XII.n O Green's Function for a Circular Disc, with ... 1900.pdf · 278 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, P moves from an infinite distance along a line above the

XII. On Green's Function for a Circular Disc, with applications to Electro

static Problems. By E. W. Hobson, Sc.D., F.R.S.

[Received 7 October 1899.]

The main object of the present communication is to obtain the Green's function

for the circular disc, and for the spherical bowl. The function for these cases does not

appear to have been given before in an explicit form, although expressions for the

electric density on a conducting disc or bowl under the action of an influencing point

have been obtained by Lord Kelvin by means of a series of inversions. The method

employed is the powerful one devised by Sommerfeld and explained fully by him in

the paper referred to below. The application of this method given in the present paper

may serve as an example of the simplicity which the consideration of multiple spaces

introduces into the treatment of some potential problems which have hitherto only been

attacked by indirect and more ponderous methods.

The System of Peri-Polar Coordinates.

1. The system of coordinates which we shall use is that known as peri-polar co

ordinates, and was introduced by C. Neumann* for the problem of electric distribution

in an anchor-ring. A fixed circle of radius a being taken as basis of the coordinate

system ; in order to measure the position of any point P, let a plane PAB be drawn

through P containing the axis of the circle and intersecting the circumference of the

PAcircle in A and B ; the coordinates of P are then taken to be p = log , 6 which

is the angle APB, and <f> the angle made by the plane APB with a fixed plane

through the axis of the circle. In order that all points in space may be represented

uniquely by this system, we agree that 6 shall be restricted to have values between

— it and 7r, a discontinuity in the value of 6 arising as we pass through the circle,

so that at points within the circumference of the circle, 6 is equal to it, on the upper

side of the circle, and to — tt on the lower side of the circle, the value of 0 being

zero at all ' points in the plane of the circle which are outside its circumference. As

* Theorie der Elektricitate- und WSrme-Vertheilung in einem Ringe. Halle, 1864.

278 Dr HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,

P moves from an infinite distance along a line above the plane of the circle up to

any point inside the circle, and in its plane, 0 is positive and increases from 0 to ir,

whereas as P moves from an infinite distance along a line below the plane of the

Fm. l.

circle up to a point within the circumference, 6 is negative, and changes from 0 to

— tt. The coordinate (f> is restricted to have values between 0 and 2ir, and the co

ordinate p may have any value from — oo to + oo , which correspond to the points A, B

respectively. The system of orthogonal surfaces which correspond to these coordinates

consists of a system of spherical bowls with the fundamental circle as common rim, a

system of anchor-rings with the circle as limiting circle, and a system of planes through

the axis of the circle. If we denote by f the distance GN of P from the axis of

the circle, and by z the distance PN of P from the plane of the circle, the system

£ cos <p, f sin <f>, z will be a system of rectangular coordinates, which can of course be

expressed in terms of p, 0, <f>. Let the lengths PA, PB be denoted by r, r respec

tively, then r/r = log p ; we have

2rr' cos 6 = r2 + r'2 - 4a2 = 2rr' cosh p - 4a2,

2aa

hence rr =—;cosh p — cos 6

Again, z .2a = rr' sin 8,

a sin 0hence z — ; •

cosh p — cos0

also since r2 + r'2 = 2a2 + 2CP1,

we have CP2 = rr' cos 0 + a2,

Curtright

Callout

Oops! log(r/r')=rho

WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 279

, nry. oCOShp + CO8 0

whence we find Cr3 = a"—=-- A ,cosh p — cos (f

hence a'sinhV

(cosh p — cos 0)2 '

thus £, z are expressed in terms of p, 0 by means of the formulae

y a sinh p a sin 0

cosh p — cos 0 ' cosh p — cos 0 '

2. To express the reciprocal of the distance D between two points (p, d, <f>) and

(j00, do, <f>o), we substitute for f, z and f„, z0 in the expression

5 = {(* - *„)2 + fa + £>a - 2ff. cos (* - *,))-»,

their values in terms of p, 0 and p0> 60; we then find

1 _ 1 (cosh p — cos 0)>> (cosh p0 — cos 0o)l

D ~ aV2 jcosh a - cos (0 - 0^ '

where cosh a denotes the expression cosh p cosh p0 — sinh p sinh p0 cos (</> — If we

suppose the expression {cosh a — cos(0 — 0o))~* is expanded in cosines of multiples of

2 f ff cos Tiv^r

6—00, the coefficient of cosm(0 — (?„) is — I .—; ——7-r,d-Jr which is equal* tottJo (cosh a — cos y)' T 1

2 V2Qm-i (cosh o) when QOT-j denotes the zonal harmonic of the second kind, of degree

7T

m — \ ; thus i = — (cosh p — cos #)* (cosh p„ — cos 2 2Q,„_i (cosh a) cos m(0— 0O), where

2 JJ 7ra

the factor 2 is omitted in the first term, for which m = 0. The series in this expres

sion for 1/D may be summed, by substituting for Qm_j(cosha) the expression

1 f °° e-mti

VfJ. (cosh u- cosh a*du< {l0C- dt P" 519>;

we find

i = ^71 (C°Sh P " C°S 6)i (C°Sh * " C°S ^ [ (cosh n- cosh a)i ^ + 22<r'"" cos *>l *.

and thus we have the formula

* = —~= (cosh p — cos <?)4 (cosh p0 - cos j —- —_^ sinhjt ^

*' ira\2 J* vco*h « — cosh a cosh u — cos(0 — 0O)

where a is given by

cosh a = cosh p cosh p0 — sinh p sinh p0 cos (</> — </>„).

* See page 521 of my memoir "On a type of spherical harmonics of unrestricted degree, order, and argument,"

Phil. Trant. Vol. clxxxtii. (1896) A.

Curtright

Callout

OK


Green's Function for the Circular Disc.

3. In order to obtain Green's function for an indefinitely thin circular disc, which

we take to coincide with the fundamental circle of our system of coordinates, we shall

apply the idea originated and developed by Sommerfeld*, of extending the method of

images by considering two copies of three-dimensional space to be superimposed and

to be related to one another in a manner analogous to the relation between the sheets

of a Riemann's surface. In oiir case we must suppose the passage from one space to

the other to be made by a point which passes through the disc ; the first space is

that already considered, in which 0 lies between — ir and tt ; for the second space we

shall suppose that 0 lies between ir and Sir, thus as a point P starting from a point

in the first space passes from the positive side through the disc, it passes from the

first space into the second space, the value of 0 increasing continuously through the value

7r, and becoming greater than ir in the second space. In order that a point P starting

from a position P0(p0, 0O, <f>0), say on the positive side of the disc, may after passing

through the disc get back to the original position P0, it will be necessary for it to

pass twice through the disc; the first time of passage the point passes from the first

space into the second space, and at the second passage it comes back into the first

space. Corresponding to the point p0, 0O, <p0 where 0O is between — tt and ir, is the

point (p0, 0o + 2ir, <f>0) in the second space, whereas the point (p0, 0o + 4:it, <f>0) is regarded

as identical with the point (p0, 0O, </>„). The section of our double space by a plane

which cuts the rim of the disc is a double-sheeted Riemann's surface, with the line of

section as the line of passage from one sheet into the other. Let p0, 0O, <f>0, be the

coordinates of a point P in the first space, on the positive side of the disc, thus

0<^0<7r; taking the expression for the reciprocal of the distance of a point Q (p, 0, <p)

from P, given in the last article, we have, since

. , , sinh ^ u , sinh ^ usinh ?< _ 1 2 1 2

cosh it - cos (0 - 0,) ~ 2 ,1 I ,a dN + 2 ,1 \ ,a 'cosh ~u — cos s (0 — 0O) cosh ^ u + cos ^ (0 — 0O)

11 f" 1 sinh | m

pn = o /fa, (cosh P ~ cos d)h (cosh P" ~ cos e$ I 7- , r= i i du

PQ 2V2?m h VcoshM-coshacoshi«-cosh0-0o)

+ o -,-»— (cosh P ~ cos (cosh Po - cos 0$ i ~r=T==v" 1 ; du >

2V2?ra K Vcoshw-coshacoshi«-cosi(^-^0-2^)

we thus see that 1/PQ is expressed as the sum of two functions, the first of which

involves the coordinates p0, 0O, <j>0 of P, and the second is the same function of the

* See his paper "Ueber verzweigte Potentiate im Raume," Proc. Lond. Math. Soc. Vol. am


coordinates p0, 0„ + 2ir, <£0 of the point P in the second space, which corresponds to P.

If Q moves up to and ultimately coincides with P, we have cosh a = 1 ; it will then

be seen that the first function becomes infinite at the lower limit, but that the second

one remains finite at that limit.

Consider then the function TF(p0) 0O, <j>0) given by

W(a>» 0o, <£o) = —i=— (°osh p - cos 0)1 (cosh p0 — cos 0O)*

2 v27ra

r.

j sinh i u

a Vcosh it — cosh a „, i 1 ., „ 1 ,a acosh ^ w — cos £ '

the above equation may be written

= TT(p0, 0O, <f>e) + W (Po, 0, + 2tt, <£„).

It is clear that the function W is uniform in our double space as it is unaltered

by increasing 0 by 4ir ; it will now be shewn that it is a potential function. We

may express W in the form

W = }_ (cosh p - cos 0)* (cosh p0 - cos f 1 =. jl + 22e-*mu cos^(0 - 0O)\ du,

2 v 2ira J > v cosh w — cosh a I * )

which may be written in the form

1 00 tn )= 2-n-a ^C0S^ P ~ 008 ^ ^C°sh P" ~ 003^ l^-* ^C°sh a) + 2 ^ @™ L(cosh a) cos (0 - 0o)j ,

since the formula

V2 Q, (cosh a) = (coshM_CQsha)i

holds for all values of n such that the real part of n + % is positive (loc. cit. p. 519).

Now (cosh p — cos 0)* (cosh p„ — cos cos s{0 — 0„) Q,_j (cosh a) is a potential form whatever

8 may be, and thus IT is a potential function, and is expressible in the form

W = (cosh p — cos #)* (cosh p„ — cos 0O)* (cosh a) + 2Q0 (cosh a) cos | (0 — 0O)

+ 2Qj (cosh a) cos (0 - 0„) + . . .J ,

the value of W {p„, 0O + 2ir, <£„) being

(cosh p - cos 0)1 (cosh />, — cos 0O)* |q_j (cosh o) — 2Q0 (cosh a) cos | (0 — 0O)

1+ 2Q_j (cosh a) cos (0 — 0O) - . . . j ;

the two expressions added together give the expansion of 1/D obtained in Art. 2.

Vol. XVIII. 36


4. To evaluate the definite integral in the expression for W, write cosh | u = a ,

cosh | a = a, cos i (0 — <?„) = t, then

f / 1 ^ rfu = V2 f

J. Vcosh«-coshacoshl _cosl(^_5o) ^ V^-a»(a;-T)

— —

V22 /tt . ,T\

where the inverse circular function has its numerically least value ; we thus obtain the

expression

w 1 (cosh p - cos 0)* (cosh p„ - cos 0o)i [tr , . _. ( 1 ... , 1 "llfr = =i 1-—r ,a a mi d o + 8in-Mcos s (0 - 0„) sech s ctM ,

7raV2 {cosha-co8(0-0o)|* |_2 I 2V 0/ 2 J J

which may also be written in the form

w=fq[1+1 sin_i !cos \ {d - e<,) 8ech I '

This expression W has the following properties:—it is, together with its differential

coefficients, finite and continuous for all values of p, 0, <f> in the double space, except

at the point P in the first space, and it satisfies Laplace's equation ; when Q coincides

with P, the inverse circular function approaches jr , and the function becomes infinite

as 1/PQ; when however Q approaches the point in the second space which corresponds

TTto P, the inverse circular function approaches — ^ , and the function does not become

infinite. The expression (1) is then the elementary potential function which plays the

same part in our double space as the ordinary elementary potential function 1/PQ does

in ordinary space.

5. In order to find a potential function which shall vanish over the surface of

the disc, and shall throughout the first space be everywhere finite and continuous

except at a point P (p0, 0O, <f>0) in the first space on the positive side of the disc

(0<#o<'""), we take the function W (p0, 0O, </>0) — TP (p0, 2ir — 0o, <£„) which is the

potential for the double space due to the point P and its image P'(p0, 27r-#0» </><>)>

which is situated in the second space at the optical image of P in the disc. This

function is equal to

1 (cosh p- cos 0)1 (cosh p„ - cos 0„)* Ttt , . _, f \ ,a u 1 11{cosh a-.co. (0-0^ L2+8m {C09 2^-^9ech2a|J

i (cosh g - cos 01 (cosh p +- co. gty rv {_ cos i + sech i n

tV2 |cosh o+ cos(0 + #„)}» \_2 { 2X 2 J J


which is the same thing as

u = ¥Q [l + w sin-1 {cos \ ^ ~ e^ 8ech \ ai ~ Fq [i + \ sin-1 {~ 008 \^e+ 6^ sech I "}

• (2),

where P' is the optical image of P in the disc. On putting in this expression (2), for

U, the values 0=tt, 0 = — 7r, and remembering that over the disc PQ = PQ, we verify

at once that U vanishes on both surfaces of the disc. If Q coincides with the point

(p0, —Bo, <j>0) the function U remains finite.

The Green's function Gpq which is a function that is finite and continuous throughout

the whole of ordinary (the first) space, everywhere satisfies Laplace's equation, and is

equal to 1/PQ over both surfaces of the disc, is given by Gpq = ^.— U, hence the

required value of Gpq is

Gpq= pQ ^-^sin-'jcosi (0-0,) sech |o|J +~ ^^sin"1 ■ -cos| (0 + 0o)sech ^ a ■

= j^q . ~ cos-1 jcos | {0 - 0O) sech | aj + Jtq • ^ cos-1 ■ cos | (0 + 0„) sech | aj (3),

the numerically smallest values, as before, of the inverse circular functions being taken.

It will be observed that in interpreting these formulae (2) and (3), the second copy of

space, having served its purpose, may be supposed to be removed.

The Distribution of Electricity on a Conducting Disc under the influence

of a Charged Point.

6. If we suppose a thin conducting disc to be placed in the position of the funda

mental circle of the coordinate system, to be connected to earth, and influenced by a

charge q at the point P (p0, 0O, <f>0) on the positive side, the potential of the system at

any point Q is qU where U is given by (2), and the potential of the charge on the

disc is —q.GpQ. We shall now throw these potentials into a more geometrical form.

We have

r i i ) cos g (0-0.)

-1 jcos 2 (0 - 0.) sech £ 4 = tan"1 i . .

cosha2«-cosJ2(0-0o)

Vlcosi(0-0o)

= tan"1

(Vcosh a — cos (0 - 0O)J

now take an auxiliary point L, of which the coordinates are p0, 0 + tt, <f>a, the upper

or lower sign being taken according as 0 is positive or negative (— 7r<0<7r). Thus L

and Q are always on opposite sides of the disc ; using the formulae of Art. 1, we find

CL>-a* = -2a'C08^, a'-<7<? = " 2aW

cosh p0 + cos 0 ' cosh p — cos 0 '

PL _\ 1 +cos(0-0o) {* fcosh p - cos 0) *

PQ (cosh a - cos (0 — 0o)j (cosh p0 + cos 0

36—2


hence

sin-1 {cos I (0 - 0t) sech ^ a} = ± tan- i^qfJ \

p. ^

V J

"

Fio. 2.

in order to determine the sign on the right-hand side, we observe that the inverse

sine is positive unless 6 lies between — (ir — 0O) and — it, that is unless Q lies within the

sphere passing through P and the rim of the disc, and is on the negative side of the

disc; thus the sign on the right-hand side is to be taken positive unless Q lies within

this spherical segment.

Similarly we find

sin- {- cos 1(6 + 6a) sech i«| = ? tan- (£| *Jg^^)

where the negative sign is to be taken unless Q is on the positive side of the disc and

within the sphere which contains the rim and the point P'. We have thus as the

expression for the potential of the system at any point Q (p, 6, </>)

2PQ

2

1 + — tantr

JPL /a'-CQ\

\PQV CL*-a'J

9

2P'QIT2

IT

when the ambiguous signs are assigned in accordance with the above rules.

The auxiliary point L may be found from the following construction :

Draw a spherical bowl through the rim of the disc on the opposite side to that on which

Q lies, and equal to a similar bowl which passes through Q; draw a plane PA'R through

P and the axis, cutting the rim in A', B ; this plane intersects the bowl in a circle ; on

this circle L lies, and is found by taking it so as to satisfy the relation

LA1 : LB = PA' : PB.


2PQ

In the case in which the influencing point is on the axis of the disc, we have p„ = 0,

hence a = p, and the auxiliary point L is on the axis of the disc at the point where this

axis is cut by the sphere through the rim and the point Q, on the opposite side of the

disc to Q; the formulae for the potential then become

v=h D + 1 sin_1 {cos \ {d ~ sech Ip}]-fq [I + 1 sin_1 f cos -2 {d + es> sech i p}_

r 2. /PL /a*-CQ*\] q V. _ 2 _. /P'i /a?-C®\\

the sign in the first bracket is positive unless Q lies in the segment ApB, and the sign in

the second bracket is negative unless Q lies in the segment Ap'B.

7. To find, in the general case, the induced charge on the disc, it is sufficient to

examine the limiting value of the potential at a point Q, as Q moves off to an infinite

distance from the disc in the direction of the axis. In the expression for — q . Gpq given

by (3), let 0 = 0, p = 0, then a=pa, and PQ, FQ become infinite in a ratio of equality;

the expression for the potential of the induced electrification on the disc has therefore

the limiting value

- ^jSq cosrI (cos \S<> sech \ P"

therefore the whole charge on the disc is

2 1

"2 -rr

which is equivalent to

cos_I^cos sech ^p^j ,

2 . (Ja*-CL>\

when I is a point in the plane of the disc which lies on the bisector of the angle APB.

This expression may be interpreted thus:—

Let PL be the bisector of the angle APB, draw the chord NLM perpendicular to

AB ; the total induced charge is

Z NPM (as-q. (»)•


QWhen the point P is on the axis of the disc, the induced charge is - q . — , where 6„

IT

is the angle subtended at P by a diameter of the disc.

When P is in the plane of the disc, the angle NPM becomes the angle between the

tangents from P to the circular boundary of the disc.

8. The surface density at any point of the disc is given by the formula

_ l_dV

9 ~ 4tt dv '

when dv is an element of normal and is given by

dv_ ± add

cosh p — cos 6 '

We thus find for the density p0 at the point (p, ir, <f>) on the positive side of the disc,

* = - £ • m i1 + 1 8111-1 (sin \e% sech i■). '

1 U.I CO3;;0„

q 1 cosh p + 1 9.

*J cosh2 - sin'

this expression can be put into a more geometrical form by introducing the auxiliary

point L (p0, 6— ir, <f>0) of Art. 6. The point L is now in the plane of the disc, and external

to the disc ; denoting this position of L by L0, its coordinates are p0, 0, <p0. We have

sin- (sin ^0 sechia) = tan- g§ sj •

which is equal to | - tan-1 sj ) ;

on reducing the second term in the expression for p, remembering that

_ a sin 0„

cosh p0 — cos #o '

we find that it becomes

2tt2 PQ> . PL, V o5^CQr '

and thus the expression for the density at any point Q on the positive side of the

disc is given by

_q_ PN_j^ PN\PQ ICL^tf _JPQ /CLj=a~>\) ,p° ~ 2tt ■ PQ> 2tt2 • PQ'lP'Lo V a2 - CQ> \PLa V a?-CQ>)} " >'

where PN is the perpendicular from P to the plane of the disc, and i0 is a point on AB

produced, such that AL0 : BL0= AP : BP.


The value /j, of the density at the point (p, — ir, <f>) on the negative side of the disc is

found in a similar manner to be

q PN-[PQ /CLf^L* _l(PQ /CLf-a>\}

pi-~wpif{PLy &-cq> ~ tan \PL, v tf=w)i { >■

Thus the densities at corresponding points on opposite faces of the disc satisfy the relation

q PN

p0 Pl~~2n' P(f

When P is on the axis of the disc, L„ is at infinity, and the formulae (7), (8) become

0=_ 9 iK-A- ™ \ Pg=, - tan- ( PQ )\

q PNi PQ . _, ( PQ \) ,„

The expressions (7), (8), (9) agree with those obtained by another method by Lord

Kelvin*.

When P is in the plane of the disc it coincides with L„; in this case we find

that the density on either side of the disc is given by

= _ JL /cpi-ai

2tt» PQ»V aa-CQ2

9. If the influencing point P is on the axis at 60, we find from (5) the following

expressions for the potential at points on the axis :—On the positive side of the disc

pk-2^Q^-^-2^FQid + do)- WheD §>$»

pLQ+ 2,W0-0o)-2^P'Q{d + 0'1 Wh6n 0<°-

On the negative side of the axis

?Q + iwTFQ ^ ~ 6^ + 2tt?PQ (<? + 6o)' When 6 + e° is Positive'

PQ + JtTPQ ^ ~ do) ~~ 2tt qpQ ^ + ^' When 0 + 00 is ne£ative-

If we denote by z0 the distance of P from the disc, and by z the absolute value of

the distance from the disc of a point Q on the negative side of the disc, the potential

at Q is given by the expression

I - (C0t- I + COt- i) - (cot- ' - COt- ■

* See his papers on " Eleotrostatics and Magnetism," p. 190.

288 Dk HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,

if z„ be given as a multiple of a, say z„ = na, the expression

—^ 7-? , foot"1 - + Cot"1 n) : foot"1 - - COt"1 W ]z + na 7r(z + na) \ a J ir {na — z)\ a /

might be used to tabulate the values of the potential at points on the negative side

of the axis. When z = 0, z = oo this expression is zero, and it will have a stationary

negative value z for some value of z which may be approximately determined by plotting

out the value of the function. Corresponding to this value of z there is a point of

equilibrium which is completely screened from the effect of the influencing point P by

means of the disc ; the lines of induction from P which pass through this point, separate

those lines of induction which end on the disc, from those which go to infinity.

The Electrification induced on a Disc placed in any field of force.

10. The potential of the electricity induced on the disc, which is connected to

earth and placed in a field of constant potential, may be deduced from the expression

(5) by taking the point P on the axis, and letting it move off to an infinite distance,

the strength q of the charge increasing so that the ratio remains finite, say equal

to — A. We can easily shew that

sin-1 (cos \ 0 sech ^ p \ = — — sin-1 ( 1 ,V 2 2r/ 2 Kr. + rJ'

where rlt r2 are the greatest and least distances of the point (0, <f>, p) from the circular

rim of the disc. We thus find for the potential of the electricity on the disc, the

2A 2awell-known expression — sin-1 , which is the potential of an insulated disc elec-

trifled freely to potential A.

11. To find the potential due to the charge on the disc when placed in a field

of force of potential fix, when a; is a coordinate measured from the centre of the

disc in a fixed direction in the plane of the disc, suppose charges of strengths q and

— q to be placed at the two points P(p0, 0, 0), P' (— pc, 0, 0) on the axis of x; the

potential of the charge induced by these on the disc is at any point (p, 0, <f>)

where cosh a = cosh p cosh p„ — siih p sinh p0 cos <£,

cosh a' = cosh p cosh p0 + sinh p sinh p„ cos <j> ;


now let p0 become very small, as P, P' move away from the origin, the expression

for the potential becomes, when higher powers of p0 than the first are omitted,

ir VCP CP")

cosh ^ p i cos ^ 6 sinh | p cos <p I

cos-1 —y~ \~p" i i i T~

I Vcosh^p/ A/cosh^p-cos^*? cosh^P

M /C0S|A ^ cos 1 0 sinh |p cos <p

+ ? icp - ci*) i cos_i —t- + *-7—7—n z1—

[ ycosh^py ^cosh'^p-cos2^ cosh'^p

now CP = Cq3^D^ ^ = ~~ > hence if q be made indefinitely great so that ^-j^ = p.,

find for the required potential

we

now

j cos I 5 sinh p

a; cos-1 ( cos 5 # sech - p) — a cos <f> .^——

v 2 cosh5 p v cosh p - cos ^

2a Vcosh p — cos 6 , a sinh p

= ' , x = cos <f> —; — ,ri + rs 1 T cosh p — cos 0

V2 cosh g p

hence we find that the potential due to the induced electricity, in a field of force of

potential p,x, is

2 (.2a 2tW(r, + r3)3 - 4a") .....--^sm-1—— \ ' (11).

12. In order to find an expression for the potential of the induced electricity on

the disc, when it is placed in a given field of force, we apply the well-known theorem

that if a is the surface density at the element dS of the surface of a conductor when

acted on by a unit charge placed at an external point Q, the potential function at Q

which has values V given at every point of the conductor is JVadS, the integration

being taken over the whole surface of the conductor. Suppose V(p, to be the given

potential function at the element p, <f> whose area we denote by dS, on either side of

the disc; the potential function at the point (p0) 0O, <f>0) external to the disc which

on the disc takes the values V(p, <f>) is then, using the expressions found in Art. 8,

(1 + cosh p) cos - 0„

1 .1 •"•'-R3

cosh2 ^ a. — sins ^ #o

tan

_, v 1 — cos 90

Vcosh a + cos 60

V(p, <j>)dS,

Vol. XVIII. 87

290 Db HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,

the value of the required function, R denoting the distance PQ. We now introduce

new coordinates r, 77, <f> instead of p, 0, <f>, these being given by

x = Jr1 + a2 sin 77 cos <f>, y = Jr* + a2 sin 77 sin <p, z = r cos 17 ;

to express 77 in terms of p, 0, we have

a? + y" = (r2 + a2) sin' 77 = (^~}—I- a*j sin2 77,

CO1— a1 z*hence cos4 77 ^—— — cos' 77=— , where CO2 = #2 + y* + z1 ; hence we have

a2 a2

cos2 ,— + _J .

On! cos # 2a2

Now CQ2 - a2 = —5 , and it is easily found that V(6V - a4)2 + ^a'z- -■cosh p — cos 0 J cosh p — cos 0

hence we have

cos

/ 1 - cos 0

V cosh p — cos 0 '

and therefore cos 17, = . / 1 cos 00 .

v cosh p„ — cos 0„

Also as P is on the plane of the disc (r = 0), we have CP = a sin 77, hence efi = \ — S|n - ,

1 — sin 77

from which we find 1 + cosh p = 2 sec2 17. Remembering that

1 _ 1 Vl + cosh p Vcosh p„ — cos 0„

-ft a V2 Vcosh a + cos #„

, / 1 — cos a

we have a / —r ^ = -V cosh a + cos 0O

cos 77 cos 770

R

and also

(1 + cosh p) cos 9 a^/1 -l- cosh p cos ^ 0O

tjcosh2 i a - sin2

V2 * 2z

■ft' Vcosh p0 - cos 0„ -ft ' cos 77 ' a V2 cos 770 aii cos 77 cos 77,, '

then since d<S = a2 sin 77 cos 77 dr/ dtp, we have for the potential function at an external

point r0, 77,,, <f>0, which has the value ^(77, <p) at the point 77, <f> of the disc, the

expression

T7- z r/" 1 -it/ j.\ fi , a cos 77 cos 770 , _, /a cos 77 003770^) , ,

here the coordinates of the external point at which the potential is found are the

elliptic coordinates given by ...

£ = r0cos 770, a; = vV02 + a2sin77l)co8</30, y = vV02 + a2sin T)0sm<f>0,

the coordinate 770 alone appearing explicitly in the expression. This formula agrees with

one obtained by Heine by a different and somewhat complicated procedure*.

* See his Kugelfunctionen, Vol. 11. p. 132.


The distribution of Electricity on a Conducting Bowl under the influence

of an external electrified point.

13. In order to adapt the method of this paper to obtain corresponding results

for the case of a spherical bowl, we must suppose the surface across which the passage

from the first space to the second takes place, to be a spherical bowl with the funda

mental circle for its rim. If the angle of the bowl is /S, we must suppose that in

the first space 6 has values from /8 — 1-ir, on the negative side of the bowl, up to yS

on the positive side, and that as we then pass through the bowl into the second

space, 6 increases from /S up to fi + 2ir, when the positive side of the bowl has again

been reached. If the convexity of the bowl is upwards, is less than ir; if down

wards, /3 is greater than ir.

The image of a point P (p0, &„, in the first space and above the bowl is the

point P1 (p0, 2/8 — $0) in the second space, and below the bowl.

The expression

V 'FQ [s + 1 siD"' H I <" " 9,> sech I "}]

VX-g'-^k {-i«+»-*>-* H] <13)

corresponds to the expression in (2); it is a potential function which vanishes over

the disc, and of which the only infinity in the first space is at P, where it becomes

infinite as 1/PQ.

The Green's function GP(j is therefore given by the formula

GPQ =^ cos-1 jcos ^ (d — 0O) sech ^ a ■

/ cosh p0 — cos#„ 1 fl 1 . , f 1,- . ... . 1 )"| ,,+ V cosh Po -L w-eo) FqI* + ^ sin_1 icos 2 + e> - 2® 8ech 2 a\J <14>-

By introducing an auxiliary point L whose coordinates are p0, 0 + /S, </>0> this

expression may be thrown into a geometrical form corresponding to (4), and the

expressions obtained by Lord Kelvin for the density on either side of the disc may be

deduced ; it is however hardly worth while to give the details of the process, as it

is precisely similar to that which has been carried out in the case of the circular

disc.

37—2

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