
ndiagram1 R indicatestheperceptible
upcommencingitsrightwardprogress
R showsthepositionreachedattime
s byanidealpo inttra e ll ingrightwardsf rom
eedofha lf thew a e- e locity . This
ndstoafairlywell-mar edperceptible
e ll inggroup.
F , i n t h e t hr e e di a gr a ms o f f ig . 3 5 a n d f f
agram1 F mar saperceptible f ront
ngcomponentgroup.Indiagrams2and
lpo intstra e ll ingrightwardsf romitatspeeds
f wa e - e l oc i ty a n d th e w a e - e l oc i ty . W e
s tu r ba n ce f a r in a d a n ce o f F , F a n d v e r y
w a e-disturbance inf ronto f f f . Thus
elsatspeedactuallyhigherthan the
dthisperceptible f rontbecomesmoreandmore
thewholegroupw iththead anceof time
f ig. 9of~ 20abo e.
bythesediagramshownearly the
cityisfoundin therears:whilethefronts
eaterandwithe er- increasingve locity .
ationsandgraphicalconstructionsof
dtocorrespondingconclusionsinrespecttothe
sion gi eninit iallyasaninfinite lygreat
dalw a estra e ll inginonedirection.
d10 showedrespecti ely attwenty-fi e
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F A T RA I NO F W A V E S
commencement a f ronte tending
ndaperceptiblerear laggingscarcelytwo
po int tra e ll ingf romtheinit ia lposit ion
half thew a e- e locity .
itia tionandC ontinuedGrow thofaTra inofTwoDimensiona lWa esduetotheSuddenC ommencementofa
V ary ing Surface-P ressure . ~~ 118
gofaf initesinuso idallyvary ing
dk eptthrougha llt imeapplied tothe
a finitepracticallylimitedspaceon
eofthedisturbance.In thebeginning
ereatrestanditssurfacehori onta l. The
f indthee le ationordepressionof thew ater
mid- lineof thew or ingforci e andat
ebegantoact.
119-126 letusconsidertheenergy
fsinusoida lw a es inastra ightcana l
ydeep withv erticalsides.Ifthe
tby anypressureonitsupper surface
lfunderconstanta irpressure w ek now
eticsthatitsmotionwill beirrotational
o lumeof thew ater: andif a tany
esurface isbroughttorest suddenlyor
rate erydepthw illcometorestatthe
faceis broughttorest.This aswe
truee enif the init ia ldisturbance isso
of thew atertobrea aw ay indrops: and
yforeachportionofthewaterdetached
nthecana l asw ellasforthew ater
ifstoppageofsurfacemotionismadefor
before itfa llsbac intothecana l.
onof thew aterisirro tational w eha e
* * * * * * * - ( 1 49 ,
. . .. . ( 149
elocity-potentia l F ha ingbeenta enas
28
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R
al( ~ 97abo e . A ndbydynamicsfor
s i n ( 6 4 o f ~ 3 8 a b o e
g z - + C . ... ... ... 1 50 .
ondition letz = 1 betheundisturbed
ethevertica lcomponentdisplacementofa
aer ta enpositi ew hendownw ards
tsurface-pressure andta e-asthe
n st a nt C . T hu s ( 1 5 0 g i e s a t t he
, t + g-dt ( , , t + ( 5 1 .
esecondandthirdmembersofthis
rbancebeinginfinitelysmall which
1 + C l t - d F ( x , 1 t a n in fi ni te ly s ma ll q u a nt it y
egligible incomparisonw ithgt1 w hichisan
ofthefirstorder.
e -disturbanceofw a e- length27r/ m
t h v e l oc i ty v , w e h a e a s i n ( 6 6 a b o e
= - k e -n z - l s in m( x - v t ... ... ... 15 2 .
151 becomes
- t - g ( . .. .. .. .. .. .. .. 1 5 ) .
ationofthefreesurface
t . . .. . .. . .. . .. . .. . . 1 5 4 ,
n v / g. . . .. . . .. . . .. . . .. . . .. . . . 155 .
1 52 w it h z = 1 w e fi nd
x - v t . .. . .. . .. . .. . .. . .. 1 5 6 .
154 gi es
2rr.. . . .. . . .. . . .. . . .. . . . 157 .
a c t i i t y t h e ra t e of d o in g w or b y
n onesideuponthewateron theother
x ) . W e h a e
- L g - 1+ ( C ] . .. 15 8 .
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W T H O F A T RA I NO F W A V E S
n d F b y ( 1 4 9 a n d ( 1 5 2 , w e f in d
v t d e -m z - [ - k m e- m z -1 c o sm ( - v t
.. 1 59 .
operations d , w ef ind
m ( - - v t + g ( - + - ( 160 .
that27r / m istheperiodictimeof the
byW thetota lw or perperiod doneby the
deof theplane( x ) uponthew ateron
h a e
T - = 2 f 2. .. . .. 1 6 1 .
mparethiswiththetotalenergy
+ P perw a e- length. Inthef irstplace
ek ineticenergy K , andthepotentia l
thedensityof thewaterbe ingta enas
2 . .. .. .. .. .. . 1 .6 2 ;
. .. . .. . .. . . 1 6 ) ,
facedisplacement.
52 w e fi nd
c os m ( x - v t . .. .. .. .. .. . 1 6 4 ;
z - 1 s in m ( x - v t . .. .. .. .. .. .. .. ( 1 65 ;
. .. . .. . .. . .. . .. 1 5 6 r e pe a te d .
- -- -2 2 .. .. .. .. k ( 1 66 ;
1 6 6
. . .. . .. . .. . .. . .. . . ( 1 6 7 ,
by (157 .
k ineticenergyperw a e- length
erw a e- length areeache ua ltothe
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R
by thew ateronthenegati eside uponthe
de o fanyvertica lplaneperpendicular
thecanal. Thusw earri eatthe
now nconclusionthatinaregularprocessionofdeep-seaw a es thew or doneonanyv ertica lplane
yperwa e-length.Thisisonly
ularprocession ad ancingto inf inity
t tra e ll ingw iththew a e- e locityv .
edanideal processionofregularperiodic
ptly tonothingataf ronttra e ll ingw ithha lf
v ; w h ic h i s O s b or n e Re yn o ld s * i m po r ta n t
octrineof " group- e locity .
sionof~ 125isv eryimportantand
two-dimensionalship-wa es.Itshows
regularperiodictra inofw a esintherear
i n e s ti g at e d in ~ ~ 4 8 -5 4 a nd 6 5 -7 9 a bo e
f thespacetra e lledby theforci e f rom
motion butthatitwouldbee actly
difyingpressureweresoappliedto
aras tocausethewa estoremain
ndof thetra in w ithout onthew hole
em orta ingw or f romthem.
ntisapplicabletoour presentsub ect
6 157below .
andf irst insteadofasinuso ida lly
agineappliedaseriesof impulsi epressures
esacertainv elocity-potentialupon
ousimpulses andletitbere uiredto
ity -potentiala tany timet a f tersome
ses.Considerfirstasingle impulseat
say atatimeprecedingthetimetbyan
locity -potentia la tt imet duetothatsingle
liert imet-q bedenotedby
. .. .. .. .. .. .. .. .. .. ( 1 6 8 .
the instantaneouslygeneratedve locitypotentia lisC V ( x , z , 0 , andtheva lueof thisatthebounding
andB rit. A ss. R eport 1877.
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F A T RA I NO F W A V E S
( x , 1 0 . Hence bye lementaryhydro inetics if I denotesthe impulsi esurface-pressure w eha e
) . . .. . .. . .. . .. . .. . .. . . 1 6 9 .
cessi eimpulsesattimepreceding
q 1 q 2 . .. q i a nd d en ot in g by S ( x , z , t t he
ocity -potentia lsatt imet w ef ind
C1V ( x , z , q , ) + CV ( x , z , q 2 + ... CiV ( x , z , q i ... 1 70 .
estobeat infinitelyshortinter alsof
ormula( 170 intothe languageof the
s: S ( x , z , t = d f ( t - q ) V ( , z , q ) . .. .. .. . ( 1 7 1 ,
no t es a n a rb i tr a ry f u nc t io n o f ( t - q ) a c co r di n g
sure arbitrarilyappliedattime( t-q ) ,
- f t - ) V ( x , 1 0 . .. .. .. .. .. . 1 72 .
dtothe surfaceattimet denotedby
i sa sf ol lo ws :
-f t V ( x , 1 0 ........... 17 ) .
or( 171 gi estheve locity -potentia l
ichfollowsdeterminatelyfromthe
din~ . 127 128. F romit bydif ferentia tionsw ithreferencetox andz andintegrationsw ithrespecttot
mentcomponentsI o fanyparticleof
natesw erex , z w henthef luidw asgi en
em moredirectly andwithconsiderablylesscomplicationofintegralsigns bydirectapplicationof
gasthatusedin( 170 , ( 171 . Thus
z , q ) i n( 1 71 , w es ub st it ut ed V ( x , z , q ) , a nd
z , q ) , w e f in d ~ an d. A nd i f we t a e
q ) and fd d V ( x , z , q ) ...... 174
, q ) i n ( 1 7 1 , w e fi nd t he t wo c om po ne nt s ~ , 4
particle ofthefluid.Confiningour
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W T H O F A T RA IN O F W A V E S
x , z , 0 , w ou l d be r e ac h ed a n d pa s se d t hr o ug h
gati etopositi e . It isclearthatthe
z , q ) a re e u al f or e u al p os it i e a nd n eg at i e
w he n q = 0 w e ha e
. .. . .. . .. . .. . .. . . 1 8 0 .
h e V ( x , z , q ) , d e f i ne d i n ~ 12 7 w h i c h
0 t o b e an y a rb i tr a ry f u nc t io n o f x , b u t re u i re s
h e n q = 0 s u gg e st s a n al l ie d h yd r o i n et i c
ing(179 w ithW inplaceofV ; and
i n g W = 0 a n d d W / d a n y ar b it r ar y f un c ti o n
siscon enientforourpresentpurpose that
= 0 an dW ( x , z , 0 - 0.. ... ... . 1 81 .
aluesofx andz , largeorsmall
,
0 a nd W ( x , z , q ) 0 .. ... ... . 1 82 .
emtheinit ia tiona lcondit ionis: -displacementz eroandinit ia tiona lve locityv irtua llygi enthroughoutthe
sultof anarbitrarilydistributedimpulsi epressureonthesurface.
iationalconditionis:-thefluidheld
epttoanyarbitrarilyprescribedshapeby
leftfreeby suddenandpermanentannulmentofthispressure.
uestionofacompletesolutionof this
mforanyarbitrary initia tiona ldata w ef indaclass
ntso lutionsinaformulaorigina llygi enin
oya lSocietyofEdinburgh J anuary 1887
g. F ebruary 1887 andusedin~ 3
maynow w ritethatformula inthefollow ing
d e p r es s io n f or V or W : R S o r { R D d e 4 Z + x )
t , w he n i se e n
, w he ni is e o dd en
t , wh en ii sod d
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R
ma y i n s t ea d o f ( 1 8 ) , t a e t h e fo l lo w in g a s
e : di I ' - t 2
B -d t i ( 1 ) e 4 ( z + x )
t , w he ni is e e n
t , wh en ii sod d
1 7 1 a n d ( 1 7 5 , r e ma r t h at i n te g ra t io n
x , z , q ) = f ) V ( x , z , t - f t V ( x , z , 0
x , z , q ) ... 184 .
adratureorotherwiseweha ecalculated
S ( x , z , t , a s g i e n b y ( 1 7 1 , f o r b ot h f or m s
6 b e lo w w e c an f i nd t h e v e r ti c al c o mp o ne n t
icleof the li uidby ( 175 , w ithout
ormula(184 a lsoshow show bysuccessi e integrationbypartsw ecanreduce
( x , z , q ) . .. .. .. .. .. .. .. 1 85
x , z , t , a s e p r e ss ed i n ( 1 7 1 .
w to ~ 1 2 8 1 2 7 1 1 8: t o m a e t h e a p pl i ed
ary ingpressureput
t - ) . .. .. .. .. .. .. .. 1 86 ;
m a e s
V ( , , 0 . .. .. .. .. .. . 1 8 7 .
fullyw or outourproblemfortw o
nofpressure correspondingtothetwo
, f d e sc r ib e d in ~ ~ 9 6 -1 1 a b o e . F o r t h is
henotationof~ 101
( x , z , t ;
t = ( , z , t = - 0 ( x , z , ... 188 .
a llthesetw ocasescasefandcase~ .
171 and( 175 , e pressingrespecti elythe
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W T H O F A T RA IN O F W A V E S
andthev erticalcomponentdisplacementof
ny time become
= d sin a( t -q ) 0 ( , z , q ) ;
= d cos o ( t- ) ( x , z , q ) . .. 1 89 ; . 0 sin
d | in d ( t- q ) - ( x , z , ) ;
- d s in o( t - q ) ( X , , ) . .. 190 .
f ig s . 3 6 3 7 3 8 a r e ti m e- c ur e s i n
ebeencalculatedbycontinuousq uadrature
ou r f or m ul a s ( 1 8 9 ( 1 9 0 .
. 3 9 b e in g s pa c e cu r e s i n wh i ch t h e
mponentdisplacementsofthewatersurface arethereforepicturesofthewater-surface( greatly
ttoslopesofcourse andmaybeshortly
es.Theirordinatesha ebeencalculated
scribedin~ 151below.Theycannot
y forsuccessi ev a luesofx by the
uadratures ifthatwerethe method
of theordinateforeachv a lueofx w ould
anindependentquadrature( d ) f rom
oftfor whichthewater-surfaceis
e . Theva luesof tchosenforf ig. 39are
+ 1 /8 7 ( i + 2 /8 7 ( i + 3 /8 7 ( i + 4 /8 T whe re
ger andrdenotes27r/o theperiodof the
retowhichthefluidmotionconsidered
a e t a e n c o = V / T r w h ic h m a e s
g = 4 a s in ~ 1 0 5 m a e s t he w a e - le n gt h
7 a l l th e c ur e s c or r es p on d t o
ormulas. Inf ig. 38 a llthecur escorrespondtosinco( t-q ) intheformulas.
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le.
”
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le.
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R
tionsof t imescorrespondtocoso( t-q )
ecur es w iththe inscriptionsa ltered
3 / 8 r ( i + 4/ 8 T ( i + 5 / 8 , ( i + 6 /8 , c o rr es po nd
t he f o rm ul a s.
resentingv e locity -potentia lsandasurface
thecur esshowsanyperceptiblede iation
ptwithinperiod1.Towardstheend of
ndby theq uadraturesshow de iations
hingtoabout1/10percent. andimperceptibleinthedrawings.Thispro esthatsinusoidalityise act
ghall timeaftertheendof thefirst
nperiod1 how nearly therise
ow s t he s a me l a w fo r S ( 0 1 t a n d
standingthev astdif ference inthe law of
representedby ( 188 , forthesetw o
nitiatingsurface-pressurecommences
m a i m um v a l u e - / 2 f or c a se ~ , a n d
theformeris2 8 t imesthe latter.
esubse uentv ariationsofv elocitypotentialshowninthefirstandthird cur esare' 954forcase
ofw hichtheformeris3 ' 00timesthe latter.
a n d fi f th c u r e s o f fi g . 3 7 s h o w a t a
engthf romtheorigin thecompletehistory
dofsurfacedisplacementthroughalltime
cationofpressureto thesurface.The
ccuratesinusoidalityofeachofthesethree
6 7 8 showsthatthecontinuationthrough
sesinusoidal.
withtheinitial agreementbetween
nd S. ( 0 1 t , t ow hi ch we al lu de di n~ 1 9 w ef in d
m ar a b le c o nt r as t b et w ee n S . ( 8 1 t a n d
oughoutthew holeof thef irstperiod. R emembering
ensity thepressure ise ua ltominusthe
ev elocity-potentialperunitoftime
d is p la c em e nt p ( 0 1 t i s a s i s sh o wn
early z erothroughoutthef irstperiod andthat
certainly stil lmorenearly z erothroughoutthef irst
enocur etorepresentit w eseethatthe
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le.
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W T H O F A T RA IN O F W A V E S
t s of t h e sl o pe s i n th e c ur e s f or S ( 8 1 t
r e pr e se n t v e r y ne a rl y t he v a l ue s o f th e a pp l ie d
hew holeof thef irstperiod . Loo now
n e ar t o z e r o i s * ( 8 1 0 , a n d h ow f ar f r om
; a n d w e se e d yn a mi c al l y ho w i t is t h at S . ( 8 1 t
hr o ug h ou t t he f i rs t p er i od a n d S ( 8 1 t i s
andissomew hatneartobe ingsinuso ida l.
ery instructi ecomparisonbetw een
a nd S ( 8 1 t . In th eb ca se f or v a lu es of x a s la rg e
pproachsomew hatnearly tothecaseofa
formsurface-pressureo eraninfiniteplane
therew ouldbenosurfacedisplacement and
he surfacewouldbeate eryinstant
ace-pressureplusthe gra itationalaugmentationofpressurebelowthesurface.Thusweseewhyit is
odicv aria tionofappliedsurface-pressure at
ce lyany riseandfa llo f thesurface le e lthere
alffromthebeginningof themotion
fo r ( 8 1 t .
h andsi thcur esof f ig. 37represent
s es o f d is t ur b an c e S . A S o , a t x = 3 2
theorigin.Ifthe frontofthedisturbance
hew a e- e locity thedisturbancesof the
commencesuddenlyatthe endofperiod4.
1 t a nd ( 3 2 1 t t he d ia gr am s ho ws t ha t
ptibleattheendofperiod4 andbeginto
dofperiod8 w hichw ouldbethee act
wasadef inite" group- e locity " e ua lto
. T he l a rg e ne s s of S o ( 3 2 1 t , a p pr o i m at e ly
rstfourperiods ise pla ineding140.
throughperiods5 6 7 8 dependson
fdisturbancesf romtheorigin asshownfor
t a nd A ( 3 2 1 t i nt he se co nd an df ou rt hc ur e s.
t c ur e o ff ig . 3 8 m ay b ec om pa re d wi th
esignationinfig. 36. Theydif ferbecause
renceinthephase ofcommencementof
whichcommencessuddenlyatits
wa r d or d in a te s i n al l t he c u r e s o f fi g s. 3 6 3 7 3 8 3 9
a luesof theq uantit iesrepresented.
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R
r esof f ig. 36 andcommencesatzero
3 8 . I f t he S , S c u r e s f or i n it i at i ng
eroweredrawn theywoulddifferfrom
sof f ig. 36inbe ingatthecommencement
bscissas insteadofbeinginclinedto it
asshowninf ig. 36. The ' cur esare
lineofabscissas butthetangency
f ig. 36 w hile it iso f thesecondorder
cur es o f f ig. 38show thew hole
= 0 andx = X , o f thesurfacedisplacement
u la s w h ic h e p s 1 t h e s ur f ac d a s i n c nt d e t o s 1 ) . .. 1 9 1
facedisplacementduetosurface-pressure
in W t ( x , 1 0 . .. .. .. .. .. . 1 9 2 .
8showsthehistory afterperiod3 , to
eriod 9 ofthedisturbanceatthe
isturbancehasnotyetbecomesinuso idal but
moste actlysinusoidalafterafewmore
e ts o f f i e c u r e s s ho w f o r ca s e b an d
alyvary ingw ater-surfaceoneachsideof the
ughtimeafter thebeginningofthe
ularregimeofsinusoida lv ibrationasfaras
thsoneachsideofthe middle.Thethird
ur eofsines. Thef irstcur erepresents
ingofaperiodf romirto( i+ 1 r. The
f irstcur e in erted representsthew atersurfaceatthemiddleof theperiod. Theothertw ocur esmay
tsofthefirst andthird accordingtothe
P s i n wt - Q c o s ac t. .. .. .. .. .. . 1 9 ) ,
cos27r / X . . . . . .. . . .. . . .. . . ( 194 ,
sof thethird fourth andf ifthcur esof f ig. 38isdouble
d indicatedonthefigure.
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F A T RA I NO F W A V E S
anscendenta lfunctionofx , ha inge ua l
p r es s ed b y ( 1 9 5 f o r po s it i e o r n eg a ti e v a l u e s
w a e - le n gt h .
= - A si n2 7r / . . 1 95
= + A si n2 7r / . .. . " '
i-amplitudeof thev ibration atany time
nning andplacefarenoughfromthe
toha ev eryappro imately sinuso ida l
nofthetranscendentalfunctionQ , and
rbothPandQ , w il lbev irtuallyw or ed
e ceedingly interestingandsuggesti e
cesrepresentedinfig.3 9. Consider
scorrespondingtoPsin cotalone andto
motionPsincot if a tany instantgi en
c o w ou l d co n ti n ue f o r e e r a s a n in f in i te
s withoutanysurface-pressure.Henceour
ssureisonlyre uiredfortheQ -motion:
y instantgi enf romx = -otox = + o
p r o i d ed t h e pr e ss u re - c o s co t ( x , 1 0 i s
dtothesurface.
maybegenera lisedasfo llows: Displacethew ateraccordingtotheformula( 19 w ithPomitted
functionofx formoderatelygreat
a luesofx , gradua lly changingintothe
siti eandnegati ev a luesoutsideany
MON( MO notnecessarilye ua ltoON .
esinusoidallyv aryingsurface-pressure
e uiredtocausethemotiontocontinueaccordingto
uponthemotionthusguidedbysurfacepressure themotion-A cos27r / X . sincot w hichneedsnosurfacepressure . Inthemotionthuscompounded w eha ee ual
e ll ingoutw ardsinthetw odirectionsbeyond
: a n d i n t he s p ac e M N w e ha e a v a r y i ng
perimposingonthemotionPsin cotan
v aryingsinusoidallyaccordingtothe
29
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R
gdynamicalconsiderationisnow
mponentofmotionneeds asw eha e
ure. TheQ-componentofmotionisk ept
ssureF ( x ) cosot w hich inaperiod
theQ -motion butw or mustbedoneto
otra insofwa estra e ll ingoutw ardsin
thisw or isdoneby theacti ityo f
theP-componentofthemotion.
estionisforceduponus. O ursolution
enusdeterminate lyandunambiguously in
casesconsidered themotionofe eryparticle
espaceoccupied. Thesynthetic
swhichw eha eusedcouldleadtonoother
o theappliedsurface-pressure but
ha e c o ns i de r ed a Q - m o t io n a lo n e k e p t co r re c t
ssure.W ouldthismotionbeunstable
uldit inasuf f iciently longtimesubside into
nthedeterminateso lutionof~ 1 5-145
Atanyinstant sayatt= 0 letthe
omponentaloneof~ 148.Letnowthe
x ) coscot besuddenlycommencedandcontinuedfore eraf ter. Itw il l accordingto~ ~ 1 5-145 produce
ompoundmotion(P Q ) w hichwillbe
otione istingattimet= 0 andthis
gi enw ithitsinf initeamountofenergy
otox = + oo andle f twithnosurfacepressure w ouldclearlyne ercomeappro imate ly toquiescence
ncefrom0 onthetwosides.Thuswe
-motiona loneof~ 148isessentia llyunstable
oesnotsubsideintothe determinate
Itw ouldsosubside if itw eregi en
initespacehow e ergreat oneachside
endistributionofdisturbancethroughany
eatoneachsideof0 le f tto itse lfw ithout
pressure becomesdissipatedawayto
andlea es asil lustratedin~ 96-11 ,
aceoneachsideof0 throughw hichthe
ndsmallerastimead ances.
intosomeoftheanalytical
actical wor ingoutofour solutions
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W T H O F A T RA IN O F W A V E S
T a i n g co s o( t - q ) i n th e fo rm ul as a nd t a i ng c as e
P c o so at + Q s in ot .. .. .. .. .. .. 1 96 ;
z , q ) ; a nd Q d sin w o ( x , z , q ) . .. 197 .
beenthusfoundbyquadratures fora ll
particularv a lueofx , by integrationbyparts
w ereadily f ind w ithoutfartherquadratures
essionsforthese enotherformulasincluded
.
Q fort= oo . U singthe
g i e n by ( 1 7 , w ef in d
c os w~ q e - m ;
d s in w e cm 2 .. .. .. .. .. . ( 1 9 8 ,
+ t X ) .
e a luationgi enbyLaplace in1810*, w e
199 .
isatranscendentfunctionofo andm
ntermsoftrigonometricalfunctionsor
ng t h e se r ie s f or s i n w i n t er m s of ( a ) 2 i + 1
2+ 16m 2by integrationsbyparts w efindthe
riesforthee a luationofQ, fort= oo
1 12 T n 2 . 1. 3 2 r . 1. . V )
m et c
i ~ ~ ~ . .. 2 00 ) ,
S X ( ( t \ I P CO
2 ..
7 + e tc .
O t
a bo e p = V / z 2 + X 2 , an dX = t an- ( x / ) .
i tu t 1 8 10 . S ee G r eg o ry s E a m pl e s p . 4 80 .
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R
re eryv alueofoV /phowe ergreat.
pgreaterthan4 itdi ergesto large
negati etermsbefore itbeginstocon erge. The largestv a lueofo^ / pforw hichweha eusedit
re sp o nd i ng t o x = 8 a n d re u i ri n g f o r th e
enty-onetermsoftheseries.B utforthis
ra ll la rgerva lues w eha eusedthe
r ie s ( 2 0 8 , f o un d i n e p r es s in g a na l yt i ca l ly n o t me r el y f or t = c a s i n ( 1 9 8 , ( 1 9 9 , ( 2 0 0 , b u t fo r a ll
eatandsmall thegrowthto itsf inal
ofthedisturbanceproducedbyour
licationofpressuretothesurface ofthe
atrest. Thecur eforirinf ig. 39has
by ( 200 forv a luesofx upto8 andby
tseriesforv a luesofx f rom5to10. The
eofthev alueswhichwerecalculated
theult imatelydi ergentseries( 208 ,
oalsowastheagreementbetween
u a d r at u re s f or x = 1 a n d x = 8 w i th v a l ue s
= 1 andby ( 208 forx = 8. It isa lsosatisfactory thattheva luesofP foundbyquadratures forx = 1
we l l wi t h th e ir e a c t v a l u es g i e n b y ( 1 9 9 ,
t t h e e p r e s si o ns ( 1 9 7 f o r P an d Q ,
iousanaly ticalmethodof treatment w e
herefore( ~ 150 a llourotherformulas to
functiondefinedasfollows:
.. . .. . . .. . . .. . 201 ,
mathematicians throughthelast
ndredyears inthemathematical
e f raction andinthetheoryofProbabil it ies. Iha eta enEasanabbre iationofGla isher stnotation
hatheca lls" ErrorF unctionC omplement "
maticaldisco ery de- 2= / 7r seemstoha ebeen
0.
ber1871.
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R
ergesforallv aluesofa- greatorsmall real
o n e r ge s i n it s f ir s t i te r ms i f 2 c2 & g t 2 i - 3
a-2isimaginary andaf terthatit
a luebeingintermediatebetw eenthesumof
andthissumw iththef irsttermof the
Theproperruleof proceduretofindthe
reeof accuracy istofirst calculateby
tseries andseew hetherornotitgi es
ugh. If itdoesnot usethecon ergent
ch bysuf f iciente penditureofarithmetical
lygi etheresultw ithanydegreeofaccuracy
nly fornumerica lca lculation butfor
o f thedesiredresultw ithoutca lculation it
emodulusesof thethreecomple argumentsof thefunctionE in( 205 , and( 206 . Theyareas
m = / T - s + t - + J
9 I
o c .. .. . . 2 0 9 ;
g ta - 0 2P
= t / -~ + /
\ 4 p p g )
. 2 1 0 ;
.................. 211 .
gq uestionsregardingthefrontof
sine itherdirection o fw hichw eha e
36 3 7 3 8 andw hichw ehadunder
- 1 114-117abo e arenow answ erable
ymathematicalmanner byaidofthe
2 06 , ( 2 09 , ( 2 10 , ( 2 11 . W h en i n th e ar gu me nt s of E i n ( 2 0 5 , a nd ( 2 06 , V / m t is v e r y gr ea t in c om pa ri so n
e tw o a dd e d te r ms i n ( 2 0 5 a r e ap p ro i m at e ly
i s r ed u ce d a pp r o i m at e ly t o i ts l a st t e rm a n d
( 1 9 0 b e co me a p pr o i m at e ly s i nu s oi d al i n
sew hent/ - isv erygreatin
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W T H O F A T RA IN O F W A V E S
andincomparisonw ithw / asw esee
u se s s ho w n in ( 2 0 9 , ( 2 1 0 , ( 2 1 1 . T h is
h e ar g um e nt s o f E in ( 2 0 5 ( 2 0 6 a n d
antre lati e ly tot.
arge andx notsosmallastogi e
ttermsof themoduluses( 209 , ( 210 ,
( 2 06 , ( 1 89 , ( 1 90 a f ul l re pr es en ta ti on o f th e
ew a e- f ront e tendingf romx = obac
thata llow spreponderanceof t/ g
du lu se s ( 2 09 , ( 2 10 . L et f or e a mp le
.
e - e l oc i ty x t i m e. .. . .. 2 1 ) .
efinedis whatinmyfirstpaper tothe
h( J anuary1887 , " O ntheF rontand
on o f W a e s i n De e p W a te r , I c a l le d t he
fn ed i n ( 4 5 o f t ha t p ap e r w h ic h a gr e es w i th o u r
fo llow ingpassagew astheconclusionof that
oaw holly f reeprocessionofw a esmay
aftertheconstitutionof thefronthas
bysuperimposinganannullingsurfacepressureupontheoriginatingpressurerepresentedby ( 12 abo e
) o fourpresentpaper , a f tertheoriginating
edsolong astoproduceaprocessionof
a es. The instructionthusgi enw ith
tweenfrontandrear hasbeenv irtually
somedif ferencesofdeta il in~~ 20-24of
paper onthesamesub ect andunder
20 1904 t. Thatsecondpapercontaineda
calculationsandgraphicrepresentations
thepresentpapercontains inf igs. 35
furtherinstalmentofsuchil lustrations.
, % ~ 9 6 -1 5 7 I h a e h a d mo s t
-t i tl e o nl y .
.
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R
mMrGeorgeGreen notonly inthev ery
tionsanddraw ings w hichha ebeen
btalsoin manyinterestinganddifficult
dinthefundamentalmathematicsofthe
ongthe methodof~ 128tocalculate
( x - v q , z , t - q ) , t he i ni ti at io n an d
lShip-w a es duetothesudden
nuedapplicationofamo ing steady
, 1 0 . W e h o pe a l so t o a pp l y ( 1 9 o f t he
mentofmyoldpromise( ~ 3 0 J une20
lwiththebeautifullyv ary ingprocessionseen
placeof astonethrownintodeep
ev ariousgraphs ca lculatedinthesepapersasrepresenting
pesofdisturbance ha ebeenanalysedandv erif iedin
fromthepointo fv iew ofgroupv elocityorLordKel in s
ase( supra p. 304 , intw opapers" O nGroupV elocity
fWa esinaDispersi eMedium " P roc. R . S. Edin.
p p . 44 5 -4 7 0 a n d " O n W a e s i n a Di s pe r si e M e di u m re s ul t in g
trbance " P roc. R . S. Edin. V o l. x x x . 1909 pp. 1-12.
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4 57 )
T IO N O T H E MA C E R EL S Y .
cia tionR eport 1876 P t. II. p. 54 reprintedfromSymons s
ag a i n e V o l . x i . 1 87 6 p . 1 1 .
pla inedthere lationof thecloudsand
andthatitw asnotessentialto theformationof
hereshouldbetw odifferenttemperatures.
thatportionsofairshouldbe mo ing
er thattheupanddownmotionshould
from theslippingofonestratumof air
ductiontherebyofw a es andthesecond
her ofthetwoportionsof airshould
saturation-thatitwouldbeclearwhen
ndcloudywhenupatitshighest .
" UeberatmospharischeB ew egungen ( 1888 andfo llow ing
Papers V o l. III. p. 289se .
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[ 4 1
MATIC A LA NDDYNA MIC A LTHEOR EMS.
gsof theR oyalSocietyofEdinburgh V o l. v .
11 â € ” 1 15 .
ationswhichtheauthorhadbeenled
tha TreatiseonNaturalPhilosophy
tare abouttopublish hemetwith
ems w hichappeartobenew andof
Asthedetailsofthe in estigationswill
e rybrief s etchonly isgi enhere .
htwire ofuniformsection
rencetracedonitssurfacepara lle lto its
diculartothislinef romanypo into f the
erse theamountof torsionortwisto f the
ny form maybedeterminedby thefo llowing
hetangenttothea iso f thew ire atapo int
radiusofanunitspherebedraw n cutting
cur e . F rompointsof thiscur edraw
sesatthecorrespondingpointsofthe bar.
eof directionfromonepointtothe other
the increaseof itsinclinationtothetrans erse
hecorrespondingparto f thew ire .
curiousconse uencesfo llow o fw hichone
benta longanycur eonaspherica l
neofreferenceliesall alongincontact
T ai t s l Y at u ra l P hi l os o ph y ~ 1 2 .
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C A LA NDDYNA MIC A LTHEOR EMS
uiresnotw ist sothatw henanapple
peeled there isnotw istinthepeel.
rrowribbandbe laidonasurface
stw istisate erypo inte ua ltothe
alsystematrest andsub ectedtoan
agnitudeandinanyspecif ieddirection it
a ethegreatestamountofk ineticenergy
secangi e it.
po intsbestruc independentlyby
inamount morek ineticenergy isgenerated
freetomo eeachindependentlyofall
connectedinanyway.
asystematrest. Letanypartso f it
yw ithgi env e locit ies theotherparts
eirconnectionswiththose whichare
esystemw illmo esoastoha eless
ngstoany othermotionfulfillingthe
ons.
Ta it sNat. Phil. ~ 311.
Ta it sNat. Phil. ~ 3 12.
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R M O F C EN T RI U G A L G O V E R NO R .
gineersinScotland Transactions V o l. x II. No . 25
ora centrifugalgo ernoristouse the
eproducedbyincreaseof.speed without
stheforcetoproducethere uisiteregulating
ayof usingthisforceforthe purpose
pressure forafrictionalarrangement
gtherotatorymotion.
ntotheInstitution isofthisperfectly
presentsnono eltye ceptinsomedeta ils
portionofits parts.Itconsistsoftwo
MM(seeP late III. ) , eachsuspendedf roma
H attachedtotheshaf t S andturningw ith
chisv ertical. Thesemassesarepre ented
alforceby astoutringof gunmetal
diameter f i edhori ontally a taboutthe
nertia . B utthegreaterparto f the
cedbypow erfulsprings P drawingthe
th e a i s . F i r m s to p s F , a r e pl a ce d l e e l
ia topre entthembeingdraw ninwards
ninchfromthe positionwhichthey
egun metalring.W henthemachine
asingve locity thego erningmassesdo
suntilthecentrifugalforce uponthem
theforceof thesprings. A v erysmall
o ethatw hichf irstdetachesthemfrom
pressagainstthegunmetalring and
sistanceimpedingfurtheraugmentation
ringsofeachmassareatav eryconsiderabledistanceapart( 5inchesinthe instrumente hibitedto
re .
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O R MO F C EN TR I U G A L GO V E RN O R
planeperpendiculartothehori onta ll ine
tothea is. Thisgi esgreaterf irmness
suspendedmass v erynearlyasif itw ere
ta lshaf t butw ithoutthefrict ionw hich
ouldentail.I tallowsthehori ontal
ntheleadmasswithoutsensibly
andtobetransmittedto therigid
esist itsmotion.Thespringwhichdraws
is ismadeupof twopiecesofstout
edandtemperedproperly placedw iththeir
soneanother andpressedagainstoneanother
her andunitingthembystoutclamps.
adaptedforpullinginsteadofpushing.
the Institutioneachmassamountsto
resetbyanad ustingscrew sothat
otherbe ingtiedinbyacord beginsto
eandthesamespeedis reached.This
minute inthe instrumentasad usted
ution and therefore asthecentreof
out41inchesf romthea is itscentrifugal
tsw eight or48poundsw eight w hich
orcew ithw hichthespringw asad ustedto
eedisincreasedbyasmallpercentageabo e
hego erningmasstobegintopressupon
w hicheachwillpresswille ceedtheforce
esamepercentage.Thus ifthespeed
hatatw hichthego ernorbeginstoact
heringw ithaforceof ' 19ofapound
tional resistanceof.0ofapoundforce
be' 105.Thusthewholefrictional
masseswillbea-of apoundactingat
tf romthea is andconsuming therefore
nd.Toincreasethe speedfurtherby
somuchincreaseofdri ingpowerasto
rsecondmore.These figuresgi ea
rof thisgo ernorwhenusedsimplyto
itionalwor donebyadditionstothe
orethanasmall increaseofspeed.The
e thatthegreatestadmissiblepercentage
gi efrictionalresistanceamountingto
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ermittedchangeofdri ingpower and
ngpowerspentinf rict iononthepi ots
inproportiontothe latter. Itw as
formiscellaneouslaboratorypurposes
inwhichappro imatelyuniformspeed
anmaybeuseful forchronoscopesin
capparatus whetherforgi inguniform
ribbonofpaper asintheMorseandother
anica l" sending instruments.
wsaplan in entedbyProfessor
andintroducedbyhiminconnectionw ith
a lgo ernor tobeappliedtothepresent
ertedintoapow erfulsteamgo ernor.
thegunmetalring andsupportingitso
rotateroundthesamevertica la isasthe
ts. B yanycon enientmechanism a
samedirectionasthatof thego ernor
ndrotationinthecontrarydirectionaugment
sed andaspringorw eightappliedto
tterdirectionw henit isnotcarried
of thego erningmasses. Thus the
the Institutiongi esthemeansof
persecondofwor toactincuttingof f
espeedaugmentsbyonepercent.
inesa idthatthisw asago ernorof
indeed asSirWm. Thomsonhadsa id itw as
stofall principlesthatcouldbeapplied
ethere o l ingmassespressaga instthe
c edthespeedw henitbecametoo
attheprincipleonwhichthe go ernor
hadnotpre iouslybeenappliedinpractice .
efficacyinpreser inganalmostuniform
tthat itwouldbepracticableto adapt
oughnotprecise ly initspresentform but
dificationsit couldbeadaptedtothat
ethe meansofregulatingthespeed
recisionw hichtheyhadheardstated and
dif ferentia lgo ernor.
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O R MO F C EN TR I U G A L GO V E RN O R
dsee beforethemeetingbro eup.
attheUni ersity inSirWill iam
oratory ofseeingcontri ancesofthis
hattheyga eresultsastouniformityof
theyhadhearddescribedinthe paper.
chtheysaw onthetable thego ernor
nfriction.Itwaseasy tounderstand
onsitmightbe madetoactupona
ecut-of fo fasteamengine.
thatwiththeappliance towhich
couldberegulatedw ithv erygreatprecision forinstance easily soastok eepthespeedw ithinaha lf
mity.Inreferenceto whatProfessor
hepossibil ityo f thisgo ernorbe ing
es ago ernorgoingatdoubletheve locity
notbesensiblydisturbedby thero ll ingof the
osedthatinusing thisasasteam
rn g wo u ld b e r e u i re d f or t h e re o l i n g
henthespeed fellshort inordertoact
onthe regulator.
therewerespringsand stopswhich
ay. HemightdoeitherwhatDrR an ine
ghtbeaf rict iona lactionof thego ernorto
wlydescendingweightalwaysthrowingon
thef rict iona lactionba lancesoro ercomesit.
e effectedbyawheelcarriedround
oughta weightwouldbethemost
chwouldberunningdow nuntilthefull
beforefullsteamwasadmittedthespeed
ausethemassestopress againstthering
weightfromrunningdownany further
oesteamfromgettingin.
R N O M I CA L C LO C , A N D A P EN D U L U M
R U N I O R M M O T IO N .
v I I . J u n e 1 0 1 8 69 p p 4 6 8- 4 70 . Re p ri n te d i n
resses i i . pp. 387- 94.
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A T IO N S O T H E CO M P A S S PR O D U C E D
THESHIP.
heB rit ishA ssociationatB e lfast 1874 f romthe
V o l . X L V I I I . N o . 1 8 74 p p . 3 6 - 6 9 .
whichhasbeenin estigatedbyAiry
isthede iationof thecompassproduced
asaconstantinclinationof theshiprounda
pro imatelyhori onta l iscalled . Itdepends
entof theship smagneticforce introduced
ch compoundedw iththehori ontalcomponente istingw hentheshipisupright gi esthea ltered
hentheshipis inclined.Regardingonly
anddisregardingthechangeoftheintensity
wemaydefinetheheeling-errorasthe
ns fortheshipuprightand forthe
sultanto f thehori ontalmagneticforces
sitionofthecompass.Thesesuppositionswouldbe rigorouslyreali edwiththecompasssupported
manner ifthebearing-pointwere
mlyinastraightline. Theyarenearly
geshiptorenderinconsiderablethe
ectuniformityofthemotionofthe
o intisplacedanywhere inthe" a iso f
eshipthecompass how e erplaced isnot
ypitching orby the ine ua litiesof the
otioncausedbyw a es. Hence supposingthecompassplacedinthea iso f ro ll ing theperturbation
gwillbe solelythatduetothe
eofA rchiba ldSmith P roc. Roy . Soc. 1874 tobereprintedina laterv o lume a lsoanarticle inPopularLecturesandA ddresses V o l. III.
h ichformedthebeginningsofcompassin estigations.
y thebestinpractice o f f indingbyobser ationtheposit ion
ohangpendulumsf rompointsatdif ferentle e lsinthe
perpendiculartothedec til lone isfoundw hichindicates
as thosefoundgeometricallybyobser ingagraduated
) seenaga instthehori on.
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I O N S O F T H E CO M P A S S B Y R O L L I NG
nta lcomponentof theship smagnetic
hecompasswould-ha eonegreat
icationofpropermagneticcorrectors
awayw iththero ll ing-error w ould
g-error.Tosetoff againstthis
wopractica ldisad antages: -one thatthe
a lw aysbelow dec w ouldnotbeacon enientposit ionfortheordinarymodesofusingthecompass
ous , that ata lle entsinshipsw ithiron
ticdisturbanceproducedby the ironof theship
chgreateratanypo into f thea iso f
ychosenposit ionsabo edec , astomore
randk ineticad antageof thea ia l
sin shipsofv ariousclassesoughtto
found thatinsomecasesthe compass
tngad antage beplacedatthea iso f
er thisposit ionforthecompasshasnot
class and asw eha eseen it isnot
rbegenerallyadoptedforshipsof all
nterestingandimportantpractical
perturbationsofthecompassproduced
-uniformmotionsofthe bearingpoint.
lemofthecompassis todetermine
ofarigidbodyconsisting ofthe
andf ly -card w hichforbre ityw illbecalled
o ableonabearing-po int w henthispo int
nmotion. Letthebearing-po inte perience
cce lerationa inanygi endirection.
rw eight o f thecompass andgW the
rec onedink ineticunits. Theposit ion
of thecompassatthatinstantisthe
restunderthemagneticforcesand
itye ua ltotheresultanto fgW anda
oppositeto thatofa.Nowtheweight
tanditscentreofgra ity so low that
carcelyaffectedsensiblybythe greatest
encedby theneedles . Hence ink inetic
ustingcounterpo iseforthecompassisre uiredw henaship
thtoe tremesouthmagneticla titudes.
3 0
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hecompass-cardis sensiblyperpendicular
apparentgra ity def inedabo e and
eneedlesisinthedirectionof the
nts inthisplane o f themagneticforces
issimplythroughtheapparent
theshipoccupiedby thecompass dif fering
n- le e l thattheproblemof thek inetice uil ibriumposit ionof thecompassinaro ll ingshipdiffersf rom
errorreferredto abo e.Thatwemay
iesofour presentproblem letthere
heshipherse lforcargo. Thek inetice uil ibriumposit ionof themagnetica iso f thecompassw illbe
ponentofterrestrialmagneticforcein
tle e l. LetK bethe inclinationof this
ra itation- le e l andb& gt thea imuth
f rommagneticnorthof the lineLL o f
planes(adiagramisunnecessary ;
hori onta landv erticalcomponentsof
rce.Thecomponentofthis forcein
e lw il lbetheresultanto fHcos/ a long
Z s i n K p e rp e nd i cu l ar t o L L ; a n d th e re f or e i f & g t , d e no t e th e a ng l e at w h ic h i t i s in c li n ed t o L L , w e ha e
n K Z s i n Ic
t a n ~ c o s IC + r - .
Ho
uestions w erec onthedirectionsasof
oles( orthenorthernendsof the
hedirectionofHcosbisa longLL northw ards
nK, w hentheshipisanyw herenorthof
isdow nw ardsintheplaneof theapparent
nsideringtheef fecto f ro ll ing the
acce leration o f thebearing-po intwill
endiculartotheship slength and
aa lle lto the length. ( Itw il l infactbe
bber-points ofthecompass-bowl.
anglesareordinarilyread intheplane
etice uil ibrium-erroro f thecompassis
b . W h e n fc i s a s ma l l fr a ct i on o f 5 7~ ' 3
stheanglew hosearcise ua ltoradiushas
amesThomson w hichisthecase
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I O N S O F T H E CO M P A S S B Y R O L L I NG
egreesof ro ll ingwhenthecompassisproperly
p pr o i m at e ly
s forthenorthernendsofthe needles
emisphereorfor thesouthernends
e towardsthesideon whichtheapparent
is( aspractica lly thecompassisa lways
ing , tow ardsthee le atedsideof theship. It
e
tosay w hentheshipheadsnorthorsouth
mount considerperfectlyregular
ra lfulf i lsappro imate ly thesimpleharmoniclaw sothatw emayput
nationof theshipattimet andnand
heheightofthebearing-pointof the
abo ethea iso f ro ll ingw hentheshipis
ontof itsacce lerationw eha e
n 2h i.
gthofasimplependulumisochronous
p w eha e
-gh/ l. i .
gtangentialtothecircle describedby
pro imate lyhori ontal andthereforethe
ityw illbeappro imately thato f the
ri ontal.
. i a p pr o i m at e ly .
eadsnorthorsouth theamountof the
rrorisappro imate ly
ompass perniciouslyusedintoomanymerchantsteamers
ghro lling e periencede iationsofapparentle e lamounting
chsideof thetruegra itation- le e l.
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le theperiodof theroll ingtobe6seconds
odof the" seconds pendulum ) ; 1w ill
esthe lengthof theseconds pendulum .
stobe14- feetabo ethea iso f ro ll ing.
sothattherangeofapparentro ll ingindicated
pointinthepositionof thebearingpointofthecompassisgreater byhalfthanthetrue rangeof
ppositionsthek inetic-e uil ibriumerror
rit ishIslandsthemagneticdipis70~ ,
e ingthenaturaltangentof thedip ise ua l
hek inetic-e uil ibrium errorforthe
thislocalityto aboutadegreeand
degreeof ro ll.
ibriumvalueof thero ll ing-errorw ill
umof thek ineticerrorin estigatedabo e
byanin estigationreadilywor ed
SmithintheAdmiraltyCompass
SectionIV . pages82-89 andA ppendi ,
thmodificationtota e intoaccountthe
ntle e l a ttheplaceof thecompass f rom
el.
s si o n " k i n et i c- e u i li b ri u m e r r or t o
estigatedabo ef romthatactua lly
ss. It ise actly theerrorw hichw ould
passwithinfinitelyshortperiodof
c needle( e itherw ithsil - f ibresuspension
theordinaryw ay ha ingaperiodof
conds showstherolling-errorv ery
te ery instanta lmoste actly theposition
. Iha ethusfoundthero llingand
asmallwoodensailing- esselthatit
oma ee actobser ationsw iththequic
eF rithofC lydeoroutatseaontheA tlantic
e ceptiona lly smooth. Thew ell- now n
eforashipofanysi ee posedtoregularw a esof
stocrest and ifmo ingthroughthew ater mo ingina
crests.
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I O N S O F T H E CO M P A S S B Y R O L L I NG
cedoscil la t ions isreadilyappliedtoca lculate
anironship theactua l" ro ll ing-error
he" k inetic-e uil ibriumerror in estigated
ecompassatany instant f romthe
theshipw ereatrestandupright
scillationifunresistedby any
ce( thedampingef fecto fcopper introducedbySnow Harrisandusedwithgoodef fectinthe
ass beingincludedinthis
theamountof v iscousresistance
briumvalueof thero ll ing-error
g.
2 r /T a n d n = 2 7 r/ T . T h e di f fe r en t ia l
s
2 E c os n t .
toe presstheeffectofregular
4n 2f -2 '
2 n f
entcommunicationtoofarto enter
orthepresentit isenoughtosay
ofv iscousresistancecanma ethe
forpracticalcon enience unlessalso
s longerthanthatofany considerablerollingtowhichtheship maybesub ected.Probablya
econds( suchasanordinarycompass
cessary forgeneraluseatsea andit
cticalq uestionhowisthisbest tobe
thesmallnessofthecompass-needles
satisfactoryapplicationofthesystem
whichAiry proposedtocausethe
pointcorrectmagneticcourseson all
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R M O F A S r RO N O M I CA L CL O C W I T HF R E E
P EN D EN T LY G O V E RN E D U N I O R M M O T I O N
HEEL.
rNo. 42( 1869 supra PopularLecturesandA ddresses
isreproduced w iththefo llowingaddit ion. F rom
876 P t. i i . pp. 49-52.
hopeheree pressedhasnothitherto
earpassedproducingonlymore orless
ousmechanica ldeta ilso f thego ernorand
untilaboutsi monthsago w hen forthe
le ceptthependulumsinappro imately satisfactorycondit ion. B y thattimeIhaddisco eredthatmycho ice
he temperaturecompensationandlead
ulumswasamista e.Ihadfallen into
oughbeinginformedthatinRussiathe
enre ertedtobecauseofthe difficulty
mperaturethroughoutthelengthofthe
tstoppingtoperce i ethattherightw ay
wastofaceit andta emeansofsecuring
peraturethroughoutthelengthofthe
ob iousmaybedonebysimpleenough
isedapenduluminwhichthecompensationis
fz incandaplatinumwire placednearly
throughoutthelengthofthependulum
f thecloc showntotheB rit ishA ssociation
an.Nowit isclearthatthe materials
should o fa llthosenototherw iseob ectionable bethoseofgreatestandof leaste pansibil ity . Therefore
numoughttobeoneof thematerials and
stronomicalmercurypendulumisa
ttobetheother( itscubice pansion
are pansionofz inc , unlessthecapil la ry
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F A S T RO N O M I CA L C LO C
surfaceleadto irregularchangesin
Theweightofthependulumoughtto
testspecif icgra ityattainable ata ll
e istobemountedinanair- t ightcase
rors ofthebeste istingpendulumsis
ariationsofbarometricpressure.The
tsitouto f theq uestionforthew eightof
ena lthoughtheuseofmercury forthetemperaturecompensationdidnota lsogi emercury forthew eight.
odcompensationcouldbegot byz incand
means mercuryought onaccountof its
y tobepreferredto leadfortheweightof
madese era lpendulums( fort idegauges w ithnoothermateria linthemo ingpartthanglassand
dk nife -edgesofagateforthef i edsupport
ma ingfourmorefortwonew cloc s
eontheplanw hichformsthesub ectof
ehadnoopportunityhithertoof testing
hesependulums buttheiractionseems
esults andtheonlyuntowardcircumstance
edinconne ionwiththemhasbeen
intwoattemptstoha eonecarriedsafely
madeby MrW hitetoanorder forthe
new cloc , it isenoughto loo atthe
perfectsteadiness frommonthtomonth
entimetreoneachsideof itsmiddle
tsonly touchedduring3 -0of thetimeby
tofee lcerta inthat if thebestordinary
sanyofitsirregularitiesto v ariationsof
orto impulsesandfrict ionof itsescapementw heel thenew cloc must w hentriedw ithane ua llygood
reregular. Ihopesoontoha eittriedw ith
atof anyastronomicalcloc hitherto
wsirregularitiesamountingto-iofthose
oc s thene tstepmustbeto inclose it
tatconstanttemperature dayandnight
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D A S PO S S I B L Y A M O D E O M O T I O N .
n Pr o c. V o l . I . M ar c h 4 1 8 81 p p . 52 0 5 2 1 P o pu l ar
V o l. I. pp. 142-146 orpp. 149-15 inla terreprint.
t leo fhisdiscoursethespea ersa id:
ynda ll sbeautifulboo , Heat aModeof
uthwhichhasmanifestedfarand wide
e greatestdisco eriesofmodern
ysadmiredit Iha e longco etedit
b y k i n d p er m is s io n o f it s i n e n to r I h a e
ning sdiscourse .
goDanielB ernoullishadowedforth
elasticity ofgases whichhasbeenaccepted
endidlyde e lopedbyC lausiusandMa w ell
swayingsofacrowdtoobser ation
eepathof anindi idualatominTait
ationofC roo es granddisco eryof theradiometer andinthev i idrea lisationof theo ldLucretiantorrents
mselfhasfollow edupthe ire planationof
ments byw hich lessthantw ohundred
erybyR obertB oy le ' theSpringofA ir
estatisticalresultantofmyriadsof
atomsmustha eelasticity andthis
nedbymotionbeforetheuncertainsound
of thediscourse ' E lasticityv iew edas
canbera isedtothegloriouscerta inty
M ot i on ' .
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O F M O T I O N
spinning-tops thechild srollinghoop
otionascasesofstif f e lastic- l i e
t ion andshow ede perimentsw ith
positions utterlyunstablewithout
nedwithafirmnessand strengthand
bybandsofsteel.A fle ibleendless
ausedtorun rapidlyroundapulley and
off thepulley andletfallto thef loor stood
itsmotionwaslost byimpactand
ef loor. A limpdiscof indiarubber
emedtoac uirethestiffnessofa
Alittlewoodenball whichwhen
erj umpedupagainina moment
ddedinj ellywhenthewaterwascaused
prangbac , asif thew aterhadelasticity
henitw asstruc byastif fw irepusheddow n
cor byw hichtheglassv esse lconta ining
stly la rgesmo eringsdischargedf rom
ure inabo w ererenderedv isible by
intheirprogressthroughtheair ofthe
ular anditsmotionwassteadywhen
proceededwas circular andwhenit
erring.W henoneringwassent
hecollisionor approachtocollisionsent
ngeddirections andeachv ibrating
bberband.W hentheaperturewas
ngwasseen tobeina stateofregular
nning andtocontinuesothroughoutits
room. Here then inw ateranda irw as
o lid de e lopedbymeremotion. May
ryultimateatom ofmatterbethus
sk inetictheoryofmatterisadream andcan
tcane pla inchemicala f f inity e lectricity
ion andthe inertiao fmasses( thatis crowds
tgi eane planationofgra ityandof
asses onthevorte theory w ere itnot
pyofcrystals andtheseeminglyperfect
70 .
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DYNA MIC S [ 46 47
nger-postpointingtowardsawaythat
mountingofthisdifficulty oraturning
ndisco ered orimaginedasdisco erable .
yof matterispossibleisthe onlyground
s instoreforthe worldanother
ledElasticity aModeofMotion.
A KINETIC THEOR YOF MA TTER .
ciationR eport Montrea l 1884 pp. 61 -622 P residentia l
ntedin PopularLecturesandAddresses
rpp. 225-229inlaterreprint.
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W O R I N G MO D E L O F T H E
ciationR eport 1884 pp. 625-628.
eB ritishAssociationatSouthport ,
methodsforo ercomingthediff icultiesw hich
Ibe lie e a llpre iousattemptstorea lise
deaofdisco eringwithperfectdefiniteness
motionbymeansof thegyroscope. Oneof
llymyselfput inpracticewithpartially
sa
easuringtheV erticalComponentof
yrostatssupportedonk nifeedges
ase withtheirlineperpendicularto
lyw heelandabo ethecentreofgra ity
w or byane ceedingly smallhe ight
he ldw iththea iso f thef lywheelandthe
hori ontal andthek nifeedgesdow nwards
rmingtheir function.Theapparatus
nifeedgeswiththeflywheelnotspinning
eamofanordinarybalance.Let now
smallk nifeedges ork nife -edgedho les
ofanordinaryba lance gi ingbearing
cuttingtheline ofthek nifeedgesas
dofcourse( unlessthere isreasontothe
municationhas so farasIk now hithertoappearedin
eport188 , p. 405 gi esthetit le ' Gy rostaticDetermination
neandtheLatitudeofanyP lace. ]
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hef ramewor appro imatelyperpendiculartothisline and forcon enienceofputtingonandoff
anordinarybalance tw overy lightpansby
ntheusua lw ay . Now w iththef lyw heel
byw eightsinthepansifnecessary sothat
e uil ibriuminacerta inmar edposition
ninclinedslightly tothehori onta lin
ef lyw heel w hetherspinningoratrest
s topressononeand notontheother
ongingto itstwoends. Now unhoo the
gy rostatandspinit replace itonits
nthetw opans andf indthew eightre uired
edpositionwiththeflywheelnowrotating
byanob iousformulaw hichw asplaced
hport gi esanaccuratemeasureof the
heearth srotation* .
pingNeedle.
rtthatthegy rostaticba lancedescribedabo e ifmodif iedby f i ingthek nifeedgesw iththe ir
s possiblethroughthecentreof
ndf ramew or andw iththefacesof the
ttheysha llperformthe irfunctionproperly
yw heelispara lle lto theearth sa iso f
nofthe flywheelinthesamedirectionas
ustasdoesanordinarymagneticdipping
titude insteadofdip anddippingthe
ow nw ardsinsteadof theendthatis
he magneticdippingneedle.Thus
eedgesbeplacedEastandWest the
itsa ispara lle lto theearth sa is and
outhend downwardsinnorthern
downwardsinsouthernlatitudes.
on andlefttoitself itwilloscillate
amelaw asthatbywhichthe magnetic
a-1TWk 2oysinI w herew denotestheba lancingw eight
uponit a thearmonw hichthisforceacts J V thew eight
adiusofgy ration o itsangularve locity y theearth s
dIthe latitudeof theplace.
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O R I N G MO D E L O F M A GN E TI C C O M P AS S 4 77
roundina imuththepositionof
esamelawas doesthatofa magnetic
ealtw ith. Thus if the lineofk nife
thegyrostatwillbalancewiththe
rtica l andifdisplacedf romthisposit ion
gtothesamelaw butw ithdirecti e
of the latitude intothedirecti ecouple
ineofk nifeedgesisEastandWest. Thus
esusthemeansof definitelymeasuring
srotation andtheangularve locityof
Ibelie e bev eryeasilyperformed
elfhithertofoundtimeto trythem.
eticConmpass .
agyrostatsupportedfrictionlessly
i s w i th t h e a i s o f th e f ly w he e l ho r i o n ta l
ustasdoesthemagneticcompass butw ith
mcalNorth ( thatistosay rotationa lNorth
orth. Ialsoshowedamethodof mounting
e itf reetoturnroundatrulyv erticala is
tionalinfluenceasnotto pre entthe
emethod how e er promisedtobe
andIha esincefoundthattheob ectof
delofthe magneticcompassmay with
ynamica lmodif ication bemuchmoresimply
dingthegyrostatbya v erylongfine
t withsufficientstabilityonaproperly
stigatethetheoryofthisarrangement
statw iththea iso f itsf lywheel
gbyav ery f inewireattachedto itsf ramewor atapo int asfarascancon enientlybearrangedfor abo e
f lyw heelandf ramew or andlettheupper
edtoatorsionhead capableofbeing
erticala isasinaC oulomb storsion
mplicity le tussupposetheearthtobenot
ngsetintorapidrotation letthe
gementsha ebeeninrecenttimestriede perimenta lly in
na ies andarenow inactualuse.
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w ire andaf terbe ingsteadiedascarefullyaspossiblebyhand letitbe le f tto itse lf . If itbeobser ed
imutha lly ine itherdirection chec this
ad thatistosay turnthetorsionhead
ositetotheobser eda imutha lmotion
endo nothingtothetorsionhead
rsea imutha lmotionsuper enes. If itdoes
yopposingitby torsion butmoregently
en thetorsionheadisleft untouched
t.Theprocess gonethroughwillha e
omwhatwouldha ehadtobe performed
tatw ithitsrotatingflyw heel arigidbody
utwithmuchgreatermomentofinertia
s hadbeeninitsplace. Theformulafor
finertiaisas follows.Denoteby
dedw eightof f lyw heelandf ramew or ,
rationroundthevertica lthroughthe
wholemassregardedfora
wheel
onof thef lyw heel
po into fattachmentof thewireabo e
f lyw heelandf ramew or ,
yonunitmass
ityof thef lyw heel thev irtualmoment
a la isis
a . . .. . .. . .. . .. . .. . .. .
Hereitis. Denoteby
f i edv ertica lplaneandthev ertical
so f thef lyw heelatany timet
posedtobeinfinitely smallandintheplane
isinclinedtothevertica lat
e t or u e r ou n d th e v e r t i ca l a i s e e r te d
suspendedflywheeland
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O R I N G MO D E L O F M A GN E TI C C O M P AS S 4 7 9
of momentofmomentumroundan
ea iso f rotationre uisitetoturnthe
angularv e locitydp/ dt w eha e
. .. . . .. . . .. . . .. . . . 2 ,
momentof thecouple inthevertica lplane
hthe angularmotiondo/dtinthehori ontalplaneisproduced. Againbythesameprinciple of
momentumta eninconnectionwith
faccelerationofangularv elocity we
- F . . . .. . .. . .. . .. . .. . . 3 ) .
see uationswefind
.. . .. . . .. ( 4 ,
ctionofHingeneratinga imutha lmotion
f asinglerigidbodyof momentof
m ul a ( 1 a s s ai d a bo e w e re s u bs t it u te d
icmodelcompass:arrangea
recedingdescriptionwitha v eryfine
tlessthan5or10metreslong( the longer
ufficientlyshelteredenclosurecon eniently
senforthee periment . P roceedprecise ly
rostattorestbya idof thetorsionhead
rooforothercon enientsupportsharing
on.Supposefora momentthelocalityof
thertheNorthorSouthpole theoperation
hegyrostattorestwill notbedisco erablydifferentfromwhatitwas aswefirstimaginedit whenthe
ot rotating.Theonlydifferencewill
ostathangsatrestre lati e ly totheearth
allconstantva lue sosmallthatthe
rtica lw il lbequite imperceptible unless
inglysmallthatthearrangementshould
sco erw hichw astheob ectof thegyrostatic
abo e thatistosay todisco erthe
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heearth srotation. Inrealityw eha e
n eniently can anditsinclinationto
rebev erysmall w henthemomentof the
ahori onta la isperpendiculartothe
lyw heelisj ustsuf ficienttocausethea is
ndwiththeearth.
ywheree ceptattheNorthor
insteadofbringingthegyrostattorestat
bringittorestbysuccessi etria lsina
udgingby thetorsionheadandtheposit ion
ethatthere isnotorsionof thew ire . In
thegy rostatwillbe intheNorthand
e uil ibriumbeingstable thedirectionof
stbethe sameasthatofthe componentrotationoftheearthroundtheNorth andSouthhori ontal
sacasetobea o idedinpractice thetorsional
eatastocon ertintostabil ity the
erotorsionalrigidity therotational
inrespecttothee uil ibriumof the
ersedf romtheposit ionofgy rostatic
r ed how e er thate enthoughthe
eatthat thereweretwostable
theposit ionofgyrostaticunstablee uil ibriummadestableby torsionw ouldnotbethatarri edat: the
ce uilibrium renderedmorestableby
osit ionarri edat by thenatura lprocess
alwaysinthedirectionoffindingby
uilibriumwiththewireuntwistedby
nhead.
orsionheadbringthe gyrostatinto
isinclined atanyangleb tothatposit ion
suntwisted itwillbe foundthatthe
aance it inanyobli ueposit ionw illbe
ngthis descriptionresultsfrom
irtualmomentofinertia represented
bo e. Thepaperatpresentcommunicated
aculationsonthissub ect w hichthrow
caldifficultieshithertofeltin any
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O R I N G MO D E L O F M A GN E TI C C O M P AS S 4 81
y rostaticin estigationof theearth s
e ledtheauthorto fa llbac uponthe
tSouthport ofwhichtheessential
nthe frameofthegyrostatinsuch a
ustonedegreeof f reedomtomo e. The
escriptionofa simplifiedmannerof
gyrostaticcompass-thatistosay
naplaneeitherrigorouslyorv ery
nta l.
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ERIMENTS.
theB elfastNaturalHistoryandPhilosophical
89 pp. 89-91( A bstract . . . . THER Eare how e er otherproblemsconnectedw ithgy rostaticswhicharefarmorediff iculttosol e . The lecturerne t
bilityofdifferentformsof gyrostats
e andordinarydiscand gimbal-formed.
gso lutionofC olumbus sproblemhow to
nd. If theeggishard-bo iledit is
ndli eatop w hereasthev iscousf luid
sitsbeingtreatedinasim ilarmanner.
ee ertoso l ethedif ficultproblemof
itwillbeby theaidofthe phenomenaof
dulatorytheoryoflight weshallsee
antdisco erydemonstratesthegy rostatic
uttheinfluencedueto rotationcould
asF aradaydisco ered isproducedby
singthroughglass betweenthepolesof
gowewereall tryingtofindsomek ind
ev ibrationsoflightandelectricity and
ebyProfessorF it Gerald( w homhewas
sent fouryearsagoatSouthportga ethe
eq uestion.Hesuggestedtheemployment
andthatsuggestionhadbeenrealisedin
w ithinthepastyear andthegapw hichw e
filled.Itis almostimpossibletogoa
sanddynamicswithouttheaid of
he reasonthelecturerisinterestedin
thephenomenatheypresentarecurious
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MENTS
l es. B utinstudy ingandreconcil ing
wsofmagnetism the lawsofe lectricity
ityofmatter gyrostaticsplayanundoubtedlyimportantpart.
hanumberof e perimentstending
staticdominationingi ingstability
etc. Oneof themostinterestinge amples
o ew reathsorrings demonstratingthe
nsodelicate amedium. Another
stheimpartingofstability towaterby
otion.Inconclusion hesaidthat
chhehadendea ouredtogi esome
anscomplete yetitw illdoubtlessintime
eanw hileany thingthattendstoad ance
desired endisworthyofour attention.
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AS E S F O R T HE M A X W E L L - B O L T Z M A N N
DISTRIB U TIONO F ENERGY .
. V o l . L . J u n e 11 1 8 91 p p . 79 - 88 N a tu r e
- 5 8.
s a rt i cl e ( P h il . M ag . 1 86 0 " O n t h e
s " enunciatesavery remar able
portanceinthek inetictheoryofgases
assemblageoflargenumbersofmutually
o f se era ldifferentmagnitudes the
thesamefore ua lnumbersof thespheres
assesanddiameters or inotherwords
es uaresof thev e locit iesof indi idua l
sthe irmasses. Themathematica lin estigationgi enasaproofof thistheoreminthatfirstart icleonthe
tisfactory butthemereenunciationof it
w asav eryva luablecontributiontoscience.
" Dynamica lTheoryofGases " Phil. T rans.
w el l f in d s in h i s e u a ti o n ( 3 4 ( C ol l ec t ed W o r s
athoroughmathematica lin estigation the
dto includeco ll isionsbetw eenB osco ich
accordingtoany law ofdistance pro idedonly thatnotmorethantw opo intsare inco llision(thatis
ncesof the irmutua linf luence simul [ Inadiscussionensuing.onthispaper theposit ionofB o lt mannandMa w ellw assupported amongothers byLordR ay le igh Phil. Mag. V o l. x x x IIi . 1892
Papers V o l. II. pp. 554-7 cf . a lsoPhil. Mag. V o l. X LIX .
c ie n ti f ic P a pe r s V o l . i . p p . 4 3 - 4 51 . T he s u b e c t i s d i sc u ss e d
ordKel ininthesecondparto f theR oya lInstitution
nthC enturyC loudso ertheDynamicalTheoryofLightand
d F e b r ua r y 2 1 9 01 r e pr i nt e d as A p pe n di B o f B a l t im or e
527.
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O L T Z M A NN P A RT I TI O N O F E N ER G Y
a well soriginaltheoremforcolliding
tudesinaninterestingand important
b e c t in ~ ~ 1 9 2 0 2 1 o f hi s p ap e r " O n t h e
neticTheoryofGases" ( Trans. R . S. E. for
StudientiberdasGle ichgewichtder
schenbew egtenmateriellenPun ten ( Sit b.
c t o b er 8 1 8 68 , e n un c ia t ed a l a rg e e t e ns i on o f
w ellasti llw idergenera lisationinhispaper
sTheoremontheA erageDistributionofEnergy
o ints ( C ambridgePhil. Soc. Trans.
shedinv o l. II. o fMa w ell sScientif icPapers
hefo llow ingef fect( p. 716 : Intheult imatestateof thesystem thea eragek inetic
rt ionsof thesystemmustbe intheratioof
eedomof thoseportions.
asbeen feltastothe complete
ofcasesforw hichthere istruth o f this
dif feringaslitt leaspossible f romMa w ell s
spheres considerahollowspherical
obuleweshallcallitfor bre itywithintheshell.Imustfirst digresstoremar thatwhathas
Clausiusandothersbeforeandafter
ityan" e lasticsphere " isnotane lastic
nandofelasticdeformation andtherefore
berof modesofsteadyv ibration into
edegreesofnoda lsub-di isionandshorter
translationa lenergyw ould if theB o lt mannMa w ellgenera lisedpropositionw eretrue beult imate ly transformedbyco ll isions. The" smoothe lasticspheres arerea lly
s w iththeirtranslationa linertia andw ith
forceate erydistancebetw eentw opo ints
eradiiof thetw oballs andinfinite
distance. WemayuseB osco ichsim ilarly fortheho llow shellw ithglobule initsinterior andsodo
stov ibrationsduetoelasticityof material
heglobule.Letus simplysupposethe
eshellandthe globuletobenothing
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coll ision andthentobesuchthatthe ir
locitya longtheradiusthroughthepo into f
heco ll ision w hilethemotionof the ir
nchanged.
sha llca lltheshellandinteriorglobuleof
ecule orsometimes formorebre ity adoublet.
ere" of~ 3 willbecalledsimplyan atom
theradiusordiameterorsurfaceof the
sordiameterorsurfaceof thecorrespondingsphere. ( Thise planationisnecessary toa o idan
curwithreferencetothe common
faction o faB osco ichatom.
umberofatomsanddoublets
df i edsurface ha ingtheproperty
componentve locityofapproachofany
ttheinstantof contactofsurfaces
edtheabso lutev e locityof thecentreof
yve locityorve locit iesinanydirection
oanyoneormoreof theatomsorof the
ngthedoublets. Accordingtothe
ldoctrine themotionw illbecomedistributed
thatult imate ly thetime-a eragek inetic
achshell andeachglobulesha llbee ua l
doubletdouble thatofeachatom.
mar ellousconclusion butIseeno reason
t. A ftera ll it isnotob iouslymore
minglyw ellpro edconclusion thatina
o ll id ingsingleatoms someofwhichha ea
assof others thesmallermasseswill
il iontimestheve locityof the larger. B ut
w ell sproof forsingleatomsofdifferent
his" DynamicalTheoryofGases re ferredto
condit ionthattheglobulesenclosedintheshells
ellsfromcollisions withoneanother
to n [ ( C o f ~ 1 8 o f " F o u nd a ti o ns o f K . T .
tthere isperfectly f reeaccessforco ll isionbetween
herof thesameorofdifferentsystems.
gationofsuchasimpleand definitecase
oubletsdef inedin~ ~ 3 -5isdesirable
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M A N N P AR TI T IO N O E N ER GY
terestingasan illustrationweretestnot
ceedinglyw idegenera lisationsetforthinthe
ldoctrine .
onlyasingleglobulew ithintheshe llo f
astnumber. Tof i ideasletthemassof the
dredtimesthesumof themassesof the
mberof theglobulesbeahundred million
beconnectedbyapush-and-pull
egi enatrest w iththespring
t andthenle f tf ree . A ccordingtothe
ldoctrine themotionproducedinit ia llyby
ributedthroughthe system sothat
ineticenergiesoftheglobuleswithin
dmillionmilliontimesthe a erage
ell. Thea eragev elocity o f theshell
d-millionthofthea eragev elocityof
ingpropositioninthe k inetictheory
rigidshe llseachw eighing1gram and
monatomicgas beattachedtothetwo
fectlye lastictuningfor andsetto
ilbecomeheatedinv irtueof itsv iscousresistancetothev ibratione citedinitby thev ibrationof theshe ll
nergyofthetuningfor isthusspent.
ublemoleculesof~ 5 supposethe
onnectedbymasslessspringswiththe
gedtowardsthecentreofthe shellwith
tothe distancebetweenthecentresofthe
w hichIga e inmyB alt imoreLectures
ionforv ibratorymoleculesembeddedin
a lenttotw omassesconnectedbyamassless
motionsinonelinetoconsider butithas
perfectly isotropic andgi ingfora ll
edlinee actly thesameresultasif
endiculartoit. When apairofmasses
i eaf i edobstacleoramo ablebody
esnote actlyperpendiculartothe
l o c it y o f a pa r ti c le " i r re s pe c ti e l y of d i re c ti o n i s ( i n t he
acon eniente pressionforthes uarerooto f thetimes
eof itsv e locity .
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it iscausedtorotate. Nosuchcomplication affectsourisotropicdoublet. An assemblageofsuch
mo ing aboutw ithin a rigid enclosing
aestatist icsbe foreachdoublet , e ua l
iesofmotionofcentreof inertia andof
o constituents
uestionsynthetically w ef inda
meprobleminthedetailsof allbutthe
o ll isionw hichcanoccur w hichisdirect
pre iouslyv ibratingdoublets orany
ouslyv ibratingdoubletagainsta f i ed
emassesofglobuleandshellaree ua l
tsoftwo impactsataninter aloftime
of f reev ibrationof thedoublet andaf ter
sseparationw ithoutv ibration j ustasif
sinsteadof thedoublets. B utinobli ue
pre iouslyv ibratingdoublets e enif
obulearee ua l w eha easomewhat
ndtheinter albetweenthetwoimpacts
eragek ineticenergiesof thetwoconstituents and con erse ly e ua la eragek ineticenergiesof thetw oconstituents e ceptinthecaseof
a l impliesthee ua lity statedinthete t. Letu u beabsolutecomponentv e locit iesof twomasses m m , perpendiculartoaf i edplane
omponentv e locityof the ircentreof inertia andrthato f
otion. Weha e
U + , - . .. .. . .. . .. . .. . .. . . 1 ; ? - t I . .. .. . .. . .. . ..
4 mm
2 = ( i ) - n . + U r . .. .. . .. . .. . .. . . ( 2
erageofU rtobez ero . Ine erycase inwhichthisis
,
u 2 ( in-m ) x Time-a .U 2( - + m ) J ...... 3 ) .
h
me-a .m u 2........................... 4
) x T im e- a . U 2 â € ” ( , I + f ) = 0 . . .. .. .. .. .. .. .. .. .. . 5 ,
pt w h en m 1 i= i , w e m us t h a e
) U 2= Time-a . .....................( 6 ,
si t io n b e ca u se a s w e re a di l y se e f ro m ( 1 , t m me 2 / m + i m )
h e k i n e t ic e n er g y of t h e re l at i e m o ti o ns u - U , a n d U - u .
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O L T Z M A NN P A RT I TI O N O F E N ER G Y
andtof indthef ina lresult ingv ibration. When
motionparalleltothe tangentplaneofthe
certa inva luedependingontheradiusof
he ll theperiodof f reev ibrationof the
i ev e locityofapproach there isnosecond
oubletsseparatew ithnorelati ev e locityperpendiculartothetangentplane buteachwiththeenergyof that
usmotioncon ertedintov ibrationa lenergy.
ll ismuchsmallerthanthe massofthe
te eryco llisionwillconsisto fa large
mse ceedinglydifficulttofindhowto
f thesechatteringcoll isions andarri eat
ultimatedistributionofenergyinany
esotherthanMa w ell sorigina lcaseof
t mann-Ma w ellgenera liseddoctrine istrue
itstruthasessentia l w ithspecia l
ases e enw ithoutgo ingthroughthe
hedetails.I canfindnothingin
cleonthesub ect( C amb. Phil. T rans. May6
hs p re i o us p a pe r s p r o i n g an a f fi r ma t i e
of~ 7.
le ttheglobulesbe init ia llydistributed
ogeneouslythroughthehollow leteach
neighboursbymasslesssprings andlet
neartheinnersurfaceof theshellbe
masslesssprings.O rletanynumberof
withinour outershell andconnectedby
entedbytheaccompanyingdiagram
myB altimoreLecturesnow inprogress.
gi enatrestw iththe irsystemsof
w ithinthem beconnectedbymassless
dinmotion asw eretheshe llso f~ 6. There
ossofenergyfrom
swhich therewasin
theult imatea erage
oletwohundred
becertainly small
matea eragek ineticenergyofthesingleshell. Itmaybe
6isfree towanderthattheenergy
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tcase anddistributedamongthem.
ntheirmotionallowingthemto ta e
nowwhentheyareconnectedbythe
posethemotionsinfinitesimal orif
aybe allforcesarein simpleproportiontodisplacements theelementarydynamicaltheoremof
show tofinddeterminatelyeachof
si simpleharmonicv ibrationsof
fromtheprescribed initialcircumstancesisconstituted.Ittells usthatthesum ofthepotential
achmoderemainsalwaysof constant
me-a erageof thechangingk ineticenergy
hisconstantv alue. Withoutfully
the600mill ionmill ionandsi coordinates it iseasy toseethatthegra estfundamenta lmodeof
cedinthe prescribedcircumstances
ndenergyfromthesinglesimple
hthetw oshellsw ouldta e if theglobules
them orw ereremo edf romwithin
itia lcircumstancesw erethoseof~ 6. B ut
theforces beingrigorouslyinsimple
ts.
ldtheybeso andif there isany
eproportionalityofforceto displacement
sitionofmotionsdoesnotholdgood.
mof fundamenta lmodes a lthough sofar
yhasnotyetbeenin estigated- . F orany
w ithagi ensum E o fpotentia land
eremustingenera lbeatleastasmany
orouslyperiodicmotionasthereare
entv ariables . B uttheconfigurationof
snownotgenerallysimilar:for different
erpositionofdifferent.fundamentalmodes
rw ithdif ferentva luesofE hasnow no
obablethat e eryfundamentalmode
edJ uly10 1891.
nc a re M e ca n i u e C6 e es t e o r a s q u o te d i nf r a p . 5 11 .
miccases thatistosay casesinwhichthere isno
fore ample aparticleconstra inedtoremainonasurfaceand
clineunderthe influenceofno" applied force.
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M A N N P AR TI T IO N O E N ER GY
sso ifMa w ell sfundamental
hesystemif lef tto itse lf in. itsactua lstate
nerorla ter passthroughe eryphasew hich
uationofenergy istrue. Itseemsto
thisassumptionistrue pro idedthe
isnote actly astoposit ionandv elocity
eofthe fundamentalmodesofrigorously
ro ideda lsothatthe" system hasnotany
suchasthose indicatedbyMa w ellfor
usthathisassumptiondoesnot hold
a w ell sfundamenta lassumption Ido
ca lw or ingsofhispaper+ anyproofo f
ea eragek ineticenergycorrespondingto
sisthesamefore eryoneof thev ariables
asagenera lproposit ionitsmeaningis
eemstomeine plicable . Thereductionof
umofs uares~ lea esthese era lparts
spondencetoanydefinedordefinable
ables. What fore ample canthemeaning
hecaseofa j o intedpendulum ( asystem
esupportedonaf i ed hori onta la isand
isf i edre lati e ly tothef irstbody
gra ity . Theconclusionisq uite
b u t is i t t r ue ) w he n t he k i n et i c en e rg y
mofs uaresof ratesofchangeofsingle
iedbyafunctionofa ll o ro f some of
e fore ample thestilleasiercaseof
t.
arlytheeasiestcaseof all motion
ne thatisthecaseof j usttw o
s a y x , y a n d k i n e ti c e ne r gy e u a l to
u a t i on s o f mo t io n a re
dV
y
lenergy w hichmaybeany functionof
o l. II. p. 714. + Ibid. pp. 714 715.
~ Ibid. p. 722.
" b " i np .7 2 .
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l y to t h e co n di t io n ( r e u i re d f or s t ab i li t y t h at i t
i t s l ea s t v a l u e be i ng f o r br e i t y t a e n a s
e d t ha t w i t h a ny g i e n v a l u e E f o r th e
tialenergiesthereare twodeterminate
thatistosay therearetw of inite
ifmbepro ectedf romanypointo fe ither
/ 2 ( E - V ) ] i n t he d i re c ti o n e i th e rw ar d s
e itspathw illbee actly thatcur e .
cases thereareonlytwosuch periodic
ousthattherearemorethantwoinother
mp le
2y 2+ c 2 y2 .
h a e
x = O )
s ( , t -f }
s. WhenEisinf inite ly smallw eha e
any finiteva lueofEw eha eclearlyan
entalmodes ande erymodediffers
fundamentalmode.Toseethislet
y p oi n t N in O X i n a d ir e ct i on p e rp e nd i cu l ar t o O X w i th a v e l oc i ty e u a l to \ / 2 E - a2 O N 2 . A f te r a
ofcrossingsandre-crossingsacrossthe
particlew illcrossthislineverynearlyatright
N . V ary theposit ionofNv eryslightly
andre-pro ectmf romitperpendicularly
y t i l l( byproper" tria landerror method
af tersti l lthesamenumberofcrossings
sese actlyatrightanglesatapo intN ,
. Letmcontinue itsj ourneya longthis
asmanymorecrossingsandre-crossings it
a n d cr o ss O X t h er e e a c tl y a t ri g ht
mNtoN ise actlyhalfanorbit
mainingha lf .
isasmallnumeric theparto f the
ssedby_c 2y2isverysmallincomparison
. Hencethepathisate ery timevery
wo primaryfundamentalnotesformu
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O L T Z M A NN P A RT I TI O N O F E N ER G Y
aninterestingproblemispresented to f ind
v a r ia t io n o f pa r am et e rs ) a e b f s l ow l y
suchthat
y -b s in ( / t- f ,
y = b/ c os ( t - f ,
ton orapractica lappro imationto it.
ossibilitiesinrespectto thiscase
r y sm al l s e em s t ho r ou g hl y t o co n fi r m Ma w el l s
uotedin~ 10 andthatit iscorrect
esmallorlargeseemse ceedinglyprobable
robablethatMa w ell sconclusion w hich
po intmo inginaplane is
.y2................. 1
sf rom 3 2. It iscertainlynotpro ed.
ceptthee uationofenergy
. .. . .. . .. . .. . .. . 2 ,
matica lw or o fpp. 722-725 w hichis
proof forit . Henceanyarbitrarilydraw n
edforthepathw ithoutv io latingthe
toMa w ell sin estigation andw emay
hsuchastosatisfy (1 , andcur esnot
a lltra ersingthew holespacew ithinthe
c 2 y2 = E . .. .. . .. . .. . .. . .. . 3 ) ,
e ll sfundamentalassumption( ~ 10 .
uestionisillustratedbyreducingit
uestionregardingthepath thus:calling0theinclinationto x ofthetangenttothe pathatany
e v e l o ci t y in t h e pa t h w e h a e
= q s i n 0 . .. . .. . .. . .. . .. . . ( 4 ,
q = V { 2 ( E - V ) } . .. . .. . .. . .. . .. . .. . .. . .. 5 .
o ta llengthofcur etra e lled
s 2 q d t = f S 2 ( E - V ) } c os 2 O d s. .. .. . 6
15becomes Isorisnot
} co s2 0 = S ds V { 2 ( E - V ) } si n2 0 . 7 ,
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C tobea llmo ingtoandf ro . The
dthee ua lbodiesA andC onitstw o
a n d k e e p e u a l t h e a e r ag e k i n e ti c e ne r gy
oeandaf terthesecoll isions tothea eragek ineticenergyofC . K
sofA beinginthespace
cludedinthea erage
of thepotentia land
e ua ltothea erage
utthepotentialenergy F
he s p ac e H i s p o s it i e , A
oursupposit ion thev e locity
e ery timeof its - - € ” H
andincreasedtothe
gmotionf romK toH.
ineticenergyofA isless
ticenergyofC
ectlyrepresentati ek indforthetheoryof temperature
ofthe assumption
so lidorli u idis
ineticenergyperatom
outasaconse uence
andw hich be lie ed
hasbeenlargelytaught
safundamentalpropositioninthermodynamics.
ppro imately
tistosay anassemblageof
oleculemo esfor
esinlinesv eryappro i- L
perienceschangesof
ncomparati e lyveryshortt imesof
y forthek ineticenergyof thetranslatory
gas " thatthetemperature ise ua ltothe
ypermolecule asf irstassumedbyWaterston
e andfirstpro edbyMa w ell.
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yisdiminishedwhenthesystempasses
ered fromaconfigurationortheconfigurationsuch thatpassagetoanyotherpermittedconfiguration
of thek ineticenergy. B y" tota lenergy o f
nwill bemeantthesumof itsk inetic
t ion. F ore erygi env a lue E
reisafully determinateorbitsuchthat
tionalongit atanyconfiguration
otalenergy E itw il lcirculateperiodica lly
m supposethenumberof f reedoms
n Q , is fully specifiedby igi env a lues
ti ely.Supposenowthesystemtopass
on Q attw otimesseparatedbyan
ethesamev elocit iesanddirectionsof
epaththustra e lledinthisinter a l: isanorbit andit isperiodically tra e lledo erinsuccessi e
oT . Tof indhow toprocurefulfi lmentof
systembestartedfromanyconfiguration
sforthe iv e locity -components( orratesof
f the icoordinates . Tocause itto
u n n o wn t i me T w e h a e i - 1 e u a ti o ns
sei-1of itsv elocity-componentsto
satthesecondasatthef irstpassagethrough
a ti o ns t o s at i sf y a n d i n v i r t u e of t h e e u a ti o n
ingv e locity -componenta lsomustha ethe
times. Thatthetota lenergymayha e
E w eha eanothere uation. Thusw e
tions amongcoordinatesandv elocitycomponents. Eliminateamongthesethe iv e locity -components
uationsamongtheicoordinateswhich
ryandsufficientto securethatQ isa
f tota lenergyE. B e ingi-1e uations
hey lea eonlyonef reedom thatistosay
path o fw hich inthe languageof
eometry theyarethee uations. Theor
n orbitoftotalenergy E.Thusis
f~ 4.
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A F I N IT E S YS T EM
rminateproblemoffindingan
hastheprescribedva lueE is ingenera l
differentperiodsfortheinfinitenumber
nedbyit.
onlytwofreedoms willhelp
f~ 6fore erycase o fanynumber
ointeddoublependulumconsistingof
B : o n e ( A s u pp o rt e d on a f i e d h or i o n ta l a i s I t h e ot h er ( B ) s u pp o rt e d on a p a ra l le l a i s J , f i e d
implicity letG thecentreofgra ityof
hetw oa es. C a llHthecentreofgra ity
betweentheplaneI andthevertica l
w esha llcall IV ; andletrbetheangle
ndthevertica l. Thecoordinatesand
minanycondit ionofmotionarep q , cb + .
system ink ineticunits willbegW z ,
mof themasses andz theheightof the ir
yconf igurationof thesystem abo eits
be placedinanyparticularposition
iredto findw hatmustbetheposit ion
wh a t v e l oc i ti e s 4 0 4 0 w em u st s t ar t A a nd
thefirstt ime9phasaga inthesamevalue d0
madeone completeturnineither
hallbew holly inthesameposit ion( q 0 k 0
am e v e l o ci t y ( 0 o j 0 ( i n t he s a me d i re c ti o n
hebeginning. Thisimpliesonly tw oe uations
= r 0 o r & l t = c o ( b e c a us e e it h er o f t he s e
ueof thee uationofenergy . A ndw e
a bl e s J 0 a n d ei t he r f 0 or 0 o ( t h e gi e n t ot a l
er40or0ow hentheotherisk now n .
nateproblemis clearlypossible unless
tgenera llyuni ue. Wemayha e
cit iesofA andB startedeachinthe
eachnegati e oronenegati eandtheother
e lo fverygreatmomentof inertia and
mallpendulumhungonacran -pinattached
yw esupposethecran tobecounterpoised sothatthecentreofgra ityofA isinitsa is it isclear
reaterorlessva luegi enforE B may
ytimes beforeAcomesagaintoits
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utit isclearthat thoughnotgenera lly
o f f indingperiodicmotionwithj ustone
eriodhasno realsolutionunlessEis
nyso lutionsforlargeenoughva luesofE
mberofsolutionsforanyfinite v alue
tionbethat notthef irstt ime but
sthroughitsinit ia lposit ion bothcoordinatesandbothv e locit iesha ethe irprim iti ev a lues. When
etota lenergy isnottoogreat theperiodic
ew illbepurelyv ibratory andthe
utifEbegreatenough A maystill
eB maygoroundandround f irst inone
ther w ithintheperiodofA sv ibration.
tatthefirst andnotatthe second
A acrossitsinit ia lposit ion bothcoordinatesandbothve locit iesha ethe irprim iti ev a lues w emay
alenergy ha estil lw ilderacrobatic
iesgoingroundand roundsometimesin
esintheother. Stillwithanyfinite
afinite numberofmodesforthemotion
thatthethirdtransito fA throughits
hefirstperiod.W ilderandwilder
hin o f if thef irstperiodiscompletedat
andsoon.
nsteinofaproblemisa ll in o l ed
maticalstatementnotincludingany
irst orthesecond orthethird or
ofAthat completesthefirstperiod.Itwill
toarrangesoas tofindatranscendental
eaninf initenumberof f initegroupsof
sof themodesoftheperiodic motions.
typresentsavastly simplerproblem
has nodoubtbeenmanytimesfound
nsintheCambridgeSenate-houseand
minations. Thecharactero f theso lutionof
mic problems isindependentof theabso lute
ergy andof0o. Itdependsonlyonthe
w hichofcoursemaybeeitherposit i e
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A F I N IT E S YS T EM
ra lso lutionf -k isclearlyaperiodic
dourquestionofperiodicity re lati e ly
Ireso l esitse lf intothis: -Duringthe
of f -b isthechangeof~ e itherz eroor
ewith27r Acorrespondingq uestion
nwhichour" system isf ree inspace
es andw ithnodisturbingforcef romother
mple inthequestionof rigorousperiodicity
diessuchastheearth moon andsun
mutuallyattractingbodies suchasthe
sideredpresently.
rdinarycloc w ithw eight andpendulum
nt affordsaninterestingillustration.F or
perfectly f le ibleandine tensible le t
edontheshaf to f theescapementwheel le ttheescapementberigidly f i edtothependulum
rigidbodyonperfect k nife-edge
irtually tw obodies eachw ithone
nt-wheel cord andw eight B the
m.Eachimpactoftoothonescapement
ndw atch fo llow edbyamutua lreco il. This
practicalcasesgoessofar asto
tion followedbyse eralmoreimpacts
ohescapes andthecorrespondingne t
eoftheescapement.B utthereisa
sandslipping bothonthenon-wor ing
esof theescapement. The lossonthe
dispensedw ith: butthe lossonthenonwor ingfacesisessentia lto thegoingof thecloc . Inour
upposeeachreco iltoe actly re ersethe
ndescapementin thedirectionperpendiculartothecommontangentplaneof thetwosurfacesattheir
supposethesurfacesto beperfectly
nfinitelygreat mutualforceattheinstant
ly inthatdirection. The j umpingaction
epstoppingthecloc andlettingitgoon
e entanyregularityofgoing.ThereforeIadd thefollowingarrangementofenergy-recei erstoannul
w or ingfacesof theescapement: -P ro long
ent-wheel andf i onit inhe lica lorder
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DYNA MIC S [ 52
rry ingatitsendadis , w ithitsf ront
ea is. A d usttheescapement-wheel
heneachof itsthirty teethstri esoneor
f theescapement thelowestoneof
tsf rontfacevertica l. Onahori onta l
onoftheshaftplace si tylittleballs
insuchposit ionsthateachshallbestruc bya
rrespondingtoothtouchesthecorresponding
t. Letthemassofeachba llbee ua lto
a l en t o f A ( t h e es c ap e me n t- wh e el & a m p c .
Eachba llstruc by itrecei esthew hole
hadbeforethe impact andlea esA
theescapement-wheelpressingonanonw or ingfaceof theescapement. F i si ty rigidstopstopre ent
i rc u ms t an c es ( ~ ~ 1 - 1 6 , g o in g t oo f a r
chthey areinitiallyplaced.Each
otted toa llow theproperdis o f the
etheballand afterwardspassclear
rm. F orbre ity thesefor edstops
ops. F i a lsosi tyotherstops( f ie ldstopswesha llca llthem insuchposit ionsthattheballssha ll
eouslyandate actly the instant( ~ 12
sthebottomof thecloc -case.
dulumofouridea lcloc w ithits
nearly tothetop tobestartedw ith
eepgo ing. F orsimplicity letthis
ecure thatwhentheweightisrun
angeofv ibrationwillst i l lbewithinthe
tionof theescapementmechanism.
-casebearigidhori ontalplanef i ed
wor bearingthew heelandpendulumin
thatw henthew eight inrunningdow n
umisate itherendof itsrange. The
the impact andthecloc goesbac w ards
turnhomef romtheirf ie ld-stopsate actly
thecord-drum: TV thedri ing-weight: k theradiusof
ghtof thewholerotatingbodyconsistingofcord-drum
af t and60armsanddis s: athe lengthofeacharm
stothepo into f itsdis w hichstri estheba ll. The
a l en t i s ( T r 2 + w 2 / a 2.
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A F I N IT E S YS T EM
nge actlye erysteptil lthew eight
yof thependulumandofthereturning
assesthroughitsinit ialposit ion. If it is
stri esaga instanunad ustedstop
edforatime w iththependulumv ibrating
llrange andonetoothofthewheel
r ingfacetofthe escapement:but
erysoon thetoothw illescape thecloc
hew eightw illrundow n andagainstri e
romit thist imenotw henthependulum
herendof itsrange.
iteorderlyactionw illf o llow and
oothwillbehoo edupby theescapement
ac w ardsabeatortw o butaf terav ery
one itw il lgo forw ardtil lthew eight
n. " Soonerorla ter thebottomw illbe
hependulumisv erynearlyatrestat
dw hense era lenergy - rece i ersare in
ehomeandstri edis satrighttimes
ac w ardsforagoodmanybeats.
thatistosayaf tersomef initenumber
ars thew eightw illstri e thebottom
erynearlyatrestat eitherendofits
ba llssoverynearly stri ingeachits
w illbedri enbac , w indingupthe
sthetopstop andimmediate ly ora f ter
ginsaga intogoforw ardandlettheweight
ectisnotthefortuitousconcourseof
otionofafinite system.
otheendof~ 12 letthetopstop
sstruc by thew eightataninstantw hen
dofitsrange.The cloc instantly
ndgoesonretracinge erystep and
thenumerousimpacts o f itsf irstforw ard
stri esthebottome actlywheneachof
ngitsf ield-stop andw henthependulum
thesameendof itsrangeasw henthe
mthefirsttime.Thusaperfectlyperiodic
.
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-periodsinw hichthecloc is
eweightrunningdow n anymoderate
slightblow onthependulum oraho lding
stoppedforsometime largeorsmall
edifference inthesubse uentmotion: t il l
tomofits range whenwefindthat
ndthestateof thingsdescribedin~~ 1 ,
nysuchdisturbanceduringaha lf -period
gbac w ardscausesthebac w ardmotion
rdmotionto followimmediately or
aterorless numberaccordingasthe
glyinfinitesimalorbutmoderatelysmall.
ustrationofthe" dissipationofenergy "
nityofattemptswhichha ebeenmade
inciple " or" theSecondLaw ofThermodynamics " ortheoriesofchemicalactiononLagrange sgenera li ed
lemof thethreebodies intw o
eLunarTheory " secondly " theP lanetary
theSun isineach casev astlylarger
ers.Inthefirstcase thetwoothers
aresonearoneanotherincomparison
fromeitherthat hisforceproducesbuta
ati emotionoftheEarth andMoon
raction.Inthe secondcase two
yunderthe Sun sinfluencewithcomparati elysmalldisturbancebytheirown mutualattraction.In
simplicity neglectthemotionof theSun s
considerhimasanabso lutely f i ed' centre
artheory supposethecentreof
h an d M oo n t o mo e v e r y ap p ro i m at e ly i n
w( withoutnecessarilyconsidering
a llerthantheEarth ataninstant
throughS gi ee ua landopposite
thelineMEso astoannultheir
ne if theyhadany andtocauseeachto
dicularly to it. If thene ttimetheirl ine
gain mo ingperpendicularlytoME
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A F I N IT E S YS T EM
oSIis rigorouslyperiodic.Thiswesee
motionsarere ersedatanyinstant
racethe irpaths andif suchare ersa l
rpendicularlycrossingthelineSI the
o thedirectpathswhichare traced
a l.
iesbegi eninline SME w e
oftheirmotionif wepro ectthem
endiculartothis linewithe actlysuch
ttimeMEisagaininlinew ithS now
motionareagain perpendiculartoEM.
as threesolutions inoneofwhich
ectionaresogreatthatMandEarecarried
inoppositedirectionsroundtheSun
oneanotherandinline onthefarside
iscaseweha ecerta inlyonly tw o
scribese ceedinglynearlyacircle
andEmo ere lati e ly tothepointI
hatappro imately incircles buttoa
ntheellipsescorrespondingtothe lunar
aria tion andquiterigorously intw o
r eseachdifferingv ery little f romthe
entreofthev ariationalellipseisatI:
pendiculartoSIande ceedstheminora is
9 6 b e in g t he s u a re o f t he r a ti o ( 1 / 1 ' 4
ofSIto theangularv e locityofME each
y fi eddirection. Therearetw oso lutions
fw hich( asintheactua lcaseofEarth
mewardsas intheothercontrary -wards
etimesasgreat asitis when
e SME andotherdimensionsthe
a easo lutionforperiodicitycorresponding
iththeorbitalcur esofMandEroundI
mcirclesand largelyfromellipses.
erta inlimit thisk indofso lutionbecomes
whollyuninterestingto followthe
esroundIfor increasingmagnitudes
hesolutionreferredtoandre ected
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beandbecomesnow more interesting but
orrespondingsolutioninwhichMand
a r e pr o e c te d s o as t o r e o l e i n t he s a me
otionof tw oplanets. Gi enSV E
oplanetsatdistancessuchasthoseof
isre uiredtopro ectthemwithsuch
uentmotionisrigorouslyperiodic.A
pro ectingthemperpendicularlyto
esthatthe irperiodsof re o lutionroundS
u a l a n d e a c tl y s uc h t ha t a t th e n e t t i me
ew ithS themotionsarerigorously
Thevelocit iesw hichmustbegi en
tbesuchthatthema ora esof the
escribedareappro imatelye ua l. This
be longsrathertotheC ometary thantothe
perpendicularly toSV E w ithsuch
egi ennumberof t imesof the irbeing
e irmotionsare forthef irstt imeagain
hedeterminatev elocitieswhichfulfil
besuchthatthe orbitsareappro imatelyellipsesofeccentricitiesnotdifferingmuchfrom those
e ma o r a e s s uc h t ha t t he p e ri o ds h a e t h e
torenderthelineofthe threebodies
arcrossingappro imatelycoincident
perpendicularcrossing.
P E RI O D I C M O T I O N B E I N G A C O N T IN U A T I O N
I O D I C MO T I O N O F A F I N IT E CO N S E R V A T IV E S Y ST E M.
. No . 26 1891 Phil. Nag. Dec. 1891.
' , . .. b e g en e ra l i e d c oo r di n at e s of a s y st e m
, f , . .. b e th e ac ti on i n ap at h ( ~ 2 a bo e
, ' , . . . t o t h e c on f ig u ra t io n ( I s . . .
E-V ) w ithanygi enconstantva lueforE
e in g t he p o te n ti a l en e rg y ( ~ 3 a b o e , o f
enfore erypossibleconf igurationof the
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RI O D I C M O T I O N
. . an d v ' , 4 , v ' , . . .. b e th e ge ne ra li e d
ofthesystemasitpassesthroughthe
b . .. a nd ( ' ' , I , . .. r es pe ct i e ly . I f by a ny
o edtheproblemof themotionof thesystem
e * ( o f w hi c h V i s t he p o te n ti a l en e rg y , w e
e n s et o f v a lu es o f r p . .. f , s b , . .. t ha t is
f un ct io n of ( 4 c q , . .. ' , b , . .. . Th en b y
ThomsonandTait sNaturalPhilosophy
w eha e
dA
= ' dX ' dr . 1
dA
- dd~ d ' = -d d ' "
ateaparticularpatht-from position
' , . .. w hi ch f or b re i ty w es ha ll c al l P , t o po si ti on
w h ic h w e sh a ll c a ll P . L e t o P o P b e a pa r t of a
f rom w hichP P isinf inite ly litt ledistant.
oP isperiodicornot pro idedit isinfinitely
i d ed O P a n d oP a r e in f in i te l y ne a r to P
w e ha e b y Ta yl or s t he or em a nd b y( 1 ,
, oX , * * . oo , X ' , . ..
+ ( ~ - + ... -o ( ' - o ' ) - o ' ( St - o ) -...
o ) ( , - + } + .. .. .. .. .. 2 .
edbymybrother P ro f . JamesThomson todenotea
u e of E t h e to t al e n er g y ( ~ 3 a b o e , t h e pr o bl e m of f i nd i ng
P toanyposit ionP isdeterminate. Itsso lutionis for
em adeterminatefunctionofthecoordinateswhich
t thetimerec onedf romtheinstantofpassingthroughP .
hecaseofa particlemo ingundertheinfluenceof no
nganinfinitestra ightline . F orasingleparticlemo ing
niformforce inpara lle ll ines( asgra ity insmall- sca le
eso lutionisduple orimaginary. F ore eryconstra inedly
isinf inite lymultiple asisv irtua llyw ellk now nbye ery
ofaB osco ichianatomflyingaboutwithinanenclosing
ery tennisplayerfortheparabo lasw ithw hichheisconcerned
mw allsorpa ement.
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YNA MIC S [ 52
choosingourcoordinatessothat
& a m p c . a r e ea c h z e r o f or e e r y po s it i on o f t he
foranyposit ionof thispath betheaction
eroatoP . Theseassumptions e pressed
ws: dA _ dA dA dA dA o
o = 0 - 0 l
#- d = ' d X / '
, i fi = 0 X = 0 . .. i = 0 X ' = 0 . .. . .. .. .. .. 3 ) .
o = , 0oX = 0 . .. O = , o 0 = , o X ' = O . .. ( 4 ;
o , o , o X ' , .. ) = A ( 0 0 , . ..0 , 0 ... ... 5
an d of ( 3 ) a nd ( 1 , ( 2 b ec om es
, ' , o ... o 00
+ 66+ + " '
1 4 0 0+ 1 5 O t + 1 60 '
+ 2 5X X + 26% X ' . . 6 ,
+ 3 6 5
/
mplicityo fnotation w esupposethetota l
system thatistosay thetotal
es * , & l t , X , i - t o b e fo u r a n d f o r
b y ac c id e nt i n ( 6 a n d ( 8 a s s ub s cr i pt s
2A _
= 1 2 o = 22 & a mp c .. .
d b y( 1 ,
( ' + 1 44 + 1 5X % + 1 6̂ '
- + 2 40 + 2 5 X ' + 2 69
3 3 P + 3 4 0 + 3 5 X I + 3 6
. + 4 4 ' + 4 5% + 4 6
4 52 + 5 X + 5 4 + + 5 6
2% + 6 4 + 6 4 x + 6 6 + 6 6
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RI O D I C M O T I O N
stodeterminethethree displacements
e t hr e e co r re s po n di n g mo m en t um s | , q , ' , f o r an y
t er m s of t h e in i ti a l v a l u es b , X ' , C ,
s up po se dk n o wn .
supposition( ~ 24 that0P oP is
letQ beapositiononitbetw eenoP
t o a o i d am b ig u it y c a ll i t o P Q o P .
entoco incide inaposit ionw hichw e
r ds l e t oP Q o P o r O Q O , b e t he c o mp l et e
t as w e h a e c a l l ed i t ( ~ 2 a b o e . O u r
infinitelyneartothisorbit andP and
osit ionsinitforw hichA hastheva lue
sareinfinitelynearto oneanotherand
i andO i+ l consideringthemasthe
hich risz ero forthe ithtimeand
e f romanearlierinit ia lepochthanf irst
hichweha ebeenhithertoconsidering.Itis accordinglycon enientnowtomodifyournotationas
= x i ' = i r ' = = , = ' ) l
l = ri + l = + i 7 7= + i = ? i l. .. .. .. .. 9 .
ethegenera li edcomponentsofdistance
tthroughr= 0 o f thesystempursuing
heorbit and:i q i ' ia rethecorrespondingmomentum-components. Withthenotationof (9 ,
mee uationsbyw hichthev a luesof these
thtimeof transitthroughF = 0canbe
forthe ithtime. Theyaree uations
daretobetreatedsecundu martem as
= ' P X i i + = pi . ( 1 0 .
P7 w ? i + l= P "
ncontinua lly increasesa longthepath. Whatismeantis
stant asregardsaction byanamountw hichise ua lto
tionintheperiodicorbit. Thesub ectmaybeelucidated
orays oflight wherethereareonlytwocoordinates
iioninaplanetrans ersetotheray : cf . ThomsonandTa it s
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( 8 modif iedby (9 , andeliminating
+ ( 1 2+ 1 5 + 4 2 p+ 4 5 X
46 = O
( + ( 2 2+ + 5 5 2p + 5 5 ) .
56 = = 0
64 + ( 3 2 + - + 6 2p + 65
66 s= 0
4 12= 21 & amp c. w eseethatthedeterminantforthee lim inationof theratiosIX 19issymmetrical
Henceitis
p 2p -2 C ( p + p -l + 2 Co.. . 1 2 ,
C arecoeff icientsofw hichthev a luesinterms
c. areeasilyw rittenout. Thisdeterminante uated
uationof the6thdegreefordeterminingp
re isanothere ua lto itsreciproca l.
a tionof thethirddegreebyputting.
.. . .. . . .. . . .. . 1 ) .
e t he r o ot s o f th e e u a ti o n th u s fo u nd . T he
pare
~ ( e -1 ; es + V ( e 2- 1 . .. 14 .
any rea lv a luebetw een1and-1 it is
cosa+ tsina . . .. . . .. . . .. . . .. . ( 15 .
- sina
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RI O D I C M O T I O N
rthef irstt imeofpassingthrough
atesandthreecorrespondingmomenta
r 7n 1 t ob ea ll gi e n w e fi nd
p l- i + + A 2 p 2 + A p -+ A p i + A p -i
' p> - i+ B 2p2i+ B 2 2- i+ B 3 p i+ B 3 ' pi. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . ( I6) , . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .
' p 1- i+ F 2 p 2i + F 2 p2 -i + F 3 p i+ F 3 ' p i
A 2 . . .F 1 F ' , F 2 F 2 a re 3 6 c oe ff ic ie nt s
t he s i e u a ti o ns ( 1 6 w i th i = 0 :
8 , m o d if i ed b y ( 9 ; w i t h i su c ce s si e l y
4 5 w it h th e gi e n v a lu es s ub st it ut ed f or q b X I
i n th em a nd w it h fo r 02 X 2 & a m p c . th ei r v a l ue s by ( 1 6 .
sthate erypathinf inite lyneartothe
ery rootof thee uationforehasa
d-1. Itdoesnotpro ethatthemotion
n isfulfilled.Stabilityorinstability
stedwithoutgoingtohigherordersof
siderationofpaths v erynearlycoincident
mb e r 10 1 8 91 .
motionand itsstabilityhasbeen
byM. Po incare inapaper " Surle
tlese uationsde ladynami ue " for
a esty theK ingofSw edenw asawarded
1889. Thispaper w hichhasbeen
sA ctaMathematica 1 , 1and2
oc h o lm 1 8 90 o n ly b e ca me k n o wn t o m e tw e l e
yley.Iamgreatlyinterestedto find
thesub ectofmycommunicationof
Society " O nsomeTestC asesforthe
nDoctrineregardingDistributionofEnergy ;
thefo llow ingparagraph:- O npeut
o isinaged unetra ecto ire fermeerepresentantunesolutionperiodi ue so itstable so it instable i lpasse
ecto iresfermees. C e lanesuf fitpas en
nclurequetouteregiondelespace sipetite
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a erseparuneinf initedestra ecto iresfermees
ra cettehypotheseunhautcaractere
Thisstatementise ceedingly interesting
w ell sfundamentalsupposit ionquotedin
thatthesystemif le f tto itself in itsactua l
soonerorla ter passthroughe eryphasew hich
uationofenergy t" anassumptionw hich
aconclusion butasaproposit ionw hich" w e
fidenceassert ...e ceptforparticular
ef i edobstacle . Itw il lbeseenthat
sis ha ingahighcharactero fprobabil ity "
w ell s w hichassertsthate eryportion
nalldirectionsbye ery tra ectory . The
in~ 1 + , asseemingtomequitecerta in
ersinfinitelylittlefrombeinga fundamental
ecessaryconse uenceofMa w ell sfundamental
whichstill seemstomehighlyprobable
casesareproperlydealtw ith.
tatement pp. 100 101: - IIyaura
titesa2distinctes.Nouslesappellerons
dela solutionperiodi ueconsideree.
nttousree lsetn6gatifs laso lution
carlesquantites4ietqiresterontinf4rieures
sentendrecemotdestabiliteau sens
a onsneglige lescarresdes: etdesq, et
entenantcomptedecescarresleresultatne
ouspou onsdireaumoinsquelesa
airementtrespetits resteronttrespetits
Nouspou onse primercefa iten
eriodi ue j ouit sinondelastabilite
astabil ite temporaire . Heretheconclusionof~ 3 1ofmypresentpaperisperfectlyanticipatedand
nterestingmanner. M. Poincare sin estigationandmineareasdif ferentastw oin estigationsof the
llbe andit isverysatisfactory to f ind
usions.
ermee o fM. Po incare isw hatIca lleda" fundamenta l
icmotion or" anorbit.
o l . ii . p . 71 4 . + [ S u pr a p . 4 92 .
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NPLA NEKINETIC TR IGONO METRY
S S S T H EO R E M O F C U R V A T U R A I N T EG R A.
M a ga i n e V o l . x x x i i . No . 1 8 91 p p . 47 1 -4 7 .
beautifultheoremof the" Spherica l
rigonometry publishedabout16 7 and
slaterbyGeneralR oy inthetrigonometricalsur eyof theB rit ishIsles w assplendidlye tendedby
f the" C ur atura Integra. Theremust
minthe" k inetictrigonometry suggested
nandTait sNaturalPhilosophy
b ( c ( d , f o r th e mo ti on o ft he g en er al i e d co ns er a ti e
umberofvariables. F orthev erysimple
mo inginaplane it iseasilyw or ed
inendea ouringtow riteacontinuationof
a lMaga ine O ctober onthePeriodic
Genera lescircaSuperf iciesC ur as auctoreC aroloF rederico
e[Gottingensi oblatseD. V III. O ctobr. MDC C CX X V II. C o llected
G 6 tt i ng e n 1 8 7 . T h o m so n a nd T a it s N a tu r al P h il o so p hy
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m w hichIhopemaybeready toappear
Hereis thetheoremmeantime.
, R C A Sbethreepathsofaparticle
e underinf luenceofaforce( -d - )
threeplacesin anydirectionintheplane
athesumof thek ineticandpotentia l
a lue( E ineachcase. Thesumof the
C e ceedstw orightanglesbyanamount
adians ise ua ltothesurface- integra lo f
t h ro u gh o ut t h e en c lo s ed a r ea A B C V 2 d e no t in g
dy2.
ma r t h a t
cionof ( x , y , f fd dysurface- integration
dsline-integrationallrounditsboundary
ria tionof * inthedirectionperpendicular
t. Hencethesurface-integralmentionedin~ 2ise ualto
. . .. . . .. . . .. . . .. . . .. . . .. . . 1 .
( 1
rmal-componentforce( N w esha llca ll it ;
e s u a re o f t he v e l oc i ty ( v 2 w e s h a ll c a ll i t .
.. . . 2 .
a t ur e ( 1 / p w e sh a ll c a ll i t , a t a ny p oi n t in
A B , B C C A . He n ce d i i d in g f ds
ngrespecti elytothesethreearcs
d d s d s , w e fi n d fo r ( 2 ,
directioninthearcA B , andsim ilarly
e theorem.
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Y O F P E RI O D I C M O T I O N .
c i at i on R ep o rt 1 8 92 p . 6 8 ( t i tl e o nl y ; N a tu r e
1 89 2 p . 3 8 4.
stigationofthissub ectwasillustrated
hichasimpleharmonicv erticalmotionwas
pporto fapendulum. Whentheperiodof
sonehalfofthat ofthenaturalmotion
uil ibriumbecameunstable andthe
edthev erticalmotionofthebobto be
motionofincreasingamplitude.Ifthe
owlessened thev erticalmotionagain
rodpoisedv erticallyinunstable
mestablebyha ingitspo into f support
monicmotion o fproperperiod inav ertical
remar edthatitw asw ellk now nto
re o l ingshaft w hendri enatacertain
andmighte enbrea , thoughathigher
comestraight. LordK el inhadnow
" O ntheMaintenanceofV ibrationsbyF orcesofDouble
P h il . M ag . V o l . x x I . 1 8 87 p p . 14 5 -1 5 9 S c i e nt i fi c P ap e rs V o l . Im .
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[ 5 5
T IONO F DYNA MIC A LPRO B LEMS.
c i at i on R ep o rt 1 8 92 p p . 64 8 -6 5 2 N a tu r e V o l . X L V I .
8 5 3 8 6 P hi l. M fa g. V o l . x x x i . p p. 4 4 â € ” 4 48 .
eridianalcur esofcapillarysurfaces
edinPopularLecturesandA ddresses V o l. I.
2 andil lustratedbyw oodcutsmadef rom
or edoutaccordingto itw ithgreatcare
errywhena studentintheNatural
ow Uni ersity suggestsacorresponding
ynamicalproblems.
ardingthemotionofa singleparticle
efo llow ingplanfordraw inganypossible
a forceofwhichthepotentialis gi en
ane. Suppose fore ample it isre uired
lepro ected w ithanygi env e locity in
ughanygi enpo intPO (f ig. 1 . C a lculate
ceatthispo int anddi idethes uare
a lue to f indtheradius
hatthatpo int. Ta ing P2
sses f indthecentreof Q 1p0
he l i ne P o , p e rp e nd i cu l ar t o
ughPo anddescribea
m a i ng P Q , e u al t o ab ou t
rthe secondarc. oC
locityfortheposition
potentia llaw and asbefore K
eshradiusofcur aturefor
componentforceforthe L
andfor thev elocity
tionofQ , . Withthis F ig. 1.
o f thecentreofcur ature C , inP1C oL
ughP1. Withthiscentreofcur ature
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T I O N O F D Y NA M IC A L PR O B L E MS
r ature describeanarcP1P2Q2ma ing
a lf the lengthintendedforthethirdarc
atureforpositionQ2 draw anarcP2PQ3 ;
e. Thisprocessiswelladaptedfor
ia landerror methoddescribedinmy
TestC ases o f theMa w ell-B o lt mannDoctrine
Energy " ~ 1 ; P roc. Roy . Soc. J une11
.
e( fig.2 hasbeendrawnwithgreat
nterestingsuccess inthe" tria landerror
nd simplestorbit bymysecretary
orthecaseofmotiondef inedby the
enononeof thelinescuttingthe
~ , andatf irstatarandomdistancef romthe
sw or edaccordingtothemethod
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dw asfoundtocutthea iso fx atanobli ue
es w ithunchangedenergy -constant w ere
sat greaterorlessdistancesfromthe
asfoundtocutthea iso fx perpendicularly .
partof theorbit andisshowninfig. 2
erto completetheorbit whichis
idesof thea iso fx andy .
motionrelatedtotheLunarTheory
oonbe infinitelysmallincomparison
h andtheearthandsuntoha euniform
heircentreofgra ity . Let( x , y be
e lati etoO X inlinew iththesun outw ards andOYperpendicularto it inthedirectionof theearth s
now ne uationofmotionre lati e ly to
g i e s f o r th e e u a ti o ns o f t he m oo n s m o ti o n
rom0( theearth o f thecentreofgra ity
,
t d . .. .. .
,
,
of theattractionsofthesunand earth
heangularve locityof theearth sradius ector. F romthiswef ind forthere lati e -energye uation
+ a 2 + y 2 - . .. .. .. .. 3 )
ant andforthere lati e -cur aturee uation
d t N d t2
” ~ -- -- ~ ~ ~ ( 4 ,
( d z + dy2 2 d 2 + dy2
onentperpendiculartothepath ofthe
, w it h
_C dV ( 5
. .. . . .. . . .. . )
( 6 .
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L U T I O N O F D Y N A MI C AL P R O B L E M S 5 1 9
n sv e locityandptheradiusofcur atureof
to t h e re o l i n g pl a ne X O Y w e h a e
a 2 y - V . .. .. .. .. .. .( 7 ,
. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . 8 .
ss andahisdistancef romtheearth
smassinfinitelysmallin comparisonwith
......... 9 ,
1 V = 2 a + m .. . .. . .. . . 1 0 ,
y2 2 r
h s m as s a n d r = V / x 2 + y 2
2 a 2 - 2 a 2 + + m . .. . .. . .. . 1 1 .
= l a n d m = b , f o r si m pl i ci t y in t h e
f llows w eha e
2+ - . . . . . .. . . .. . . .. . . .. . . .. . . .. . 12 ,
. . .. . .. . .. . .. . .. . . 1
.................................. 15 .
2 a nd ( 1 ) , G . W . Hi ll h as w it h fo ur
foundx andye plicitly intermst forthe
ase whichgi esthesimplestorbit
o l i n g pl a ne X O Y ; o f wh i ch t h e on e wh i ch
iationf romthewell- now n" v aria tiona l
unartheory isasymmetrica lcur ew ith
gcuspscorrespondingtothemoonin
supposedthistobethemoste tremede iationf romthevaria tiona lo a lpossible foranorbitsurrounding
hisMethodesNou ellesde laMdcani ue
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2 , admiringj ustly themannerinw hichHill
ment" studiedthesub ectof f initeclosedlunarorbits po intsout
respondingto
ding Hil l s w rongly
uspedorbit. MrHillte lls
ticism.Thelabour
ccurateanalytical
e sloopedorbits by
dprobablybev erygreat.
t itm ightinterest
stoapplymygraphic --
t leastoneof }
ts inourPhysica l( and
ry intheU ni ersityof /
presentsa loopedorbit
outaccordinglybyMr7
eO f f icia lA ssistanto f
losophy fromthe
1 5 a b o e . T he i n it i al v a l ue s
r e w erex = 2
- 1 0 a nd t he re fo re F i g . 3
8 .
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E V E R Y P RO B L E M O F T w o F R E E DO M S I N
NA MIC STO THEDR A WINGOF GEODETIC
C E O F G I V E N S PE C I I C C U R V A T U R E .
c i at i on R ep o rt 1 8 92 p p . 65 2 6 5 ; N a tu r e V o l . X L V I .
8 6 .
aseof two- f reedommotionispro ed
pondingcaseofthemotionof amaterial
edynamics w ithanygi env a luefor
theresultantve locity q , a tanypo int
ow n fu nc ti on o f( x , y , b ei ng g i e n by t he e u at io n
tentia la t(x , y ; ande eryproblemdepends
f ds( theMaupertuis" action ) isa
t S o f the infiniteplane f indasurface
nitesimaltriangleA B ' C drawnonithasits
acorrespondingtriangleA B C inthef ie ld X ,
odenotingtheva lueofq atanyparticular
theplane. B y theprincipleof leastactionw esee
nS , correspondingtopathsonS are
mniccaseofmotion o faparticleonS ,
ompleterepresentati eofthemotionon
nderforcew ithanyarbitrarilygi enfunction
danyparticulargi env a lue E forthetota l
article .
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atthesurfaceS , tobefoundaccording
a n d th a t it s s pe c if i c cu r a t ur e ( G a us s s n a me f o r th e
lcur atures atanypo intise ua lto*
n o f th e f in d in g o f S . A s o ne e a m pl e
efulnessofthis methodindynamics
cmotionof anunresistedpro ectileis
eodeticli nesonacertainfigure of
e plicite uationise pressedintermsof
redby thetransformationfromorbitsontheplaneto
S . F orak inetictriangleontheplanethee cessof the
nglesistheareamultipliedbyV 2logq ( cf . supra p. 514 .
stheareamultipliedby theGaussiancur ature. B ye uating
lows. On thesek inetictransformations cf . Larmor P roc.
1 8 84 D a rb o u , T h eo r ie d e s Su r fa c es V o l . II . l i r e v .
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F " M E RC A TO R S " P R O J E C TI O N P E R O R M E D
CA LINSTRU MENTS.
o l . X L V I . S e p. 2 2 1 8 92 p p . 49 0 4 9 1.
erali ingMercator sPro ectionis
ommunicationtoSectionAofthe
recentmeetinginEdinburgh entitled
ProblemofTwoF reedomsinC onser ati e
fGeodeticLines onaSurfaceofgi en
A nabstracto f thispaperappearedin
mer commonlyk now nas" Mercator
me , ga etothew orldhischart now of
g at i on . I n it e e r y is l an d e e r y ba y e e r y
n e i f n ot e t e nd i ng o e r m or e t ha n t wo o r
orfarthernorthandsouththan a
rthreedegreesof longitude isshow nvery
eshape: rigorously so if ite tendso er
n infinitesimaldifferenceoflongitude.
ointersectinglineson thesurfaceof
orouslywithoutchangein thecorrespondingangleonthechart.
imaginedas beingmadebycoating
bewithathinine tensiblesheetof
fore ample( forsimplicity how e er
tensiblebutinelastic -cuttingaw ay
ttedfromthechart cuttingthesheet
thatof 180~ longitudefromGreenwich
ngthesheete erywheree cepta longthe
ea llthecirclesof la titudee ua linlengthto
e uator andstretchingthesheetinthe
thesameratioasthe ratioinwhich
stretched w hilek eepingatrightangles
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themeridiansandtheparallels.The
e laidoutf la torrolledup asapaper
edMercator schartforabodyofany
erical isaflat sheetshowingforany
be drawnonapartof thesurfaceof
glineswhichintersectat thesameangles.
te dimensionscanonlyrepresentapart
a finitebody ifthebodybesimply
ay if ithasnoho leortunnelthroughit.
chor ringcanob iouslybemercatori ed
een forthecaseof theglobe thattwo
i ethew holesurface anditw il lbe
ochartssuffice foranysimplycontinuous
ere tremely itmayde iatef romthespherica l
a l fo r 1 84 7 i t s e d it o r L i o u i l le g a e a n
n accordingtow hich if thee uationofany
en aseto f l inesdraw nonitcanbefound
hesurfacecanbedi idedinto inf initesimals uares by these linesandthesetof l inesonthesurface
ngles. Now itisclearthatifw eha e
surfacethusdi idedinto inf initesimal
atistosay di idedinto inf initesimals uares
uarestogether a llthroughit w ecan
toonesi eandlay themdow nonaf la t
t withitsfouroriginalneighbours and
ofsurface ismercatori ed. E ceptfor
o lution orane ll ipso id orv irtually
i ou i l le s d i ff e re n ti a l e u a ti o ns a r e of a v e r y
eonly recentlynoticedthatw ecanso l e
withanyaccuracydesiredifthe problem
w hichit isnot bya idofavo ltmeter
orothermeansofproducinge lectriccurrents
be mercatori edinthin sheet
essthroughout. B y thinImeanthatthe
llf ractionof thesmallestradiusofcur atureofanyparto f thesurface.
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T IO N O F M E RC A TO R S P R O J E C TI O N
of thesurface N S andapply the
atthesepoints.
ee lectrodesof thevo ltmeter tracean
ascloseasmaybearoundoneelectrode and
ne F , asnearasmaybearoundtheother
setwoe uipotentials E I tracea large
differente uipotentials. Di ideoneof the
F ] i n to n e u a l pa r ts a n d th r ou g h th e
nescuttingthew holeseriesofe uipotentialsatrightangles. Thesetrans erse linesandthe
thew holesurfacebetw eenEandF into
Ma w ell E lectricityandMagnetism ~ 651 .
toonesi eandplacethemtogether
Thusweha eaMercatorcharto f thew hole
ationcorrespondtothenorthand
scharto f theworld andourgenera li ed
fillingtheessentialprincipleofsimilarity
ybeconstructedfor asphericalsurfaceby
opointsnot necessarilythepolesatthe
er. If thepo intsN Sare inf inite lynear
ngMercatorchartforthe caseofaspherical
ographicpro ectionof thesurfaceonthetangent
f thediameterthroughthepoint C
nthis casethee uipotentialsand
sonthe sphericalsurfacecuttingNSat
ingit respecti e ly .
o thersurfacew emaymercatori eany
A B C D b o un d ed b y f ou r c ur e s A B , B C
anotheratrightanglesasfo llows. C utthis
metallicsheet totwoofits opposite
f o r in s ta n ce f i i n fi n it e ly c o nd u ct i e b o rd e rs .
vo lta icbattery totheseborders and
uipotentiall inesbetw eenA B andDC .
w eenconsecuti ee uipotentialsintos uares ,
achdistantf romthene tby thesame
raw cur escuttingperpendicularly thew hole
Thesecur esandthee uipotentia ls
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YNA MIC S [ 57
nto inf initesimals uares. E ua li e the
getheronthef la tasabo e.
ticalinstrumentsbywhichwecan
satrightanglesto asystemalreadydrawn
hematicalinstrumentsaltogether and
i idingintos uaresbye lectrica linstrumentsasfo llow s. R emo etheconductingbordersf romA B , DC
eborderstoA DandB C applyelectrodes
rs andasbeforedraw ne uidifferent
ondsetofe uipotentials andthef irst
area intos uares.
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TOR C HA R TONO NESHEETR EPRESENTING
YCO MPLEXLY CO NTINUO U SCLO SEDSU R ACE .
l . X L V I . O c t . 6 1 89 2 p p. 5 41 5 42 .
dbyanyperforation itssurface isca lled
w e ercomplicateditsshapemaybe. Ifa
forations ortunnelst itsw holebounding
pe lycontinuous" ; duple lyw henthere is
n+ 1 -ple lyw hentherearenperforations.
p ofnanchor-rings( or" toroids )
re lati eposit ions isacon enientand
ofan ( n+ l -ple lycontinuousclosed
X X
aticsof thegenera lproblem seeR iemann Gesanmmelte
" T h eo r ie d e r Ab e l s c he n F u n c t io n en ( 2 . L eh r si t e a u s
" Nachlass " F ragmentausderA na lysisSitus a lsoB etti
. i . ( 1 8 70 - 1 a l so F o r sy t h s T h eo r y of F u n c t i on s a n d
adeepho llow notthroughw ithtw oopenends. The
propriatefortheapertureof ananchorring.Neither
be ingune ceptiona llya a ilable Iamcompelledtousethe
n.
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aq uadruple lycontinuousclosed
hinsheetmeta l uniformastothic nessand
ity throughout. Toprepareforma inga
titopenbetw eenperforationsC andB ,
space inthemannerindicatedat2 1 and
cti eborderstothe twolipsseparated
apply thee lectrodesofav o lta icbattery to
fmo ablee lectrodesofav o ltmetertrace
av ery largenumber( n-1 o fe uidifferente uipotentialclosedcur esbetweenthe+ and-borders.
ee uipotentials intopartseache ua lto
perpendicularlyacrossittothe ne t
ideof it andthroughthedi isiona l
utt ingthee uipotentialsatrightangles.
am-lines. Theyand the( n+ 1 closed
ingthe inf inite lyconducti eborders di ide
minfinitesimals uares ifnmbethe
chw efoundinthee uipotential. The
wthegeneraldirection oftheelectric
f thecomple circuit eacharrow
metalshelloneitherfar ornearside
paper.
tream-linesintheneighbourhoods
e d i n or d er o f t he s t re a m1 2 3 , 4 w e
ipsthereis onestream-linewhich
yononesideandlea esitperpendicularly
cIca llthef lu -shed- line(or forbre ity
the liptowhichitbe longs. Thestream- lines
-shed onitstw osides passinf initely
ofthelip andcomeininfinitelynear to
u -shedonitstw osides. LetF 1 F 2
ownonthediagram bethepo intsonthe+ terminal
s he d s of t h e li p s 1 2 3 , 4 p r oc e ed a n d
bethepo intsatw hichthey fallonthe- lip.
& a m p c . d e no t e th e p oi n ts o n t he f o ur l i ps a t w hi c h
by the irf lu -shedlines.
pre iousarticle ( " Genera li a tionofMercator sPro ection" ) , in~ 3 , andinlastparagraphbutone aremanifestlywrong andmust
htherulegi enfordi idinginto inf initesimals uares in
r ec t ed i n t e t p . 5 25 s u p r a.
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A RTO F A C YC LICA LLYC ONNEC TEDSU R A C E529
P , 1 , P 4 1 4 p b e th e di ff er en ce s of p ot en ti al
r om S 1 t o T , T t o S 2 . .. S 4 t o T4 a n d T4 t o
nedifferencesofpotential.W eare
Mercatorchart. Wemightindeedha e
orateconsiderationsandmeasurements
eofmypre iousarticle butthechart
inf initecontractionate ightpoints the
T1 . . . S4 T4. Thisfault isa o ided
thewholesurfaceonafinite scalein
thefollowingprocess.
beofthinsheetmetal ofthesame
ityasthatofourorigina lsurface andon
a r f o u r p oi n ts A h h 2 h , h 4 a t c on s ec u ti e
erenceproportionalrespecti elytothe
neswhichwefind betweenF 1andF 2
a nd F 4 F 4 a n d F , o n th e + l i p of o ur o ri gi na l
h 2 h h 4 d ra w l in e s pa r al l el t o t he a i s o f
nte ua ltothetota lcurrentw hich
he -lipthroughthe originalsurfacebe
esentcylinderbya v oltaicbatterywith
sonthe cylinderv eryfardistantonthe
Mar onthecy lindere ightcircles
t d is t an c es c o ns e cu t i e l y pr o po r ti o na l t o lI p 2 1 2
4 a nd a bs ol ut el y su ch t ha t 1l p & a m p c . a re e u al t o th e
alsfromoneanotherin order.
themeta lbetweenthecirclesK1 and
K 5 a n d K 6 K 7 a n d K 8 o n th e pa ra ll el s tr ai gh t
h , h 4 r e sp e ct i e l y. E n l a rg e t he s e ho l es
sothatthealteredstream-lines
, h 4 ( t h e s e po i nt s s up p os e d fi e d a nd v e r y
the irf lu -sheds. Whilea lw aysmainta ining
heholesandaltertheir positionsuntilthe
potentia lintheirl ipsbecome15 12 1 , 14
ntialbetweenthelips insuccession
. Inthuscontinuouslychangingtheho leswe
sarbitrarily butto f i ourideas w e
waysmadecircular.This ma esthe
e ceptthedistancef romthecircleHof the
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chmaybeany thingw eplease pro idedit
nto thediameterofthecylinder.
thusproposedisclearlypossible and
ue.Itis ofahighlytranscendental
aproblemformathematica lanaly sis butan
a landerror gi esitsso lutionbye lectric
uiteamoderateamountoflabourif moderate
eenf ina llyad ustedtofulf i lourconditions draw byaidof thevo ltmeterandmo ablee lectrodes the
abo ethegreatestpotentia lo f l ip1 andfor
ta lo f l ip4 andbetw eenthesee uipotentials w hichwesha llcallf andg draw n-1e uidifferent
hestream- lines ma inginf initesimal
ordingtotherulegi enabo einthe
oundthatthenumberofthe streamlinesism thesameasonour originalsurface andthewhole
uaresonthecylinderbetweenfandg
roughatfandg cutitopenbyany
andopenitoutf la t. Wethusha ea
yfourcur escuttingoneanotherat
dedintomninf initesimals uares correspondingindi idua lly tothemns uaresintow hichwedi ided
firstelectricprocess.Inthis chart
an scorrespondingtothe lips1 2 3 , 4
ere ise actcorrespondenceof the irf lu shedsandneighbouringstream- lines andof thedisturbances
ee uipotentials w iththeana logous
originalsurfaceascut forourprocess.
tricalproblemwasa necessityforthe
w hichIha ebeenoccupied andthisis
gitout thoughitm ightbeconsideredas
elf .
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N.
DUC TIONO F ELEC TR IC C UR R ENTS
GR A PHWIR ES.
ciationR eport 1855 P t. II. p. 22.
ys. Papers V o l. II. A rt. l x v . pp. 77 78.
R OU GHSUB MA R INEC A B LES ILLUSTR A TED
TEDTHR OU GHA MODELSU B MA R INE
E D B Y M I RR O R G A L V A N O M E TE R A ND B Y
ersinScotlandTrans. V o l. xv I. March18 187 , pp. 119
ys. Papers V o l. II. A rt. l x x v . pp. 168-172.
TR A TIONSO F THEMA GNETIC A NDTHE
R Y E F E C TS O F T R AN S PA R EN T B O D I ES
HT.
. V o l . v I I I . J u n e 12 1 8 56 p p . 15 0 -1 5 8 P h il . M ag .
5 7 p p . 1 9 8 â € ” 2 0 4.
Le c tu r es A p pe n di F p p . 56 9 -5 8 . i
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ND W A V E S I N A ST R ET C HE D U N I O R M C H A IN
RO STA TS .
Soc. P roc. V o l. v I. A pril8 1875 pp. 190-194.
rotating fly-wheel frictionlessly
eable f ramew or orcontainingcase. A
einwhichnot onlythefly-wheelbut
ymmetrica lroundthea iso f rotationof
ernategyrostatsandmasslessconnectinglin s andlettheconnection
r e j o i n t st a t e ac h c + 2
simplicity at
esof thegy ro- - - - -
ne linew hen
ight.This /Ci+
uil ibriuma is.
so f the
glin srespec- /
a n d X a n d /u t h e
thea iso f G-1
culartoit
o fagy ro- / i
landcase included
tof inertiao feachf ly -wheela lone roundits
tobetheangularv e locityoncegi ento
mainingalwaysthesamebecauseofthe
ots. Insteadof f irst in estigatinginf initesimalmotionsingeneral w eshallf irstta etheparticular
t limitedtobeinginfinitesimal.Then
iesimal bycomposit ionofcircular
en t s c f . Ro u th s A d a n ce d Ri g id D y na mi c s ~ 4 1 9. C f. a l so
aturalPhilosophy ~ 109.
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N
ntrarydirections andwithdifferent
tothe generalsolutionoftheproblemof
lengthofsuchachain tobeplaced
openplanepo lygon and theendsof
in g h el d f i e d b y un i e r sa l f le u r e j o i n t s l e t
otionperpendicularlytothisplanethat
esasarigidpolygonrotatingroundthe
w ithagi enangularv e locityn: re uired
andtheforcesonthef i edends sothat
tse lf maycontinuere o l inginthe
ingandgy rostaticlin s. . . Ci Gi C i+ l
idenotethe inclinationsofC iandGito
ds andlety ibethedistanceof thecentre
ne. Weha ethegeometricalre lation
4+ sinO i + c sin.+l......( 1 .
ro f theuni ersa lf le ure j o int( Thomson
e a ch g y ro s ta t ic l i n m o e s a s if i t s a i s w er e
ningthef i edends andthere j o inedtoa
ersa lf le ure j o int. Hencethe instantaneous
bisectstheanglew r-O ibetw eenthe line
o iningthef i edends. Itsangularv e locity
isis2n sin10i.The componentsofthis
ndinaplaneperpendiculartothea iso f
um-a is are
cos10i orn( 1-cos0i andnsin6i.
ntsofmomentumare
1 - co sO i a nd .. n si nO i .
to fmomentumofGi( caseandf lyw heel roundthea iso fGiis
cos oi + X ' w.
sn O i r o un d t he e u i li b ri u m a i s a nd
eplaneof thechain w eha e forw hole
omentumroundthelastmentionedline ,
1 - c os 0 i + X ' o s in 0 i -/ in s in 0 i co s 0i .
ouldbechangedhenceforth.
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H AI N O F G Y RO S T A T S
angularv elocitynina planeperpendiculartothee uilibriuma is andtheremustthereforebea
n ( 1 - co s0 i + X ' o s in i -F L n si n6 1c os O i ,
isperpendiculartotheplaneof the
thea iso fGi. Thedirectionof thiscouple
hastotendto increasetheangledi. We
wnthee uationsofmotion( ork inetic
hecomponentparalle lto thee uil ibrium
econnectinglin s mustbethesamefora ll.
onsofmotionparalle lto thee uil ibrium
eP: sothatPseciisthepull inthe
TheappliedforcesonGiarethepullso fCi
eso l ingthemweha e: Para lleltoe uil ibriuma is. Perpendiculartoe uil ibriuma is.
r i
P t an i + 1 l
ntreof inertiao fGi w eha ef ina lly
oe uil ibriuma is
i p e r p en d ic u la r t o e u i li b ri u m a i s
4 i + + t an A i . I g c os i
briuma isanddirectiontendingto increase
ofcentreof inertiao fGi
- t a n i = 0 . .. . .. . .. . .. ( 2 ,
( 1 - c os O i + X ' } s i n O i -/ Ln si n0 ic os O i
s di ( t an r s+ j + t a n i } . . . 3 ) .
2 , ( 3 ) , a p p li ed t o ea ch g yr os ta ti c li n , g i e a s
e ar e o f un n o wn q u a nt i ti e s % , 0 i y i i f
chaintobea gyrostaticandtheother
sothattherebethesamenumberof thetw o
sandinclinationsareinfinitelysmall
dif ferences asappliedbyLagrangeto
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N
nsofa" l inearsystemofbodies ( acaseof
becomesw hen w = 0 iscon eniently
1 , ( 2 , a n d ( 3 ) , w h en w e c an n e gl e ct t h e
, b e c om e
9 ~ + c.. 4 ,
' - ) = 0 . .. . .. . .. . .. . .. . .. 5 ,
P~ g IO - B i + ~ ( ' 3 r j + ~ r ) . ... ... .. 6 .
perationsuchthat
.. . . .. . . .. . . .. 7 ,
fi.
e
3
P g
a n d e l im i na t in g ' b e t we e n ( 4 a n d ( 5 , w e f i n d
p + P g( p ~ + 1 2 CP I yi = o
. .. 8 ;
course w ithO ior' 3 isubstitutedfory i.
ha etheq uadratic
0 . .. . .. . .. . .. . .. . . 9 ,
- A n2
2. .. . . .. . 10 .
o r V 1 1 e = s i n Ia . 1 .. .
ecomes
- 1
so lutionof (8 is
a . . . .. . . .. . . .. . . .. . 12 ;
ordinateof thecentreof inertiao fG
n a s z e r o , w e ha e
.. . .. . .. . .. . .. . ) .
+ B s in ~ - - - . . . . .. . 1 4 ;
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N S I N A C H AI N O F G YR O S T A TS 5 7
tesof inertiao f the lin slieonaheli
l e ng t h is
. . .. . . 15 ,
of particlesinthewa e-length1.
n........................... 16 ;
enotetheve locityofpropagationof the
w a emadeupbypropersuperposit ionof
e
/ a .. . .. . .. . .. . .. . .. . .. . 1 7 ;
,
. .. .. .. .. ( 1 8 ;
' n _ 2 P c /
w -tL 2 m
' n e- _ e n2
se pressionbecomeeache ualto
small thatistosay w henthew a e
nte lygreatincomparisonw iththedistance
e ighbouringmolecules andthee pression
.. . . .. . . .. . . .. . . .. . . ( 20 ,
ocityofpropagationofw a esinauniform
c be ingthemassperunito f length and
b u t v e r y sm a ll w e ha e a p pr o i m at e ly
_ _ 1 7 2 + c 2
77 - ( g )
e length. A ndby theappro imate
r V , o r n l/ 2 7r m n 2 = 4 - r 2 P ( g + c / 1 2 ap p ro i m at e ly . A ls o b e ca u se e a ch l i n i s v e r y s m al l i n al l i ts l i ne a r
nw ith1 p / ml2andX ' / m12areeachvery
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ablewithg2/12.Hencethesecond factorof
o imately
twofactors st i l lappro imate ly
I
+ c / l= 0 a pp ro i ma te ly . Th en
( g c
P g + c
m
5 1 8 84 .
tionnow consideredsupposes n tobev erygreat
4 1 8 8 ) .
R Y O F L I GH T .
eA cademyofMusic Philade lphia underthe
inInstitute Sept. 29 1884.
a n l i n In st it ut e V o l . LX X X V I II . No . 1 88 4 p p. 3 2 1 â € ” 3 4 1
i . 1 88 4 p p . 91 - 94 1 1 5- 1 18 .
uresandA ddresses V o l. i. pp. 300- 48.
A N D G R E E N S D O C T RI N E O F E X T R A NE O U S
NDYNA MIC A LLYF R ESNEL SK INEMATIC S
R AC TI O N .
. P ro c . V o l . x v . D e c. 5 1 8 87 p p . 21 - 3 ; P h il . M ag .
1 8 88 p p . 11 6 -1 2 8.
altimoreLectures pp.228-248.
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SISF O R ELEC TR O-MA GNETIC INDUC TION
RC U I T S W I T H C O N S EQ U E N T EQ U A T IO N S O F
IX EDHO MOGENEOU SSO LIDMA TTER .
ciationR eport 1888 pp. 567-570 Nature
p .5 6 9- 5 71 .
alformulastillneededforcalculation
uid motion w hichforbre ity Ica ll
Tertiary def inedasfo llow s: Half theve locity
mericallyanddirectionallywiththe
emolecularspin atthecorresponding
( short butcompletestatement the
ry istw icethespininthePrimary and
elocity intheTertiary isthespininthe
ertiarythemotionisessentially
andineachof themwenaturally
compressible f luidasthesubstance. The
rbitrarilyrestrict byta ingitsfluid
edtheproblem: Gi enthespinin
n to f indthemotion. Hisso lution
ntialsof threeidealdistributionsof
ingdensit iesrespecti e lye ua lto1/ 47r
nentsof thegi enspin and regarding
ialsasrectangularcomponentsofv elocity
n ta ingthespininthismotionasthe
edmotion. A pplyingthissolutionto f ind
ndary f romthev elocity inourTertiary
elocitycomponentsinourPrimaryare
ldistributionsof gra itationalmatter
rbre ity tosignify incompressible fluid
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N
especti e lye ua lto1/ 47rof thethree
ourTertiary.Thispropositionispro ed
5below bye pressingtheve locitycomponents
thoseofourSecondary andthoseof
hose ofourPrimary andtheneliminatingthev elocitycomponentsofSecondarysoastoha ethose
ofthoseofPrimary.
edso lidorso lidsofnomagnetic
ofelectricmotionin whichthereisno
andthereforenoincompleteelectric
hesame anycaseofe lectricmotionin
ectric currentagreeswiththedistributionofv elocityinacase ofli uidmotion.Letthiscase
numericallye ua lto47rtimesthee lectric
Tertiary.Thev elocityinourcorrespondingSecondaryisthen themagneticforceoftheelectric
heve locity inourPrimary isw hat
ca lledthe" e lectro -magneticmomentumatany
urrentsystem andtherateofdecrease
ycomponentof thislastv e locityatany
ingcomponentofe lectro -moti e force due
tionoftheelectriccurrent systemwhen
nge. Thise lectro -moti e force combined
e if there isany constitutesthew hole
anypo into f thesystem. HencebyOhm s
lectriccurrentatanypo intise ua lto
multipliedintothesumofthe correspondingcomponentofelectrostaticforceandtherateof decrease
espondingcomponentofv elocityof
s ym bo ls l et ( u 1 v l w I , ( u , v W , w 2 ,
w ) d e no t e re c ta n gu l ar c o mp o ne n ts o f t he v e l oc i ty a t
, y z ) o f ou r P ri m ar y S e co n da r y a n d Te r ti a ry .
d - d1 ul
' d dy. ... .. 1 ,
e ll- now nelementary theoremV 2V = -47rp.
n et i sm ~ 5 1 7 ( p o s t sc r i pt ( c .
eism ~ 604.
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U A T I O N S F O R E L E CT R IC P RO P A G A TI O N 5 4 1
d 2 du .
= â € ” - -- W 3 = ...... 2 .
d ' d dy
w 2 fr om ( 2 b y ( 1 , w e fi nd
d l 2 d u ..
) 2 _ ( ^ d t -+ d y2 + t d 2 ) ^ } & a mp c .... 3 ) .
~ 2 o f incompressibility inthePrimary
s
v 3 = - V 2 V 1 w = - V 2 w1 .. .. .. .. . 5 ,
v i i . ( N o e m be r 1 8 46 o f m y Co l le c te d
a lPapers( V o l. I. ,
. .. . . .. . . .. ( 6 .
6
edproofof ~ 3 .
denotethecomponentsofe lectriccurrent
n t h e e le c tr i c sy s te m of ~ 4 s o t ha t
4 T r y= v 3 = - V 2 l 4 rw = W 3 = - V 2 w .. . 7 ,
4 , g i e
electro-moti eforcedueto changeof
4 V 7 r V - 2 d .. - 2. . ( 9 ,
dt
electrostaticpotentia l w eha e forthe
cmotion( . 4
dd
-1 ,
ofthespecificresistance.
a te r ni o ni c r ea s on s t a e s V 2 t h e n eg a ti e o f m in e .
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N
accordingto~ 4 w emay
ent put
/ + = . .. ... ( 1 ,
x cd
a le nt s to ( 1 0 ,
K ; d t 2 ) . .... 12 .
iminationofT maybeillustrated
ample a f initeportionofhomogeneoussolid
e( a longthinwirew ithtw oends ora
s ol i d gl o be o r a l um p o f a ny s h ap e o f
ogeneousthroughout , withaconstant
edthroughitby electrodesfroma
rsourceofe lectricenergy andw ithproper
o leboundary soregulatedastok eepany
aate erypo into f theboundary w hile
late throughtheinteriorbyv arying
riortoit.There beingnochanging
posit ionof~ 4 Pcanha enocontributionf romelectrif icationw ithinourconductor andtherefore
. .. . . . 1 ) ,
nd ( 1 1 , g i e s
.. . . ( 14
( 1 4 w e h a e f o u r e u a ti o ns f o r th r ee u n n o wn
ecaseofhomogeneousness( cconstant are
ree because inthiscase(14 fo llow sf rom
4 issatisf iedinit ia lly andthepropersurface
pre entanyv io lationof itf rom
ntthroughoutourf ie ld thefoure uations
remutua lly inconsistent f romw hichitfollows
nchangingnessofe lectrif ication(~ 4 isnot
ngandimportantpracticalconclusion
are inducedinanyw ay inaso lidcomposed
te lectricconducti it ies( piecesofcopper
pe f i e d t og e th e r in m e ta l li c c on t ac t t h er e
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U A T I O N S F O R E L E CT R IC P RO P A G A TI O N 5 4
inge lectrif icationo ere ery interface
conclusionwasnotat firstob iousto
obyanyoneapproachingthesub ect
mathematicalformulas.
heterogeneousnessuntilwecome
rificationandincompletecircuits letus
ntehomogeneousso lid. A s( 8 ho ldsthrough
rsupposit ionin~ 4 andasK isconstant
dthroughallspace andtherefore = 0 w hich
dw
- â € ” ; = _ -.. . 1 5 .
d t
esssimplythek nownlawofelectromagneticinduction.Ma well se uations( 7 of~ 78 ofhis
becomeinthiscase
2 & a m p c .. .. .. .. .. .. . 1 5 ) ,
thin , accordingtoanyconce i able
ctricconducti ity w hetherofmeta ls or
r es i ns o r w a , o r s he l la c o r i nd i a- r ub b er o r
s orso lidorli u ide lectro lytes be ing as
dforcompletecircuitsby thecuriousand
to menotwhollytenablehypothesis
610 , forincompletecircuits.
uggest forincompletecircuits
y inge lectrif ication issimply thatthe
-moti eforcedueto electro-magnetic
du/ dt & amp c. Thus forthee uationsof
p ly t o k e e p e u a ti o ns ( 1 0 u n ch a ng e d w h il e
b ut i ns te ad o fi t ta i ng ' V / 2 du d _ dw = V d d p
d = d t. .. .d p ( 1 6 ,
d t
the electricdensityattimet andplace
a n d ' v ' d e no t es t h e nu m be r o f el e ct r os t at i c un i ts i n t he
efundamenta lpostulatethatrateofchangeofe lectricdisplacementoperatesascurrent andsoma esa llcurrentsf low ef fecti e ly incomplete
nowonlyofhistoricalinterest representsprobably
fMa w ell sscheme promptedbyHert ' sthenrecent
a es.
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N
ectricq uantity.Thise uatione pressesthattheelectrificationofwhichT isthepotential
nanyplace accordingaselectricity
more inthanout. Wethusha efour
n d( 1 6 , f or ou rf ou ru n n o wn s u v , w P a nd
olutions withnothingv agueordifficult
l ie ew henunderstood by the irapplication
rtoconce i able idea lproblems suchas
ryortelephonicsignalsalong submarine
andlines electricoscillationsina finite
form transferenceofelectricitythrough
p c . & a m p c . T hi s h o we e r d o es n o t pr o e m y
tisre uiredforinformingusastothe
ctsofincompletecircuits and as
ed it isnoteasy to imagineanyk indof
decidebetweendifferenthypotheses
netry ingtoe o l eoutofhisinner
hemutualforce andinductionbetween
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R ENC EOF ELEC TR IC ITYWITHINA
L I D CO N D U C T O R .
c i at i on R ep o rt 1 8 88 p p . 57 0 5 7 1 N a tu r e
.5 71 .
onandformulasofmypre iouspaper
, a n d ta i n g p to d e no t e 47 r t im e s th e e le c tr i c
p ac e ( x , y z ) , w e ha e
2f du d dwA.*
a + d t. . .. . .. . .1
y d j
, w T b y t hi s f ro m ( 1 0 , w e f i n d o n t h e
t
. . . .. . . .. . . .. . . .. . 18 .
ryconditions whenafinitepieceof
b e c t i n o l e s c on s id e ra t io n o f it v , w
u a ti o ns ( 1 7 a n d ( 1 2 m u st b e t a e n i nt o
esub ectisaninf initehomogeneoussolid
w enow suppose ittobe ( 18 suf fices. It is.
remar thatthisagreesw iththee uation
ouse lasticf luid foundfromSto es s
irw ithv iscosity ta enintoaccount and
, w g i e n by ( 1 7 a nd ( 1 0 , w he n p ha s
ewiththev elocitycomponentsofthe
ndnaturalenoughsuppositionbemade
actsonlyaga instchangeofshape andnot
mewithoutchangeofshape.
me
s . .. . .. . .. 1 9 ,
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R OPA GA TIO N [ 68-70
utionin( 18 ,
. .. . .. . .. . .. . .. . .. . . 2 0 ,
* * 1 20. * * * * * * * * * * * * ( 2 0 ,
r2 + ) + ( . . . .. . . .. . . .. . . .. . 21 .
h e q u a dr a ti c ( 2 0 f o r q ,
.. .. .. .. .. . 2 2 .
totheSectionnumericalillustrations
cillatorydischargeweregi en.
N S O F F O U R I ER S L AW O F D I F U S IO N
A G RA M O F C U R V E S W I T H A B S O L U T E
S.
c i at i on R ep o rt 1 8 88 p p . 57 1 -5 7 4 N a tu r e V o l . x x x v I I I.
y s. Papers V o l. III. A rt. x c ii i . pp. 428-4 5.
LIGHTNINGC ONDUC TO R SA TTHE
N.
c i at i on R ep o rt 1 8 88 p p . 60 - 6 06 N a tu r e V o l . x x x v I I I.
46 .
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O N A N D R E R A CT I O N O F L I GH T .
o l . x x v I . p p. 4 1 4- 4 25 N o . 1 8 88 a n d pp . 5 00 5 0 1
a l ti m or e L ec t ur e s p p . 17 4 3 5 1 - 5 4 4 0 7.
TY A NDPONDER A B LEMA TTER.
ne e rs J o u rn a l V o l . x v I I I . 18 9 0 p p . 4- 7 ( I n au g ur a l
1 88 9 .
s.Papers V ol.III.Art.cii. pp.484-515.
M F O R T HE C O N S T IT U T I O N O F E T HE R .
oc . P ro c . V o l . x v I I . Ma r ch 1 7 1 8 90 p p . 12 7 -1 2 .
ys. Papers V o l. III. A rt. c. pp. 466-472.
I SC O U S L I Q U I D E Q U I L I B R IU M O R M O T IO N
D E Q U I L I B R IU M O R M O T IO N O F A N
A LLEDF O R B R EV ITY" ETHER ; MEC HA NIC ALR EPR ESENTATIONO F MA GNETIC F O R C E.
dPhys. Papers V o l. III. A rt. x ci . May 1890
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7 5
IMENTSF O R C OMPA R INGTHEDISC HA R GE
H RO U G H D I F E R EN T B R A N CH E S O F A
LOR DK ELV INandA LEX A NDER
ciationR eport 1894 pp. 555 556.
emetallicpartofthe dischargechannel
wo linesofconductingmeta l eachconsistinginparto fatest-w ire theotherpartso f thetw olines
ape materia l andneighbourhood o f
pectto facilityofdischargethrough
.
asnearlyasweha ebeenhitherto
a landsimilar andsimilarlymounted. Each
tinumwire of' 006cm.diameterand
etchedstraightbetweentwometalterminals
.O neendofthe platinumwirewas
assmounting theotherw asf i edtoa
armformultiplyingthemotion.The
de elopedinthetest-wirebythe
itse longation theamountofw hichw as
traced by theendof themultiply ingarm
a mo ingcylinder. TwoofLord
e lectrostaticv o ltmeters suitablerespecti e ly for
000and1 500 w erek eptconstantlyw ith
theouter coatingsoftheleyden and
heinside coatingsoftheleyden.
thertomadethetwow irestobe
eenofthesamelength. When theywere
utofdif ferentdiameters thetesting
w astobee pected thatthetest-w ire in
hic erwirewasmoreheatedthan the
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EDISC HAR GEINDIV IDEDC IR CU IT
h.In acontinuationofthee perimentswehopetocomparehollowandtubularwires ofthesame
ndsamelengthandsamemateria l.
non-magneticmateria l- fo re ample copperandplatino id-o f thesamelength butofvery
astoha ethesameresistances thetesting
earlye ua l.
rimentsthetestedconductorswere
each' 16cm. diameter 9metreslong and
w h ic h i t w il l b e ob s er e d i s v e r y s ma l l in
msin eachoftheplatinumtest-wires.
w asco iledinauniformheli o f forty
m. diameter.Thelengthoftheheli
edistancefromcentretocentreofne ighbouring
middle oftheothercopperwirewas
mtheceil ing andthetwohal espassed
epointsofj unctioninthecircuit.
wireinthis channelwasmorethan
hetest-wireinthe channelofwhich
se enty-onevarnishedpiecesofstra ight
within theglasstube whichwasas
Thismadethetestingelongationten
rchannel.
w hichw eha emadehasbeenbetw een
conductors.The lengthofeachwas
erof the ironw irew as' 0 4cm. andits
ms. Thediameterof theplatino idw irew as
sstance6 82ohms. Eachof thesew ireswas
eadf romtheceil ing attachedto itsmiddle
neof thetestedconductors . F ourteen
e se enw iththetest-wiresinterchanged
sin whichtheywereplacedforthefirst
leshowsthemeansof theresultsthus
gardingtheelectrostaticcapacitiesof the
o ltagesconcernedintheresults.
ars connectedtoma ev irtuallyone
rofarad w erechargedupto9 000vo lts and
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N
dedchannel. Theenergy therefore in
ewas11105 x 106ergs.Ineach ofthe
o ltsw erefoundremaininginthe j arsaf ter
he lastfour1 400.
incms.
n Energyused
channelcontain-Inchannel containingplatinoid ingiron
2 x 1 0 6 er g s - 0 17 9 4 0 1 22 6 , ) ~ , , - 0 18 6 1 01 8 29 - 0 1 2 47 - 01 2 9
4
4x 106ergs -0182 ) -01276
01828 01244
eelongationofthe test-wireswas
mtheprecedingdescription somewhat
ngtoseethatthemeanresultsinthe
84megalergsofenergyusedaresonearly
etw ocircuitsare inthetw ocases
d1-46. Theconclusionthattheheatingef fect
iththeplatinoidwireisnearly one-anda-halftimesasgreatasthatofthe test-wireinserieswiththe
g notonly initse lf butinre lationto
e se ceedingly interestingandinstructi e
ai epathsforthedischargeof leyden- ars
nLightningConductorsandLightning
renotdecisi e inshow inganygenera lsuperiority
hesamesteadyohmicresistance bute en
emingsuperiorityof theironforefficiency
en- ar. Ourresult isq uitesuchasmight
rome perimentsmadeeightyearsagoby
edinhis paper" O ntheSelf-induction
undConductors .
I I . 18 8 6 p . 4 69 .
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5 51 )
R YO F R E R A C TION DISPER SION
ISPERSION.
c i at i on R ep o rt 1 8 98 p p . 78 2 7 8 ; N a tu r e V o l . LV I I I .
46 5 47 .
altimoreLectures p.148.
N D U L A T O R Y T H EO R Y O F C O N D EN S AT I O N A LR A RE A C TI O N A L W A V E S I N GA S ES L I Q U I D S A N D SO L I D S
L W A V E S I N S O L I DS O F E L EC T RI C W A V E S
C A PAB LEO F TR A NSMITT INGTHEM A ND
V I S IB L E L I G HT U L T R A -V I O L E T L I GH T .
c i at i on R ep o rt 1 8 98 p p . 78 - 7 87 N a tu r e V o l . L I X .
. 56 5 7 P h il . M Ia g . V o l . X L V I . N o . 1 8 98 p p . 49 4 -5 0 0.
ectures pp. 148-162.
O N A N D RE R A CT I O N O F S O L I T A RY P L AN E
N TE R A C E B E T W E EN T W O I S O T R O P I C
I D S O L I D O R E T H E R.
c . Pr o c. V o l . x x I I . De c . 19 1 8 98 p p . 3 6 6 - 7 8 P h il .
eb. 1899 pp. 179-191.
a lt imoreLectures ~ ~ 112-121.
SELLMEIER SDYNA MIC A LTHEOR YTO THE
R O D U C E D B Y S O D I U M - V A P O U R .
c . Pr o c. V o l . x x I I . F e b . 6 1 8 9 9 p p . 5 2 - 5 1 P h il .
r ch 1 8 99 p p . 3 0 2 - 0 8 .
ectures pp. 176-184.
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RO P A G A TI O N [ 8 0 8 1
NO F F O R C EWITHINA LIMITEDSPA CE
D U C E S PH E RI C AL S O L I T AR Y W A V E S O R
CW AV ES O F B O THSPECIES EQ U IV O LU MINALANDIRRO TATIO NAL INANELASTICSO LID.
o l . X L V I I . M ay 1 8 99 p p . 48 0 -4 9 ; V o l . X L V I I I . A ug u st
O c t . 1 8 99 p p . 3 8 8 - 9 ; a l so r e ad a s a P re s id e nt i al
a thematica lSociety cf . V o l. xx x I. J une8 1899
ectures pp. 190-219.
PR O DUC EDINA NIN INITEELA STIC SOLID
R U G H T HE S P AC E O C C U P I ED B Y I T O F A
N L Y B Y A T TR A CT I O N O R R EP U L S I O N .
. P ro c . V o l . x x I I I .J u l y 16 1 9 00 p p . 21 8 -2 5
1900 pp. 181-198 C ongresInternationa lede
st io n d e 19 0 0 V o l . II . p p. 1 - 22 .
ectures A ppendi A pp. 468-485.
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ETHER F O R ELEC TR IC ITYA ND
o l . L. S e pt . 1 90 0 p p . 3 0 5 - 0 7 .
shedinthe lastnumberof the
ofw hichthisisacontinuation Il im ited
hematicaldynamics andmerely
ffindingin itane planationofthe
heU ndulatoryTheoryofLightreferred
agraphs( ~~ 1 18 . Thefo llow ing
tanceofasupplementarystatement
enora lly totheC ongresInternational dePhysi ueatameetingheldinParislastWednesday
temptationtospea ofefforts
roperassumptionsforincludingsomethingofthealliedsub ectsmentionedinthefootnoteon~ 1.
icity w hich fo llow ingLarmor Ia t
it ine itablyoccurstosuggestaspecia l
theconditionstatedinlines 12-22
twouldbeanatom whichbyattraction
paceoccupiedby itsv o lume anda
beanatomwhich by repulsion rarefies
spaceoccupiedbyitsv olume.The
r outsidetwosuchatomsbythe
hichtheye ertontheether within
arentattractionbetw eenaposit i eanda
ngof thesectionsiscontinuousw iththato fNo. 81.
this maybetheresinouselectrification butitmay
.Itmustberememberedthatv itreouselectrificationhas
i emere lybecause it isitw hichisgi enby the" prime
rdinaryelectricmachine.
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N
dapparentrepulsionbetweentwoelectrons
egati e .
ttractionsandrepulsionswould
iminisheddistancethanaccordingto
in erses uare . Thislaw w hichwe
ndC a endishtobetruefore lectric
cannotbee pla inedbystressinether
orhithertoimaginedpropertiesofelastic
mplehypothesis assumingactionatdistances
sofether e plainsitperfectly.Consider
py inginf initesimalv o lumesV , V ' , a t
pothesisisthat theyrepelmutually
V '
. . .. .. .. .. .. .. .. .. ( 1 ;
hedensit iesof thetw oportionsofether
naturaldensityofundisturbedether.
ulsionor attractionaccordingas( p-1 ,
ameorofoppositesigns andz ero ife ither
ansthatetherofundisturbednatura ldensity
actionnorrepulsionfromanyotherportion
blesA epinus doctrineof themiddle
commonly referredtoasthe" onef luidtheoryofe lectricity butnow insteadofe lectricf luid w e
elasticso lidper adinga llspace. A ccordingto
similarelectricatomsrepeloneanother
inv irtueof forcebetw eeneachatomand
t andmutualrepulsionorattraction
ith nocontributi eactionofthe ether
andbetweenthem.
ngthusfreedfromtheimpossibletas
rostaticandmagneticforce is( w emay
ompetenttoperformthesimplerdutyof
cealone.
yinsuperableobstacleagainst.
racticalreali ationhasbeenthe
manywellk nowncasesofmagnetic
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RY
les whetherdueto steelmagnetsor
ringthat inourmostdelicatee perimentsinv ariousbranchesofscience ponderablebodieslargeand
emo edf reelyby forcesof lessthana
ness o famill igram how canw econce i e
eymo etobecapableof thestress
ssionofforce betweenflatpolesofan
gpers uarecentimetretomorethantwo
inessofak ilogram Thisdiff iculty is
pothesiswhichI ha edescribedto
o e . Wemaynow supposethedensityof
se sub ectonly tothe lim itationthatit
disturbsensibly theproportionalityof
ity indif ferentk indsofmatter pro edby
p e ri m en t f o r le a d b r as s g l as s & a m p c .
fK epler sthirdlaw forthedif ferent
bablywemightsafely ifwewishedit
rto beasmuchas 10-6. Iamcontent
tosuggest10-9. This w iththeve locityof
etrespersecond ma estherigidity ( be ing
e l o c it y e u a l to 9 . 10 1 d yn e s pe r s u a re
mewhatgreaterthantherigidity ofsteel
lynotforw antofstrengththatweneed
ceofethertotransmitmagneticforce
opefulo f see ingso l edsomeof the
sw hichmeete eryef forttoe pla in
duction andelectromagneticforce and
eelmagnet bydefinitemechanicalaction
guityusethesimpleword" w eight here becausethis
s andispracticallyusedmoreoften tosignifyamass.
ea inessofamass.
eticf ie ldhithertomeasuredis Ibe lie e thato fDubo is
netismtothisCongress[ seeunderNo. 81 inw hichhe
weentwosmallplaneend-facesofsoftiron polesofa
Thisma estheattractionpers uarecentimetreof
2 - 87 r o r a pp r o i m at e ly 2 . 1 07 d y ne s o r 2 0 k i l o g r am s .
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Y INGMETHODF O R STR ESSA NDSTR AIN
. P ro c . V o l . x x I . J a n . 20 1 9 02 p p . 97 - 10 1
. 1902 pp. 95-97 A pril 1902 pp. 444-448.
stressandstrain hithertofollowed
hasthegreatdisad antagethatit
stra in tobe inf initely small. A sa
eincon eniencethatthespecifying
tiallydifferentk inds( inthenotation
f g s i mp l e el o ng a ti o ns a b c s h ea r in g s .
o idedifw eta ethesi lengthsof the
onof theso lid orw hatamountstothe
le thethreepa irso f face-diagona lso fa
especify ingelements. ThisIha ethought
s butnotti l lto -day ( Dec. 16 ha e
n enientlypracticable especia lly for
genera li eddynamicsofacrysta l.
olid tobeahomogeneouscrystalof
utfromita tetrahedronAB CD ofany
thethreenon-intersectingpairs( AB ,
D , ( C A B D o fi ts si e dg es be de no te db y
( 3 q , 3 q ' ) , ( 3 r 3 r ) . ............. 1 .
, q ' ) , ( , . .. .. .. .. .. .. .. .. . 2
trahedron sim ilartoA B C D formedby
a 3 , y 8 t h e ce n tr e s of g r a i t yt o f t he f o ur
ngaf igureboundedby threepa irso fparalle lplanes is
hy butthe longerandlesse pressi e para lle lepiped is
eadofit bymathematicalwritersandteachers.Ahe ahedronwithitsanglesacuteandobtuse iswhatiscommonlycalled bothinpure
ography arhombohedron.Aright-angledhe ahedronis
Gree orotherlearnednameishithertotothefrontinusage
a lhe ahedronisacube.
enceforthca llthecentreofgra ityofatriangle oro f
scentre.
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IC A T IO NO F STR ESSA NDSTR AIN
C D A D A B , A B C r e s pe c ti e l y s o t ha t we
= , 8 y r = y c/ p = y 8 q ' = a S r = / . Con si de rn ow
untsofwor doneby thesi pa irso fba lancing
stress-componentsdescribedin~ 2
ntsv ary fore ample theba lancing
w hena/ increasesf romptop+ dp. a ll
, r p , q ' , r r e ma i ni n g co n st a nt . F o r t h e
maysupposetheopposite forces P tobe
steadofbe inge uablydistributedo erthe
cethew or w hichtheydo isPdp and
ci ng p ul ls Q , R P , Q ' , R , d o no w or .
apply tothefacesA DC B DC e ual
e ua llydistributedo erthem. Thesetwo
astressora stress-component.
achof thef i eotheredgesapplyba lancing
uttingit. Thusw eha eina llsi
eltothesi edgesofthetetrahedron
Q ' ) , ( R R ) . ... ... ... ... .. 3 ) ;
forces appliedastheyareto the
areba lancedinv irtueof themutua lforces
henitsedgesareof thelengthsspecified
o , q o q 0 , r o r o , b et he v a lu es of th es pe ci fy in g
hennoforcesareappliedtothefaces. Thus
eva lues o f thesi lengthsshownin
presentthestra inof thesubstancew henunderthe
) .
ew henpullsuponthefaces each
aregradua lly increasedtotheva luesshown
e of t h is p r oc e ss w e ha e
Q d + Q ' d ' + Rdr + R d r . .. 4 .
w e p r es s ed a s a f un c ti o n of p p , q , q ' ,
w = dw dw R
' . , d = r dR
d d r d . .. .. . .. . 4 .
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N
tionofthemolardynamicsof an
enera lpossiblek indaccordingtoGreen s
dintermsof thenew modeofspecify ingstresses
lythestateof strainspecifiedby
t th e t et r ah e dr o n of r e fe r en c e A o B o C oD o f o r th e
andstress bee uila tera l( thatistosay
of~ 2 ( 1 le tIo feachedge
o = r o .
sphericalsurfacetouchingeachof thesi
eatK0 thecentreof thetetrahedron
mustbethemiddlepointsof theedges.
eneousstra in , to thecondition( p q , r
i n w hi c h Ao B o C o Do b e co m es A B C D . T he i n sc r ib e d
sanell ipsoidha ingitscentreatK the
touchingitssi edgesatthe irm iddle
sfully andclearlythestateof strain
p , q ' , r . I t is w ha t is c al le d th e " s t ra in
ellipsoidtouchingthesi edges
ious. (1 ThroughA B andC Ddraw
ralle ltoC DandA B anddealsim ilarly
fnon-intersectingedges.Thethree
sfound constituteahe ahedronw hich
lipsoidtouchingthesi f acesatthe ir
wA , B K , C , D , a nd pr od uc et oe u al
' , K C , K D b ey on d K . W e t hu s fi nd f ou r
C , D , w hi ch w it hA B , C D a re th ee ig ht co rn er s
chw efoundbyconstruction( 1 . A circumscribedhe ahedronbeingthusgi en theprincipa la esof the
ntation arefoundby theso lutionofacubic
he strain-ellipsoid whichisin
andw hichhasthead antagethatinits
' N a tu r al P h il o so p hy ' ~ 1 5 5 ' E l em e nt s ' ~ 1 6 .
stingtheoreminthe geometryofthetetrahedron:Ifanellipsoidtouchingtheedgesof atetrahedronhasitscentre atthecentreof
ntsofcontactare atthemiddlesofthe edges.
' N a tu r al P h il o so p hy ' ~ 1 0 0 ' E l em e nt s ' ~ 1 4 1 .
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IC A T IO NO F STR ESSA NDSTRA IN
eusoutsidethe boundaryofour
isasfo llow s: -Inthee uila tera ltetrahedronA oB o C oDodescribe f romitscentreK0 aspherica lsurface
aces.Ittouchesthesefacesat their
chesthefourthface andatitscentre .
edeterminate one-so lutiona l problemto
attheir centresanythreeofthefour
B C D andha ingitscentreatK, this
re thefourthfaceofthe tetrahedron
dforthehomogeneousstrainbywhich
onofso lidisa lteredtothef igureA B C D.
odofspecifyingstrainandstress
arymethodfor infinitesimalstrains
sses:-LetX denotethelengthofeach
tetrahedronof re ference A oB oC oDo and
ubeofw hichA 0 B 0 C o Doarefour
gthehe ahedronfoundbyapplyinge ither
5tothetetrahedronA oB oC oDo . The
f thiscubeareeache ua ltoX , andtherefore
cubebeinfinitesimallystrainedsobhatits
, h ( 1 + f , h ( 1 + g ; a nd s ot ha t th e an gl es
realteredfromright anglestoacute
ngrespecti e lybya b cf romright
e f g a b c i n t he n o ta t io n o f
toin theintroductoryparagraphabo e.
metryoftheaffair weeasilyfindthe
oftheface-diagonals whichaccording
ar e ( p - 1 X , ( p - 1 X , ( q - 1 X , e tc . a n d
:
b
- b . . .. . .. . .. . .. . .. .. . .. 5
etwospecificationsof anyinfinitesimal
nddenotinge+ f+ gbys w ef ind
' + , r - 6 = 2s .. .. .. .. .. .. .. . 6 .
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N
c e f g i nt er ms of p q , r p , q ' , r , w e ha e
- ' ; c = r -r ; l
s - - ' + 2 g= s -r -r + 2 ( 7 .
edtoproduceaninf initesimalstra in
c i n a h om o ge n eo u s so l id o f c ub i c cr y st a ll i ne
bythefollowingformula:
g 2 + 2 3 ( f g+ g e+ e f + n ( a 2 + b 2+ c 2 . . .. 8 .
lymodifiedbyputting
n = I - A ( - . ) . .. .. .. .. .. . 9 ,
ul modulusandn , nthetw origiditymoduluses. Withthisnotation( 8 becomes
g 2 + 2 n [ ( f -g 2 + ( g - e 2 + ( e - f 2
. .. .. . 1 0 .
earingsparallelto thepairsofplanes
chisthesamething changesof theanglesof
efacesfromrightanglesto acuteorobtuse
gidity re lati e tochangesof theangles
hefacesfromright anglestoacute
compressibilitymodulusisk .U sing
w e ha e
q + q ' - -r ) 2 + ( r + r -p -p ) 2
) 2 + n ( p _- p 2 + ( q _ q ) 2 + ( r - r 2 ... 11 .
ETHER EA LTHEOR YO F THEV ELO C ITY
L I Q U I D S A N D SO L I D S .
c i at i on R ep o rt 1 9 0 , p . 5 5 P h il . M /a g . V o l . v I . O c t . 1 9 0 ,
ectures x x . pp. 46 -467.
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b it s 5 1 ; r e du c ed t o
5 1
p r es s ur e d ue t o 1 88
on98
ccompany ing282
es77
eadingw a eson
onstancy50
470
ing464 gy rostatic475
n52
place s intida l
, 5 2 1
, motion steady through f ree
0 1 , p o te n ti a ls 4 1 s e . 5 6
aperson551
propagationin3 0
3 0
odifiedforcyclic
ai e f luidpressure
esshedof f217
fmotion472
andstrain
hofdisrupti e548
tioninopen
ctor545
onoffluidin 19
abil ityofv ortices
ples174
ar
minimum181 to li uidgy rostat
cularpartit ion484
a i m um f o r gi e n i mp u ls e
g e n
ron547 duties
ortices stabil ity
ids97 dueto
dicfunctions35
ip p le s 9 1
ifuga l460 pendulum
w a e s 3 0 4 4 0 1
s ta bi li ty 1 3 , 1 8 ;
eriments482
y rostats5 3
2 o f c ha i n 5 3
motionin211
n determinate
ton o f2 ;
27 , incyclicmotion6
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luesof appropria te
91
cmotion
trigonometry51
s 4 6
ousf luid disturbed3 3 0 , motion R eyno lds criterion
e - mo t io n i n tu r bu l en t
h eo r y of 3 1 7
eticana logy94 99
of f loating1 5
yof474
n g e ne r al i e d 5 2 ,
gy484 495
mination516 521
n s ta b il i ty o f v i s c ou s
al c l oc 4 6 , 4 7 0
490 stabil ity515
o n r ip p le s 9 1
2
fv iscousf low 33 0
toanobli ue lymo ingplane
3 6
onthecriterionof
1 3 3 5
mwa es79
nimumv elocity91
pass464
id f igureof189 , w ater gra itationa loscil la tionsof141
uid54
n wi n d an d wa e s 8 1
a tt e rn 4 1
3 3 6
7
negati e
mo t io n o f i n l i u i d
to n 72 , i n m o i n g f l ui d f o rc e s on 9 ;
4
cyclic
insimple
2
a itatingf luid
dperiodicf luid
menta l
iscousf low betw eentwo
itha
0
agationinturbulentmedium308
79
lsin5 2
partition484
v orte of free
269 Airy s
eory
basin
54 , in inlandsea ef fecto fearth s
e s
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e fl u id s l ip s 1 87 , o f f re e mo b il i ty 9 7 , r i ng f l ui d c on e c te d b y 7 , r i ng i n er t ia o f 9 e n er g y
rings interactiononco ll ision
f fl o w of 6 , , r i ng t r an s la t or y m ot i on o f 6
suredueto impact