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TM 5-237DEPARTMENT OF THE ARMY TECHNICAL MANUAL
SURVEYING
COMPUTER'S
MANUAL
HEADQUARTERS DEPARTMENT OF THE ARMYOCTOBER 1964
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iid~~_~_,~~~~~_~~_~~~_~.~~~~~~~~~_ _____~;r
This manual contains copyrighted material.
*TM 5-237
TECHNICAL MANUAL
No. 5-237
CHAPTER 1.
2.
3.
Section I.
II.
III.
IV.
V.
VI.
CHAPTER 4.
Section I.
II.
III.
IV.
CHAPTER 5.
Section I.
II.
III.
IV.
V.
VI.
CHAPTER 6.
Section I.
II.
CHAPTER 7.
Section I.
II.
CHAPTER 8.
Section I.
II.
III.
CHAPTER 9.
Section I.
II.
CHAPTER 10.
Section I.
II.
III.
IV.
CHAPTER 11.Section I.
II.
III.
IV.
HEADQUARTERSDEPARTMENT OF THE ARMY
WASHINGTON, D.C., 30 October 1964
SURVEYING COMPUTER'S MANUAL
Paragraphs
INTRODUCTION--___--____----__-----------1-4
ASTRONOMIC TABLES-- -______________________ 5-9
ASTRONOMIC OBSERVATION COMPUTATIONS
Conversion of time----------------------------------10,11
Computation of azimuth-----------------------------12-16Determination of latitude-----------------------------17-26
Determination of longitude--------------------------_ 27-31
Computation of latitude and longitude from observationsmade with the astrolabe-____--____________________ 32-35
Astronomic results----------------------------------36,37
DISTANCE MEASUREMENTS
Tape measurements--------------------------------- 38-42Tachymetry measurements---------------------------43-46
Measurements using light waves-____ --------------- _ 47-49
Measurements using electromagnetic waves---------------50-53
TRIANGULATION
Preparation of data for adjustment--------------------- 54-61Quadrilateral adjustment (least squares method)-----------62-65
Geographic position---------------------------------66-72
Adjustment of triangulation net----------------------- 73-76Special problems-------------------------------------77-81Shore-ship triangulation_________ -_____ -_________ 82, 83TRILATERATIONPreparing data for adjustment-- ----------------------- 84, 85Trilateration adjustment by the method of least squares-__ 86-90TRIANGULATION-TRILATERATION COMBINA-
TIONPreparing data for adjustment------------------------ 91-94Adjustment using combined measurements--------------- 95-97GEOGRAPHIC TRAVERSEIntroduction -------------------------------------- 98-100Adjustment of traverse (least squares method) ----------- 101-105Adjustment of traverse (approximate method) ----------- 106-108RESULTS OF HORIZONTAL CONTROL SURVEYSTabulation of results ------------------------------- 109-112Description of horizontal control station---------------- 113, 114DIFFERENTIAL LEVELINGDifferential level line--------------------------------115-118Adjustment of a level net---------------------------119-121Description of vertical control station-----------______ 122 123Computation of tide observations--------------------- 124-126TRIGONOMETRIC LEVELINGAbstract of zenith distances------------------------- 127 128Trigonometric elevations from reciprocal observations---- 129-131Trigonometric elevations from nonreciprocal observations-- 132-134Adjustment of trigonometric elevations----------_---- 135 136
*This manual supersedes TM 5-237, 21 May 1957.
Page
34
10153665
8092
95105106118
131167175188221232
236236
238239
245245255
263266
270271279282
287291295298
CHAPTER 12. ALTIMETER LEVELING Paragraphs Page
Section I. Single-base and leap-frog methods- -------------- __ 137-140 300II. Two-base method .--------------------------------- 141-143 304
CHAPTER 13. THE UNIVERSAL TRANSVERSE MERCATOR GRID
Section I. Mathematics and construction of the UTM grid-----_ 144-151 312II. Conversion of geographic coordinates to UTM coordinates_ 152, 153 316
III. Conversion of UTM coordinates to geographic coordinates__ 154, 155 319IV. UTM grid azimuths -------- ----------------------- 156, 157 321V. UTM scale fact.r --------------------- --- --------- 158, 159 325
VI. Zone to zone transformation on the UTM grid----------160-163 327
CHAPTER 14. HORIZONTAL CONTROL USING UTM GRID
Section I. Position computation on the UTM grid ---------------- 164-166 333II. Triangulation on the UTM grid_ - - - - -------- _ 167, 168 336
III. Traverse adjustments on the UTM grid---------------169, 170 354
CHAPTER 15. COMPUTATIONS ON THE UNIVERSAL POLAR
STEREOGRAPHIC GRID
Section I. Universal polar stereographic transformations ----------- 171-173 360
II. UPS scale factor and convergence_--- --------------- 174, 175 362
CHAPTER 16. OTHER GRID SYSTEMS
Section I. Transverse mercator projection .--- -------------- _ 176-178 363
II. Lambert conical conformal projection_----- -_ ---- _ 179-181 366
III. State plane coordinate systems in Alaska- _ _ 182, 183 368
IV. World polyconic projection_ ------------------------ 184-186 369
CHAPTER 17. GRIDS AND DECLINATIONS FOR MAPS
Section I. Dimensions of a grid__ ------------------------------ 187, 188 372
II. Grid and magnetic declination-----------------------189, 190 376
CHAPTER 18. DATUM OR SPHEROID SHIFT BY TRANSFORMA-
TION OF GRID COORDINATES---------------- 191,192 378
19. REDUCTION OF GRAVITY OBSERVATIONS- --__- 193-197 383
APPENDIX I. REFERENCES- --------------------------------------------- 394
II. CHARTS AND GRAPHS---------------------------------- 397
III. TABLES------------------------------------------------- 403
GLOSSARY.------------------------------------------------------- -------- --- 450
INDEX----------------------- ---------------------- --------------- 457
CHAPTER 1
INTRODUCTION
1. PurposeThis manual is published to serve as a reference
and guide for the survey computer in accomp-
lishing geodetic and topographic survey com-
putations; to standardize the methods and pro-
cedures for completing these computations; to
standardize tabulation procedures for numerical
figures and results for use in records or for dis-
semination; and to familiarize survey computers
with the accepted methods of performing survey
computations.
2. ScopeThis manual contains descriptive material,
references, sample solutions, and sample tabu-
lations for all types of computations that are
not usually completed in the field notebooks and
which may be encountered in military topo-
graphic surveys. The instruction for completion
of each of the computations contains both de-
scriptive material and a detailed solution. Unless
otherwise stated, all grid coordinates will be
Universal Transverse Mercator Grid coordinates.
3. ReferencesTM 5-236 contains many of the tables used in
performing the computations included in thismanual. Basic topographic surveying methods
are discussed in TM 5-441. Other references are
included in appendix I.
4. Accuracy
a. The solutions found within this manual are
designed to meet any foreseeable need of the mili-
tary survey computer. Most of the included
computations, unless otherwise stated, will meet the
requirements for first order. Computations designed
for first order accuracy may be adapted to lowerorder surveys by relaxing some of the refinements.
b. Users of this manual are encouraged to
submit recommended changes or comments to
improve the manual. Comments should be keyed
to the specific page, paragraph, and line of the
text in which change is recommended. Reasons
should be provided for each comment to insure
understanding and complete evaluation. Com-
ments should be forwarded directly to the Com-
mandant, U.S. Army Engineer School, Fort
Belvoir, Va., 22060.
_ __ I
CHAPTER 2
ASTRONOMIC TABLES
5. Introduction
The astronomic determination of direction,latitude, longitude, and time depends upon theapparent movement of stellar bodies. Astronomictables, in general, list the directions and positionsof planets and stars, their rates of apparentmovement, and certain numerical quantitiesnecessary to convert their apparent motions tomore usable information for any given instant.The most commonly used astronomic tables arethe American Ephemeris and Nautical Almanac(AE&NA), Apparent Places of the FundamentalStars (APFS), and the General Catalogue of33342 Stars by Benjamin Boss. The first twosets of tables are published annually while thelatter has been published for the epoch of 1950.
6. ,The American Ephemeris and NauticalAlmanac (AE&NA)
The American Ephemeris and Nautical Almanacis published annually by the U.S. Naval Observa-tory and is distributed by the Army as a TechnicalManual, TM 5-236-XX, with the third number
indicating the year for which the data is applicable.The volume has undergone two important changesduring recent years. One is the deletion of theten-day stars while the other was the introductionof Ephemeris time which is a more precise time.Corrections, necessary to convert Universal timeto Ephemeris time, are included in the Ephemeris.
a. Tables Included in the AE&NA.(1) AT, reduction from Universal time to
Ephemeris time.(2) Universal and Sidereal Time for Oh UT.(3) Sun (year)-for Oh Ephemeris Time.(4) Besselian and Independent Day Numbers.(5) Mean places of stars.(6) Table II-for finding the latitude by the
observed altitude of Polaris and azimuthof Polaris at all hour angles.
(7) Table VIII-Sidereal time to mean solartime.
(8) Table IX-Mean solar time to siderealtime.
(9) Table X-Conversion of hours, minutes
and seconds to decimals of a day.(10) Table XI-Conversion of time to arc.(11) Table XII-Conversion of arc to time.(12) Table XIII-Interpolation constants.(13) Table XIV-Second-Difference correc-
tions.
b. Use of Ephemeris Time. Starting with theAE&NA for 1960, the tabular argument in thefundamental ephemerides of the sun, moon, andthe planets is Ephemeris time. Ephemeris timeis the uniform measure of time as defined by thelaws of dynamics and determined in principlefrom the orbital motions of the planets; specificallythe orbital motions of the earth as represented byNewcomb's "Tables of the Sun". Universal timeis defined by the rotational motion of the earth, andis determined from the apparent diurnal motionswhich reflect this rotation. Because of variationsin the rate of rotation, Universal time is notrigorously uniform.
c. Universal and Sidereal Times. Beginningwith the 1960 Ephemeris the Sidereal time atOh Universal Time and the Universal Time at Oh
Sidereal time, which formerly were included inthe Ephemeris of the Sun, are tabulated both forthe mean equinox of the date, and for true equinox,with the short period terms of nutation included.
d. Tables of the Sun.(1) The date column. The dates found in this
column represent the instant at 0 h
Ephemeris Time on the date indicated.(2) The apparent right ascension column. The
apparent right ascension (RA) is givenin units of time for each day. The light-faced type, to the right and between thelines, gives the tabular differences in
seconds. Linear interpolation is made
by multiplying these terms by the frac-tion of a day elapsed since Oh, and adding
the result to the right ascension given
for Oh of the date. Note that the sunalways increases in right ascension and
the tabular differences are positive.
(3) The apparent declination column. Theapparent declination (S) is given in the
third column. Interpolated values are
determined in the case of right ascen-sions as in (2) above. However, both
the declination and tabular differences
may be either positive or negative.
Care must be exercised in preserving theproper sign. Both the apparent right
ascension and declination of the sun con-
tain the effects of long-period terms of
nutation and aberration, and define theapparent place (as the observer sees it)
of the true sun.(4) The horizontal parallax column. Parallax
is the displacement in the position of aheavenly body due to observations being
made from the surface of the earth in-
stead of at its center. Horizontal paral-
lax is the angle at the center of the sun
subtended by the earth's equatorial ra-
dius. For solar observations, the true
parallax is equal to the product of thehorizontal parallax and the cosine of the
altitude, or the sine of the zenith distance.
(5) The semidiameter column. The amounts
shown in this column are correct for
horizontal semidiameter but, due to the
effects of refractions, they will vary
slightly for vertical semidiameter.
(6) The equation of time column. The sixthcolumn gives the equation of time stated
as apparent minus mean. Tabular dif-
ferences are also furnished. Note that
both the equation and the difference may
have either algebraic sign.
e. Table VIII. Sidereal into Mean Solar Time.Table VIII of the Ephemeris gives the quantities
that must be subtracted from a given time inter-
val expressed in sidereal units in order to obtain
the same interval expressed in mean-time units.
Or it represents the amount a mean-time clock
would lose as compared to a sidereal clock over a
given sidereal interval. The main table is given
at intervals of 1 sidereal minute for the entire day,the hour being given at the head of the column
and the minutes down the left-hand side. The
corrections for seconds are given at the extremeright. Interpolation is made as in table IX (J
below). The sum of the quantities from the mainand seconds table is then subtracted from the
sidereal interval to obtain the corresponding
mean-time interval.
J. Table IX. Mean Solar into Sidereal Time.This table of the Ephemeris is more frequently
used than table VIII. It gives the quantities
that must be added to a given interval expressed
in mean-time units in order to obtain the same
interval in sidereal units. It is the amount a
sidereal clock gains with respect to a mean-time
clock over a given mean-time interval. Table IXhas the same form as table VIII. In order to
convert a mean-time interval to a sidereal inter-
val, first enter the main table in the column for
the hour, and on the line corresponding to the
last full mean-time minute. Next, the table to
the right is entered using the seconds of mean time
over the last full minute. Any interpolation for
desired fractions of a second is made mentally.
The final quantity from the seconds table is then
added to the quantity taken from the main table.
This gives the correction to be added to the mean-
time interval.
Example: Find the sidereal-time interval corre-
sponding to a mean-time interval of:
1 7h 14m
12.7
From main table, cor-
rection for 1 7h 14 m =
From second table, cor-
rection for 1237 =
Sidereal interval
2m 495860
03035
- 1 7 17m 025595
g. Mean Places of Stars. Mean places of 1078
stars are given in this table for the instant of the
beginning of the Besselian year. This date
occurs when the sun's mean longitude is 2800
and falls very close to the beginning of the calendar
year. The civil date is given in decimal days at
the top of the page. The mean place does not
coincide with the apparent place for the same
date, but constitutes a base for the application
of corrections in order to find the apparent place
at any given date. The formulas for this reduction
mean to apparent place, are given in the section
of the American Ephemeris devoted to the use of
the tables, under Stars. This reduction is seldom
required of the computer. The mean-place tables
are useful in preparing observing lists, and for
any purpose where a close value of the star's
position is not required.
h. Latitude From the Altitude of Polaris. This
table affords a means of determining the latitude
when the altitude (h) of Polaris and the local
sidereal time (LST) are known.
(1) Three corrections; ao, al and a2 are
extracted from table II. The arguments
are the local sidereal time, the approxi-
mate latitude, and the month of obser-
vation respectively.
(2) The observed altitude is corrected for
refraction. The latitude is determined
by taking the algebraic sum of the
corrected altitude and the three cor-
rections from table II.
i. Azimuth of Polaris. Table II is also used to
determine the azimuth of Polaris. The procedure
is as follows:
(1) Extract the values for bo, bl and b2 from
table II, using the local sidereal time
(LST) of observation, the latitude and
the month of observation respectively,as the arguments. These corrections
are added algebraically.
(2) Multiply the quantity (bo+bl+b 2) by thesecant of the latitude to obtain the azimuth
of Polaris, as referenced to the Pole.
j. Besselian and Independent Day Numbers.
These day numbers, used for reducing mean to
apparent place, are coefficients of the effects .on
the stars position caused by the processes of
precession, nutation and aberration. . They are
computed by trigonometric means from the mean
coordinates of the star and are dependent only on
the Greenwich Ephemeric Time Date. Factorsfor proper motion are abstracted directly from the
star catalogue. Either the Besselian or theIndependent system may be used in the reductionsalthough the Besselian system is more convenient
for mass production.
k. Table XII. Conversion of Arc to Time.This table is often useful in avoiding division by
15. In successive columns, the time equivalents
of degrees, minutes, seconds, and decimal secondsof are are given.
Examnple: Convert 32044'42'.'15 to units of time.320 -2h 08m
44' = 2m 56
42" = 2.8
0'.'15 = 001
Sum =2h 10m 58f81
1. Table XI. Conversion of Time to Arc. Thistable is the inverse of table XII. The first part isa table in arguments of hours and minutes, fromwhich is taken the are equivalent of the evenminute. The right hand section gives the areequivalents of seconds and 100ths of seconds oftime.
Example: Convert 2h 10m 58.81 to are units.
2 h 10m =320 30'
58 -= 14' 30"
0.81= 12'.'15
Sum=320 44' 42'.'15
m. Table X. Conversion of Hours, Minutes,
and Seconds to Decimals of a Day. This table isuseful in finding the decimal day equivalent to the
UT for the purpose of entering the tables for theapparent places of stars. The equivalent of thehours and minutes is taken from the main table.That of the remaining seconds is given in theright hand column. This latter refinement israrely necessary.
n. Table XIII. Besselian Interpolation Con-
stants. The use of this table is explained in
paragraph 9d.
7. The Apparent Places of the Fundamental
Stars (APFS)a. Introduction. The tables in the Apparent
Places of the Fundamental Stars are the result of
international cooperation which has reduced the
duplication of certain parts of the ephemerides
published by contributing nations. The data
contained in this publication is limited to tables
of the stars position and certain auxiliary tables.
b. The Main Table. The main table in the
APFS lists the apparent positions at upper transit
of 1483 stars, between 810 north and 810 south
declination, at 10-day intervals throughout the
year. The column headings contain the catalog
number, name, magnitude and type of spectrum
of each star. Right ascensions are listed in
units of time to thousandths of seconds. Declina-
tions are in units of arc to the hundredths of
seconds. In both cases, the seconds values are
followed by the tabular differences. At the foot
of each page are found the mean place, secant,and tangent of 6, and factors for computing short
period terms of nutations for each star. The
dates when two transits of the Greenwich meridian
occur, during the same mean-time day, are also
given.
c. The Table of Circumpolar Stars. Immediately
following the main table is the table of Circum-
polar Stars, which lists the apparent positions of
52 circumpolar stars for every upper star transit
at Greenwich. The date refers to the civil day.
Interpolation is made from the Greenwich HourAngle of the star at the time of observation. The
one-day interval, between tabulations, permits the
inclusions of short period terms of nutation within
the tabulated values. Right ascensions are given
to two decimal places only, this being in the order
of the uncertainty of circumpolar star positions.
Otherwise the tables are similar to the 10-daystar tables.
d. The FK 4 System. The "Apparent Places of
Fundamental Stars" for 1964, and subsequent
years, contains the 1535 stars in the Fourth
Fundamental Catalogue (FK4). This volume
provides the mean and apparent places of 10-day
and Circumpolar stars together with tables fortheir reduction.
e. Table I. Table I in the APFS furnishesfactors for computing the short-period terms forthe 10-day stars. The equations for accomplish-ing this are found at the foot of the pages of thetable. The other necessary coefficients are tabu-lated under each star in the apparent-place table.
j. Table II. The sidereal time of Oh, UniversalTime and the long- and short-period terms of theEquation of the Equinoxes are given. Theapparent sidereal time is the sum of the mean
sidereal time plus period terms.
g. Table III. Table III provides the conversionfactors, mean solar to sidereal time and is identicalto table IX of the AE&NA.
h. Table IV. Table IV is used for convertingintervals of sidereal time to mean solar time andis identical to table VIII of the AE&NA.
i. Table V. Table V is used for reducing hours,minutes, and seconds to decimals of days and is
similar to table X of the AE&NA with the excep-
tion that table X is a six-place table while table V
is a five-place table.
j. Table VI. Table VI lists second difference
corrections for use with linearly interpolated
values. This table is somewhat different from
the table of Besselian coefficients. The quantities
given are for the term B" (A'+ A'), the symbols
being explained in paragraph 9d. The arguments
are the fractional part n, and the double second
difference (a' A'+a'). The latter factor is at a
tabular interval of 5 units of the last place of the
fraction. The correction is always of opposite
sign to (A'o'0 A').k. Table VII. Table VII is used for correcting
the time of transit for the effect of diurnal aber-
ration. The correction is rarely needed.
8. General Catalogue of 33342 Stars ByBenjamin Boss
This catalogue consists of five volumes, the
first being used for instructions and appendixes
and the other four for mean places of stars. This
catalogue is principally used by the geodesist in
the determination of latitude. Mean to Apparent
Place reduction is accomplished by means of the
Besselian or Independent Day numbers from the
AE&NA. Statistical and historical information
contained in these tables is explained in the first
volume.
9. Interpolation
a. Introduction. As the computer will have
an almost continuous need for interpolation in the
use of various tables, this paragraph will review
linear and double interpolation.
b. Linear Interpolation. Geodetic surveys and
astronomy will rarely have need for more than
simple linear interpolation. This assumes that
the function varies as a constant ratio, that is, as
a straight line, between tabular values. Most
functions are actually curves when plotted on the
coordinate axis. Hence, a linear interpolation is
subject to some error. The amount of the error
depends on the sharpness of the curve and the
spacing of the tabular values. All good tables are
so arranged that the errors are nearly always
negligible.
(1) When the interpolated value is taken as
the proportional part of the difference
between successive values given in the
table, it is said to be interpolation on the.
chord. The tables of the American
Ephemeris which give these tabular
differences in smaller type are so ar-
ranged that the interpolation is along
the chord.
(2) Another form of linear interpolation is
said to be on the tangent. In this case,the small type figure is the rate of change,or slope of the tangent at the value of
the function as given in the table. This
form is more accurate than interpolation
along the chord, provided not more than
a half-interval is taken. These forms of
linear interpolation are shown in figure
The function is represented in the figu
by the curve PQ, at which point its
values are given in the table. It is c
sired to interpolate for the y value wh
x=0.8. Point A represents the tr
value on the curve. Point B represen
the value found by interpolation on t
chord, the error being B-A. Interp
lation along the tangent from the near
tabular value Q gives a result at
The error C-A is less than B-
Point D is found by interpolating alo
the tangent from P, and the error
greater than that obtained on the cho:
1.
Ire
yde-
hen
'ue
Its
he
)o-
est
C.
Figure 1. Forms of linear interpolation.
c. Double Interpolation. Double interpolation
becomes necessary when a function is subject to
two variables instead of one. The requirements
of most double interpolations are met by a series
of three chord interpolations; and the use of chord
interpolations is recommended for this purpose.
For example, the parallax of the sun varies during
different dates of the year, and varies with the
altitude of the various observations. The process
of interpolation is as follows:
(1) Example: Determine the parallax for
June 10, 1964 when the observed altitude
is 33031'19".
(2) Parallax-from table VIII (app. III)Date Altitude Parallax Date Altitude Parallax
June 1 360 7"02 July 1 360 7:01June 1 330 7"28 July 1 330 7"26
(3) Step 1-Interpolate for parallax on
June 10 at 360 altitude. Use nearest
tabulated value. p= 07"'02- 1/3(07'.'02-07'01) =07"017
(4) Step 2-Interpolate for parallax on
June 10 at 330 altitude. p=0728-1/3
(07'28-07.26)=7'273
(5) Step 3-Interpolate for parallax at
altitude of 33031'19 ' . p=07.273-1/6
(07"273-07'.017)= 07"230= 07'"23
-A. d. Besselian Interpolation. This method is re-
ng quired only in the highest class of computations.
is Interpolation to second and higher differences
rd brings successively closer approximations of thetrue value on the curve. The tabular differences
taken from the table are known as the first
differences. The successive differences between
Q the first differences are called second differences,and so on. The customary designation of these
differences, to second differences only, appears
below.
(Function)Tabular value
F-1
Fo
lst diff.
0134
2d dif
AO'
Fa
The desired value F,, lies at a fraction n, between
F0 and F 1. Bessel's formula is generally preferred.
It is written as:
F,F--o+n1/2-,-B (Ao-'+A')--[B" ', 2'] + ....
The quantity in the brackets represents the third
difference, given for the sake of completeness.
Only the first three terms are needed for second
differences. The terms B", B" ', and so on, are
known as the Besselian interpolation constants.
B" and B" ' are given in table XV of the American
Ephemeris for values of from 0 to 1. Their values
are:
B" =n(n-1)2(2!)
B" , n-1)(n1)(-)3!
and may be so derived in the absence of the
tables. The terms A'/2, A ', and A"', are taken
from a tabulation as shown above.
Example: Find, with respect to second differ-
ences, the apparent declination of the sun at UT
June 1.67, 1963.
Date Decl.
May 31 (F_1)+210
46'44"6
June 1
June 2
June 3
(Fo) + 21-55'26 8
(Fl) + 22003'46"2
(F2) 22o11'42:4
The fraction n==0.67. Hence, from table XV,B" is -0.0553. B"' if required would be
-0.0063. Applying the formula:
F,= +21055'26.8± 0.67(+499.4)
-0.0553(-46.0) = ±22103'9.
This interpolation may be simplified with littleloss of accuracy by taking the term, B"(' ± +A')from- table VI of the Apparent Places of Funda-
-x499.4(=B')
-23.2(=oz '
+ 476.2(=Ai')
mental Stars. For the above example, n=.67and ('" +±A')46.0. The correction takenfrom table VI is ±25 in units of the last decimal
place. This, added to n®1,, which is equal to
334:'6, gives 337.1 as the interpolated difference.
The final value is +2201'03'9, as by the first
solution. Care should be taken that the double
second difference and the figure taken from the
table be in the proper decimal place.
Ist dif. 2d dif.
CHAPTER 3
ASTRONOMIC OBSERVATION COMPUTATIONS
Section I. CONVERSION OF TIME
10. Kinds of TimeThe geodetic computer will be concerned with
three kinds of time in astronomical and solar
computations. They are apparent sidereal, mean
solar, and apparent solar.
a. Apparent Sidereal. Apparent sidereal time
is generally used in astronomical computations.
Various expressions of sidereal time will confront
the computer and the most common are listed
below.
(1) Greenwich Sidereal Time (GST) is ap-
parent sidereal time at the zero meridian
of longitude near Greenwich, England.
Greenwich Sidereal Time is zero hours
at the instant of upper transit of the
Greenwich meridian (0°X) by the ap-parent motion of the vernal equinox.
(2) Local Sidereal Time is sidereal time at
the local meridian, e.g., the meridian of
a survey station, and is zero hours at
the instant of upper transit of the local
meridian by the apparent motion of the
vernal equinox.(3) Mean Sidereal Time is not used in
astronomical computations.
b. Mean Solar. The mean solar day is meas-ured by the fictitious mean sun between successive
meridian passages. The solar year is identical in
length to the sidereal year, but due to the ap-
parent movement of the sun, it contains 1 dayless. The mean solar day is therefore about 3
minutes and 56 seconds longer than the sidereal
day.- -Mean solar time is that used in everydaylife.
(1) Local Mean Time (LMT) is mean solar
time at the local meridian and is the
hour angle of the mean sun measured
westward from the local meridian. Local
Mean Time is 1200 hours at the instant
of upper transit by the mean sun acrossthe local meridian.
(2) Standard time is mean solar time at anadopted central meridian for a 15° wide
time zone. In the United States, forexample, the central meridians of the
time zones are 750, 900, 1050, and 1200
West of Greenwich corresponding toEastern (EST), Central (CST), Moun-
tain (MST), and Pacific (PST) time
zones respectively. For any particulartime zone the standard time is 1200
hours at the instant of upper transit bythe mean sun across the central meridian
of the zone.
(3) Daylight Saving Time is standard timeplus one hour adopted for the general
convenience of the public during themonths of the year having the longestperiod of daylight hours, i.e., in the
United States from April to October.
(4) Universal Time (UT) is mean solar time
at the 0 ° meridian, and corresponds
essentially to Greenwich civil time
(GCT). There are three categories of
Universal Time called UTO, UT1 andUT2.
(a) UTO is mean solar time determined
astronomically by individual observa-tories and referenced to the Greenwich
meridian by application of difference
in longitude. UTO is not corrected
for polar motion.
(b) UT1 is obtained by applying the cor-
rection for polar motion to the un-
corrected Universal Time (UTO) by
the observatory. The correction to
the JT2 signal to obtain UT1 and
UTO is published in Time Service
publications of the major observatories.
UTi is equal to UT2 minus S, where
S is the extrapolated seasonal varia-
tion in speed of rotation of the earth.
(c) UT2 is Universal Time (UTO) corrected'for polar motion and for extrapolated
seasonal variation in speed of rotationof the earth. Time service bulletinsof the major observatories publish the
correction to be applied to the time
signal in order to obtain UT2.
c. Apparent Solar. Apparent solar time is keptby the actual sun. An apparent solar day is theinterval between two successive meridian passagesof the sun, and varies in length by about 30minutes during the year, due to irregular apparent
motion of the sun. Apparent time is necessaryin computing some observations on the sun.
d. Ephemeris. Ephemeris Time (ET) is theindependent variable in the gravitational theoriesof the Sun, Moon, and planets. If it is desired toconvert Ephemeris Time to Universal Time, the
following relationship may be used: UT=ET-
AT. AT is the amount ET is ahead of UT andits value is published in the American Ephemeris
and Nautical Almanac.
11. Conversion of Time
It is frequently required to convert one kind of
time to another. This is done by the following
processes:
a. To find the sidereal time of a given mean
solar time (fig. 2), use the tables in part I of theAmerican Ephemeris or table II in the Apparent
Places of Fundamental Stars and DA Form 1900
(Conversion of Mean Time to Sidereal Time).These tables give the apparent sidereal. time
corresponding to oh, Universal Time, for each day
of the year. This is the mean solar time of thebeginning of the day (midnight) at the Greenwich
meridian.
(1) Find the Universal Time (UT, also called
GCT) and date by adding algebraically
the longitude in hours, minutes, and
PROJECT. CONVERSION OF MEAN TIME TO SIDEREAL TIME1 2-21 - 32 (TM 5-237)
LOCATION ORGANIZATION
MARYLAND A INc.
DATE 9 July /963
LONGITUDE VV 77 04 20.628
HOURS MIN. SECONDS HOURS MIN. SECONDS HOURS MIN. SECONDS HOURS MIN. SECONDS
1. Recorded sLCa1std. time 2 07 52.093
2. .(Wetseh (Chronometer) correction(F-, S+) - 0 0o 03.100
3 {. ted Local std. time 5 . 1 7 48.993
4. Longitude or time zone difference(W+, E-) 5- 00 00.000
5.1 Universal time (UT) (3+4)
/0 Jol /963 2 07 48.9936. Sidereal time of Oh UT for
Greenwich date 9 0 47.67/
7. Corertionfor sidereal gain
+ 0 00 20.9
8. Greenwich sidereal time (GST)(5+6+7) 1 /6 556/
9. Longitude(hr., min., sec.) (W+, E-)+ 5 a8 /7.375
10. Local sidereal time (LST) (8-9)16 08,40.186.
COMPUTED BY DATE CHECKED BY DATE
WR..am m.. - AAAS /3 Jul /963 . - AMS '4 July /963
DA PORM 1®oADAI FEB 57 U. S. GOVRNMENT PRINTING OFFICE :157 0-420017
Figure 2. Conversion of Mean Time to Sidereal Time (DA Form 1900).
seconds of the place from Greenwich tothe given mean time. If the given timeis a standard time, add the number ofhours corresponding to the time zone ofthe place. Longitudes and times west
of Greenwich are positive, and east ofGreenwich are negative. If the sum ismore than 24 hours, subtract 24 hoursand add 1 day to the date.
(2) Enter the table for the Greenwich dateand find the sidereal time of Oh UT fromthe table in the American Ephemeris, orin table II of the Apparent Places ofFundamental Stars. This term is alsoknown as RAMS+ 1 2 h (right ascension
of mean sun). The sun's right ascensionis measured from the upper meridian,while the beginning of the day is referredto the lower meridian. Hence, it isnecessary to add 12 hours to the sun's RA.
(3) Since the sidereal units are shorter thanmean time units, the sidereal time willconstantly gain with respect to meantime, and a correction for this must beapplied to the interval between Oh UTand the UT of the observation. This isfound in table IX, American Ephemeris,or table III, Apparent Places of Funda-mental Stars. The tabular differencesare minutes of mean time. An auxiliarylisting in the right hand column of eithertable gives the correction for additionalseconds in the mean time interval.
(4) Add the UT found in (1) above, the
sidereal time of Oh from (2) above, andthe total correction from (3) above.
This gives the Greenwich sidereal time
(GST) of the given mean time.
(5) Subtract the longitude of the place from
the GST to obtain the local sidereal time
(LST) of the given mean time.
b. To find the local mean time (LMT) of a
given sidereal time (fig. 3), use tables as noted and
DA Form 1901 (Conversion of Sidereal Time To
Mean Time).
(1) Add the longitude of the place to the local
sidereal time to obtain the GST.
(2) Subtract from the GST the sidereal time
of Oh UT for the date to obtain thesidereal interval since Oh UT.
(3) Subtract the correction, sidereal to meansolar time for this interval from tableVIII, American Ephemeris, or table IV,Apparent Places of Fundamental Stars.This gives the UT.
(4) Subtract the longitude of the place fromthe UT to obtain the local mean time, orsubtract the time zone correction toobtain local standard time.
c. To find the apparent solar time of a givenmean time (fig. 4), use tables in part I of theAmerican Ephemeris, or any other solar ephemeris,and DA Form 1902 (Conversion of Mean Time toApparent Time).
(1) Add the longitude to the given local meantime to obtain UT.
(2) Take from the table the equation of timefor the date. This value applies to OhUT. Note proper algebraic sign.
(3) Multiply the daily change of the equation
of time by the fraction of a day elapsedsince 0h UT.
(4) Add algebraically the amounts from (2)and (3) above. The sum is the equationof time for the given time.
(5) Add algebraically the equation of time tothe local mean time to obtain the localapparent time.
d. To find the local mean time of a given localapparent time (fig. 5), use the tables. The tablesare made for mean time units, and since apparenttime is given, a first approximation must be madefor obtaining the equation of time.
(1) Find the Greenwich apparent time (GAT)by means of the longitude as above.This will seldom differ from the UT bymore than 0.01 day.
(2) Subtract the equation of time for Oh UTcorrected for the elapsed interval in ap-parent time for the GAT. This gives aclose approximation of the UT.
(3) Recompute the equationl of time for theelapsed interval of mean time.
(4) Subtract this value from the LAT toobtain the LMT.
(5) The equation of time is the same at anygiven instant for all points in the world.
PROJECT 991CONVERSION OF SIDEREAL TIME TO MEAN TIME2-2/-32(TM 5-237)
LOCATI ON 4A YZIDORGANIZATION
LOCAL DATE 9 July /963 ________ ______ __________
0 0 0 / n 0 r
LONGITUDE w 77 04 20.6~28________
HOURS MIN. SECONDS HOURS MIN. SECONDS HOURS MIN. SECONDS HOURMIN. SECONDS
1. Recorded local sidereal time (LST) /7 0 686___________ __ ____
2. Watch correction (F-, S+) - O 00 03.602
3. Corrected LST / 7 08 43.294 __
4. Longitude (W+, E-) +S 08 /7.37,5
5. Greenwich sidereal time (GST) 22 /7 o. 9(3 +4) 22 _17 _00_66
6. GST of 0' UT for Greenwich 08 47571/0 Jul 1 963date* /9 04 51.0/4 9 July /963 ___
7. Sidereal interval since 0° UT 08'S 9(5-6) 03 /2 09.655____
8. Correction for mean time lag 30.8 34
o 00 31.jj481_______
9. Universal time (UT) (7-8) 74 26
03 /1 38/74
10. Ebeesgitd di e or (time zone) s 00 00.000
11. Lzzeft Ma. 4 _ (LT 5
T 07 42.24
Local std.time (L STD T)(9
-10
) 22 // 38. /74*If Greenwich date is doubtful use local date for trial computation. At step (11) determine correct Greenwiich (late and if necessary rew ork comuputation from step (6) to end.
COMPUTED BY DAECHECKED BY DATE~ ~.odsww4.a - AIMTE,3 July /963? G. i. Teo~,, - M /4 July 1963
DaA FORM 90FBD /U .GVRMN RNIGOFC 970008
Figure 3. Conversion of Sidereal Time to Mean Time (DA Form 1901).
PROJECT CONVERSION OF MEAN TIME TO APPARENT TIME2 - 2 /- 32 I(TM 5-237)
LOCATION ORGANIZATION
DATE /8 MA Y /963 ________
0 0 I 0 i r 0 / IFLONGITUDE W 83 48 24 __ ____
HOURS MIN. SECONDS HOURS HIN. SECONDS HOURS MSIN. SECONDS HOURS HIN. SECONDS
1. Recorded local mean time (LMT)____ ____ ____ ____ ii 56 17.8 _ _
2. Weatch correction (F-, S+) __ 00 03.1 ________________
3. Corrected local mean time (LMT)
_____ ____ ____ __ _ II 56 /4.7 _ _ _
4. Longitude (TIME)_____________ + +5 35 /3.6 ___ __
5. Universal time (UT) (3+4)
_____________ 1 7 31 28.3
6. Equation of time for O1° Greenwichdate 0 0-3 41.0
7. Variation of equation of time forGreenwich date - 0 00 02.0
8. Fraction of day elapsed (5) _- 24
____ ___ ____ ___ 0.73 W-
9. Correction to equation of time(7X8) - 0 00 01.5
10. Corrected equation of time (6+9)_____________ 0 0 3~ 39.5_____
11. Local apparent time (LAT) (3±10)
_______ _______ ___ 1/ S9 54.2_ _ _ _ _ _ _ _ __ _ _
COMPUTED BY DATE CHECKED BY DT
" u, - 95 20 MAY 196.16 .T.rn s - q1I 2/ MAY 1.96 3FORM ADAI FEB 57190
Figure 4. Conversion of M1/ean Time to Apparent Time (DA Form 1902).
U. S. GOVERNMENT PRINTING OFFICE: 19570-{20810
CONVERSION OF APPARENT TIME TO MEAN TIME
Loca2 Date 18 May 63
LocaL Appatent Time (LAT)-------------------Longitude o6 Station, degnee--------------------Long~ctude of Sttion, houtha--------------- --- +
12h0d100b83 48 24 W5h35 13.6
Recotded LAT------------------------- - 2hm0Watch Cotec.tion ------------------------------------ 00Colvreated LAT---- -----------------------rr- -Longitude Vi . enence------------' ----------- + 5 35 13.6Gkeenwi ch Appaxent Time (GA) ------ ------------ 13Equation of Timem 18 May (&%om Tabte)----------+ 3 41.0Covkec/t.Zon Jon In.tehvaL= -2.00 (17.6/24)--------- -- 0 01.5Co'uleted Equation of Time------------------rr97FApp'ox.Lmacte tln&,veu1a Tcme--------------rrrrr-- 17 31 34.1CoAuection Jon Inte'wa= -2.00(17,53/24) --------- - 0 01.5F Znae CouLeeted Equation of Time------- --------+ 3 39.5
Co'uected LAT--------------------------------- 12 00 00.0Equation o Time---r----------------------------+ 3 39.*5LocaL Mean Time (LMT = LAT-Eq, ob Time)----------TF520
Figure 5. Conversion of apparent time to mean time.
Section II. COMPUTATION OF AZIMUTH
12. Method of Computationa. The observation of astronomic azimuth con-
sists of observing the angle between a mark oil the
earth's surface and a star or the sun. The com-putation consists of calculating the azimuth of thecelestial body at the time of observation, thensubtracting the measured angle from this value toobtain the azimuth of the mark.
b. The calculation of the azimuth of the staror sun involves the solution of the spherical tri-angle whose vertices are the pole, the observer'szenith, and the body observed. This triangle isknown as the astronomic triangle or the PZS tri-angle (fig. 6). Since the body is apparently mov-ing, the time of the observation must be known
except in some special cases.c. If the angles of a spherical triangle are desig-
nated A, B, and C, and the sides opposite theias a, b, and c, just as is customary in plazie trigo-
nometry, a fundamental formula can be derived
for the solution of the triangle when any three of
its elements are known.
cos acos b cos c+sin b sin c cos A
All other formulas for the solution of the spherical
triangle may be derived from this fundamental
equation.
d. In the astronomic triangle, the angle at the
zenith, between the pole and the celestial body,is the azimuth of the body, hereafter designated
A. The angle at the pole, between the zenith and
the body, is the hour angle, designated t. The
angle at the star or sun, usually denoted by q, is
the parallactic angle. The parallactic angle will
seldom be used in astronomic computations.
e. The side of the triangle opposite the azimuth
angle, A, is the arc of the hour circle between
(90-0)
ZENITH DISTANCE()
(90-d)
Figure 6. PZAS triangle.
OZ
Q
I-
LIW
/
the pole and the star (or sun) and is known as
the polar distance (p) or codeclination (90 °0-).
It is obtained by subtracting the star's declination
from 900. In most cases this subtraction need
not be made, since the cofunction of the declination
itself can be used instead. The side opposite the
hour angle at the pole is the are of the great circle
between the zenith and the star. This is known
as the star's zenith distance, designated by the
Greek letter . The zenith distance is either
observed directly, or its complement, the altitude
(h), is subtracted from 900. The cofunction of
the altitude is frequently used in place of the
required function of the zenith distance. The
third side, lying opposite the parallactic angle, is
the arc of the observer's meridian between the
pole and the zenith. It is obtained by subtracting
the observer's latitude from 900, and is sometimes
known as the colatitude (900-0). In nearly all
practical formulas, the cofunction of the latitude
is used.
Jf. The hour angle (t) is obtained from the
recorded time of the observation and the right
ascension of the body observed. In the case of
a star, the local sidereal time (LST) is required.
This may be found by observing the altitude, or
the time of transit across the meridian of known
stars; or from radio time signals, provided the
longitude of the station is well known. The hour
angle (t) equals the LST minus the right ascension.
The hour angle is measured westward from the
upper meridian from 0" to 24". For convenience,the t angle is limited to the first 2 quadrants
(0h to 12") on the computing forms, and is con-
sidered as measured both west and east from
the upper meridian. The latter direction is
considered negative. Should the hour angle,found by subtracting the right ascension from the
LST, fall between 12" and 24 h, it is subtracted
from 24 h to obtain the negative t angle.
g. The right ascensions and declinations of the
stars are given in the Apparent Places of Funda-
mental Stars. These are given at intervals of
Universal Time (UT). Hence, the local observed
time must be converted to UT before taking out
the right ascensions and declinations, as explained
in paragraph 9. The right ascension and declina-
tion of the sun are given in the American Ephemeris
and many other publications.
h. Since only three parts of the astronomic
triangle are required in order to compute the
azimuth, different combinations may be observed
in the field. Thus, we may be given (1) the lati-
757-381 0 - 65 - 2
tude and declination, and observe the altitude;(2) the latitude and declination, and observe(indirectly) the hour angle; (3) the declination,and observe the altitude and hour angle. Thereare also special cases, such as observations atelongation or culmination, when the star's positionis found by trial without knowing the time.
i. The computer frequently must apply somecorrections to observed values. Since the star
(or sun) is generally observed at considerable
altitude, an error is introduced in projecting its
direction downward to the horizon whenever the
axis of the telescope is not truly horizontal. For
the inclinations involved, the correction, c, is-
C"-=i" tan h
in which c and i are in seconds of arc; i is the
inclination of the telescope axis as determined by
the readings of the plate level or a striding level;
and h is the altitude of the star. This correction
is not required in third-order computations.
(1) Computing the inclination correction. In
order to compute the inclination, i, the
sensitivity value of the level bubble in
seconds of arc per division, and the
displacement of the bubble in divisions
from its level position must be known.
The scale should be read at both the left
and right hand ends of the bubble on
both the direct and reversed pointings on
the star.
(a) If the scale reads continuously from
one end of the tube to the other, the
record appears as-
Direct -Reversed-- ---
Left Right
07.5 16.817.7 08.5
10.2 08.3+01. 9
The final figure +01.9 is the inclina-
tion factor, and is actually four times
the mean displacement of the bubble
for the two pointings. It is found as
follows: The smaller value is sub-
tracted from the larger in each of the
columns, indicating readings taken
at the left and right ends of the bubble.
The difference in the right hand column
is then subtracted algebraically from
that in the left. That is, if left is
greater than right, the inclination is
positive; if right is greater than left,
17
it is negative. In case a striding level
is used, it should be reversed on the
axis during each pointing, D and R.
A record identical to the above, and
computed in the same manner will be
obtained for each of the pointings.
The final inclination factor is then the
algebraic mean of the inclinations ofthe two pointings.
(b) Occasionally, a record will be made byreading the scale outward in both,
directions from its middle, as-Left Right
Direct__________ 05.0 04.3Reversed --------- 05. 2 04. 0
10.2 08.3+01. 9
In this case, the columns are added;then right is subtracted from left, as
before. When observing on Polaris,some observers may mark the columns
west and east instead of left andright. The inclination factor mustthen be multiplied by the level factor,
d tan h, in which d is the value of a
division of the level scale in seconds of
are, and h is the altitude of the object:observed. The value d for the in-strument used must be furnished the
computer. The final figure,
iX- tan h
is the correction which must be added
algebraically to the circle reading
taken on the star (or sun).
(2) Correction for refraction. An inclined ray
of light is subject to bending in passing
through the earth's atmosphere, as a
result of which all observed objectsappear too high. This bending of the
ray is known as refraction, and varies in
amount with the angle the light ray
makes with the vertical, the temperature
of the air, the barometric pressure, and
to a lesser degree, the relative humidity.
The humidity can be disregarded in all
work unless specifically required. Table
V, appendix III is used in finding the
mean astronomic refraction and the
corrections to be applied to the mean
for varying temperatures and pressures.
To use this table, proceed as follows:
(a) Enter the table at the apparent zenith
distance of the object, and by inter-
polation, find the mean refraction.
This value applies to a standard
temperature of 100 C. (500 F.) and a
barometric pressure of 760 millimeters
(29.9 inches) of mercury.(b) From the table of corrections for
temperatures other than 500 F., deter-mine the multiplier (CT) of the mean
refraction for the observed tem-
perature.(c) From the table of corrections for
barometric pressures other than 29.9
inches, determine the multiplier (CB)
of the mean refraction for the observed
barometric pressure.
(d) Find the refraction correction, r,by multiplying together the meanrefraction and the two factors.
r=rmXCBXCT
(e) The computer should judge whether
the class of observation requires the
corrections for nonstandard atmos-
phere.
13. Observation on a Close Circumpolar Starat Elongation
a. The stars ordinarily used for observation on
a close circumpolar star at elongation are Polaris
and 51 H.Cephei in the northern hemisphere,and a Octantis in the southern hemisphere. The
reduction formula is:
cos 0sin AE O
or
sin AE=sin p sec o
in which p is the polar distance (90- ) of the star.
The procedure (fig. 7) is as follows:
(1) Obtain 0 for the date from the Apparent
Places of Fundamental Stars.
(,2) Solve the formula.(3) Apply correction for diurnal aberration,
if warranted by precision desired.
cos A cos qDiurnal aberration = 0.32 cos A
cos h
plus in the northern, and minus in the
southern hemisphere.
(4) Subtract the observed angle, mark to
star, correcting reading of circle on thestar for inclination.
(5) The azimuth of the star is measured eastor west from the meridian, accordingto whether eastern or western elongationwas observed. The field notes shouldstate which, or at least record a time fromwhich it can be determined.
(6) The above formula is exact and may beapplied to any star at elongation (method1, fig. 7).
b. The approximate formula for close circum-polar stars only is:
A '=p" sec .
This formula is adequate for computing mostazimuths observed by this method. AE and p arestated in seconds of arc. 5 is obtained as aboveand subtracted from 900 to obtain p (method 2,fig. 7).
c. In table II, AE&NA, the azimuth of Polarisis obtained by determining the product of thequantity (bo+bl+b 2) and the secant of the lati-tude. The factor "bo" varies with the localsidereal time, the factor "bi" varies with the lati-tude and the factor "b2" varies with the monthof the year. All values must be interpolated asaccurately as possible (method 3, fig. 7).
d. In observations on a close circumpolar starnear its point of elongation, it is possible to obtainone direct and one reversed pointing so near tothe point of elongation that the observations maybe computed as if made at the instant of elonga-tion. In most cases, additional pointings areneeded, particularly if a repeating theodolite isbeing used, since the required pointings cannot beobtained within the time limit. Pointings at
some distance from elongation may be easilyreduced if accurate time is available. The formu-las given in a above are used for the computations.The time of each pointing on the star should berecorded. The formula for reduction to the in-
stant of elongation is:
A'=405,000 sin 1" tan AE(T--T) 2
in which AE is the azimuth of elongation, T is the
observed time, and TE the time of elongation,T-TE being expressed in minutes of sidereal
time.
(1) The correction or reduction to elongationfor Polaris can be obtained from table
VI in appendix III. Along the left
margin are the minutes of sidereal time
and along the top are the azimuths ofPolaris from north. In most cases,double interpolation is required to ex-
tract the desired value.(2) The correction will always reduce, nu-
merically, the angle between the posi-tion of elongation and the meridian.The mean correction for the star is ap-plied to the azimuth of elongation before
subtracting the angle, mark to star.e. For the determination of the Local Hour
Angle at Elongation, the formulas are-
tan 4cos t=tan-tan o cot S.
The body is on the meridian at the instant when
the local sidereal time and the right ascension of
the body are equal. The body is at western
elongation at this instant plus the time interval
represented by te. Eastern elongation takes place
at culmination minus the time interval repre-
sented by te or plus the quantity (24h-te). Ex-
ample: If the observer's latitude is 38°39'33'8
and the declination of Polaris is 89005'15'2,
cos to is equal to the product of tan 38°39'33'8
X c o t 89°05'15'2 = 0.79998800(0.01592651) =
0.01274102. te= 89016'12"= 5h57m04.8S. Polaris
will be on the meridian when its right ascension
and the local sidereal time are equal, or at
1h0 8 m5 8 19 local sidereal time. Western elongation
comes 5 h5 7 m0 4 .8s later and eastern elongation
5"5 7 m0 4 .8S earlier. The conversion from sidereal
to civil, or standard, time is explained in para-
graph 11b.
14. Observations on a Close Circumpolar Star
at Any Hour Angle
a. The advantages of this method are that the
star may be observed at any time it is visible and
an unlimited number of observations may be
taken. The two common methods of determining
the azimuth are known as the direction method
and the hour angle method. Both methods use
the same basic formulas which are as follows:
tan A=- s tcos ( tan 5-sin . cos t
sin 0 cos t-cos 0 tan 5sin t
tan A=-cot ~ sec 0 sin t (a1-a
COMPUTATION/ OF AZIMUTH USING A CIRCUMPOLAR
STAR AT ELONG AT/ON
Stat ion : Tap 9( 380 39'33."8 N
A- 78°44' 373 W
Posit&on o1 Polaris : RA : 01"57"°5515S6: t 89° 04, 52
Date: 9 Jul/y /963
Time : 0/ h08 "58.~9 L stdT
Solut7ion7 by f~ormen
(89 °05' /52 )
(38°39" 3 3 .8)
S/n AE
AE
MarA t'o Star
A s trono,c Az im u th
4/a SI N AE = -o c
0.0/S 92449 os$
0.78087338
0.0203 9318
= #0/ o/0' 06.7
/58 32 /6.1
2020 37'50.6
Method O) Solutior by 7'arrrna/a AE =/O$ "sc!
6S (89o05'/5S'2)
9o° 9- (89°05'/52)= 0 054'44. =3284.'8
Sec 0 (38°39133."8) =/ 2806 /735
AE 12806 /735 (3284.8) =4206.15 = 0/'0'06.6Mark to Star /58* 32' /6.1
Astrono.*nc Azimutht 202' 37' 50'1,
Solution~ by Table Zi, Arnerican Ephemeris
38° 39' 33.8 ( 38 °40 )
-. 9 July /963
4, =~4' Sec 0 .2806 /735
= 70.1778 = a- 0/*/0"/C7
/5-80 32' /6.
2020° 375S4.
Figure 7. Computation of azimuth using a circumpolar star at elongation.
Me t hod O:
Co~s 6
Cos 0
Metfho cl Q
LST
Date
,61 = 0.0
b2 =- 0.3
rot~ = + 54.'8
AE =/2806 /735 (54.'8)
Mark to Stqr
Astror'o/',c. Azirmath
Where: A= Azimuth of star, as reckoned from theobserver's meridian in a clockwisedirection
a= cot 5 tan 4 cos t
t=The local hour angle, reckoned west-
ward from upper culmination.
Tables for log (-a) are found in TM 5-236.
b. The direction method is named for the typeof theodolite used in the observations; this methodis the one most commonly used for high order
astronomic azimuths. Computation can be made
with natural functions, or logarithmic functions(fig. 8).
(1) Correct the mean recorded times of eachposition or set for the chronometer error.
When a sidereal chronometer is used, thiswill give the local sidereal time (LST),the chronometer correction being ob-tained from observations taken at the
station, or from radio time signals andthe station longitude. If a mean-time
chronometer or watch is used, it is cus-tomary to obtain the correction to localstandard time. This is then converted
to UT, thence to GST, and by applyingthe longitude, to LST.
(2) Obtain the right ascension and declination
of the star from one of the ephemeridesfor the date and UT of observation, using
the mean epoch of a series of observations
which should not extend beyond a period
of 4 hours.
(3) Subtract the star's right ascension from
LST to obtain the hour angle of the
star (t) and convert to units of arc.
(4) Solve the formula for A, the azimuth of
the star.
(5) Determine the. correction for curvature
(table VII) when applicable and apply
to A to give the correct azimuth of star.
This correction numerically decreases the
value of A.
(6) Determine the level correction and cor-
rect the circle readings on the star. If
the altitude is not observed, it may be
computed in the case of Polaris from
Table II, AE&NA, or for any star from
the formulas:
sin h=sin 4 sin 8+ cos 4 cos 5 cos t
cos h= cos S sin t cos 6 sin t
-sin A -tan A cos A
Computation of h to the nearest minuteof are is sufficient.
(7) Subtract the corrected reading on thestar from the circle reading on the mark.
(8) Add algebraically the corrected azimuthof the star from North and 1804 to (7)to obtain the azimuth of the mark fromSouth.
(9) Abstract the results of all positions or
sets, apply the rules for rejection, andtake the mean of the acceptable observa-tions. Record this information on DAForm 1962 (fig. 9).
(10) Determine the probable error of obser-vation.
(11) Apply correction for diurnal aberration.(12) Apply correction for elevation of mark
by formula:
C=+0.000109 h cos2 4 sin 2Awhere h= elevation of mark.An accurate sea level reduction chartmay be used.
(13) When the x and y of the instantaneousnorth pole are known for a given date,the correction to be applied to the astro-nomic azimuth to reference the azimuthto the mean pole is computed by theformula: Aa=(x sin X-y cos X) sec 4wherein West longitude is considered
positive.
c. The hour angle method normally is used forlower order azimuths when the time of observa-
tion is less accurately determined. Solution can
be made using natural function or logarithmic
functions (fig. 10).
15. Observations on East-West Starsa. Basic Considerations. When high order
astronomic azimuths are needed and close cir-
cumpolar stars cannot be seen, East-West stars
which reach elongation at approximately 150
altitude may be observed between approximately
7%° and 22% altitude. At latitudes of less
than 20, stars crossing the prime vertical at
approximately 300 altitude may be observed.The declination of the observed stars should befour to five times the observer's latitude for the
elongating stars and approximately one-half the
latitude for stars crossing the prime vertical.
For lower order azimuths, the altitudes of east-
PROJECT LOCATION -AZIMUTH BY DIRECTION METHODA LAD (TM 5-237)
ORGANIZATION MARK LATITUDE 0~) LONGITUDE MX STATION
US MoS MAP 67S 0267 5 08 28.86 NP(AMS /958)CHRON. NR. INST. (NR.) LEVEL VALUE (d) ECC.- (INST.) (SIGHAI OBSERVER G. CIVIL DAY
/2460 T3 63010 16.462 29.953 , FE.. APR.///
Date19 63 , position / 2 3 4Chronometer reading /0 07 48.3 /0 /4 35.0 /0 33 66.4 /0 40 /6.7
Chronometer correction -o0 295 -08 29.6 - 08. 29. 7 - 08 297
Sidereal time 9 659 /88 /0 06 05.4 /0 25 26.7 lo 3/ 47Z0
RA(a) of POLARIS (star) / 356 450 / 56 45.0 / 6,6 450 / 56 450
HA(t) of star (time) 8 02 33.8 8 0920.4 8 28 41.7 8 35 02.0
t of star (arc) /20 38 270 /22 20 06.0 /27 /0 255 /28 45 30.0
Decl. (a) of star 89 05 32.59 89 05 32.59 89 05 32.59 89 05 32.59Sin0 Cos ~ Ten i Cos 0Tan a
Constants for star . 628 65203 .777 68672 .63.122 643 49.089 64119
Sin t + .860 37903 + .844 93526 1- .796 80684 .77979344
Cos t 50o96472 -. 534 86859 - .604 23412 -626o03690
Sin m 006 t -. 32039547 -. 336 24622 -. 379 85300 -. 393 5937
Cos tan i-sin o cos t 494/0 03666 4 9425 88 741 4946949419 49483 200"6
-Tan A- sin tusNtuaa-iu*..st .0/741304 .0/709499 .0/61/0703 .0/575875
A (Az. of star from N.)t -0 5S9 513 -0 58 458 -0 55 22.0 -0 54 /0.2
Difi. in time between D. & R. / 3S/ 322 //9
Curvature correction-----
Altitude of star (h) 380 28' 69" 380 2/44~ 380 23" 53' 380 22' 45'
d tan h(level factor) 1.284 1.283 /.280 1.279
Inclination + 1.2 + 2.3 +415 +3.3
Level correction +01.5 +03.0 t+0/.9 t-04.2
Circle reading on star 170 35 09.8 /8 / 36 4 7/ 203 4/ 12.4 215 40 39.2_
Core. reading on star /70 35 /1.3 /8/ 36 50.1 203 41 /4.3 2/5 40 43.4
Circe reading on Mark 00 00 /1. 3 // 00 44.6 33 0/ 47 0 44 .59 592
Diff. (Mark minus star) 189 25 00.0 /8 923 54.5 /8 920 32.7 /8 9 /9 /5.8
Corr. Az. of star, from N. t - 59 5I/3 -S8~ 45.8 - 55 22.0 - .54 /0.2:">;<": :, .. :1800 00, 00"'.0 1800 00' 00".0 1800 00' 00".0 1800 00' 00'.0
Azimuth of MAP0 / if 0 / f 0 0 ,
(clockwise from south) 8 25 08.7 8 25 08.7 8. 25 /0.7 8 25 05. 6To the mean result from the above computation must be applied corrections for diurnal aberration, elevation of mark,and eccentricity (if any) of station and mark. Carry times and angles to tenths of seconds only.* Give volume and page of record for eccentricity, if any. t Minus, if west of north.
COMPUTED BY DATE / CHECKED BY DATE
I . ' -. 9/S 2 OCT l3 Q Rb. i.ko - 7IWS IIOfC.63
DA I FE571903 GPO 008848U. S. GOVERNMENT PRINTING OFFICE :1957 0-120800
QD Natural functions, DA Form 1903
Figure 8. Computation of azimuth using a circumpolar star at any hour angle (direction method).
PROJECT LOCATION IAZIMUTH BY DIRECTION METHOD (LOGARITHMIC)MARYLAND I.(TM 5-237)
ORGANIZATION MARK LATITUDE (0)01 LONGITUDE WX STATION
UJSAMS MAP 38057 02.67 j5f08"'28.s86 NP(AMS 1958)-CHRON. NR. INST. (NR.) LEVEL VALUE(d) ECC.' (INST.) (SIGNAL) JOBSERVER .G. CIVIL DAY
12460 T35S3010 6.462 2 9953 m F E. P11 APR. /1. 13/Date 19 63 , position / 2 3 4Chronometer reading 10 07 48.3 /0 /4 35.0 /0 33 6.4 /0 40 /6.7Chronometer correction - 08 295 - 08 29.6 -08 2,97 -- 08 29. 7Sidereal timne 9 59? /8.8 /0 06 06.4 /0 25 26.7 10.3/ 470RA(a) of PO LA RIS (star) / 6 4.5.0 / S6 45.0 / S56 45.0 / 56 45. 0
HA(t) of star (time) 8 02 33.8$ 8 0.9 20.4 8 28 41.7 8 35 02.0
t of star (arc) 120 38 27.0 /22 20 06.0 /27 /0 25.5 /28 45 30.0
Decl. (8) of star 89 05 3259 89 05 32.59 89 05 32.59 89' O5 32.5'9
Log cot a 8.199 8/5 8. 199 8/5 8.199815 8.1598/5Log tan - 9.907 606 9. 907 606 9.907 606 .9907606Log cos t 9. 707 2 7 6 N 9 728 2 4 7 N 9 78/ 2 05N 9 796 60O NLog a(to bplaces) 7 814 70 N 7 835 67 N 7 888 6 3 N 7.904 02 N
Log cot a 1/99 8/5 8.199 815 8.199 8/5 8.199 816Log sec m 0. /0 9 / 95 0. /0 9 /95 0.109 195 0.1091/95Log sin t 9 934 690 9 926 823 9 90/ 363 9.891 .980
L____ 9. 997/175 9 997 036 9 996 653 9.996632Log (-tan A)(to 6 places) 8.240 875 8.232 869 8.207 016 8.1917 522
A (Az. of star from N.)t -0 59 5/.3 -0 58 4S8 -0 55 22.0 -0 54 /0.2Diff, in time between D.&dzR. /. m .. S. . s.m. S.
35/35 2 02 / /8Curvature correction ---
Altitude of star (hx) 38 88 59f 802 4f235' 3 2 3tan h(level factor) A.284 1.283 1.280 1.279
Inclination f 1.2 +2.3 f /5 433Level correction +01.5 40-3.0 + O/. 9 t 04.2
Circle reading on star /70 35 .9 8 /8/ 36 471/ 203 41 /2.4 215 40 39.2Corr. reading on star /7035 /13 /8/ 36 5'0.1/ 203 41 /4.3 2/5 40 43.4Circle reading on Mark 00 00 //. 3 /100 44.6 33 -0/ 47.0 4455S9 59.2
Duff. (Mark minus star) /89 25 00.0 /89 23 54.5 /89 20 32.7 /89 1/9 1/58Corr. Az. of star, from N. t .- s9Y 61.3 -- 58 45.8 - 55 22.0 - 54 /-0.2
'ys .;9k ,. <,> >.sf ": 80 0, 0011.0 1800 00' 0011.0 1800 00' 00"1.0 1800 00' 0011.0
Azimuth of M A P o if 0" o if 0 i f 0 if1
(clockwise from south) 8~ 25 08.7 8 25 08.7 8 251/0.7 8 25 o056
To the mean result from the above computation must be applied corrections for diurnal aberration, elevation of mark,and eccentricity (if any) of station and mark. Carry times and angles to tenths of seconds only.* Give volume and page of record for eccentricity, if'any. t Minus, if west of north.
COMPUTED BY DATE CHECKED BY DATE
o~~ 4D25 2 OCTr 63 D.k. flj~a ..- /7fls 120AC 63
DAtF 5 57190 U.. S. GOVERNMENT PRINTING OFFICE :1957 0-420843
Q Logarithmic functions, DA Form 1904
Figure 8-Continued.
AZIMUrH SUMMARY
PROJ®ECT TABULATION OF GEODETIC DATA(TM 5-237)
LOCATION, 044r /4rd RANIZATION
STATION
NP(AS,158)APR- //9/ AY 24. 08/
______8025 V 8® 2s V
/ 08. -2.3 08.4 -2.0 _ _ _ _ _ _ _ _ _ _ _
2 08.7 -2.3 o6.6 o2___________
1 /0.7 -4.3 o6./ +0.3 __ _ _ _ __ _ _ _ _ _
4 056 f08 052 + 1.2
S 103.9 + 2.6" 07.5
6 06.8 .-0.4 o6. t. .Fv2 7.973
7 07.0 -0.6 o4.3 +2.1
807.0 - o.6 o8.3 - /1.9 .6745 3Y2(3)
9 03.6 +2.8 06.4 079.73 ~
/005S7 + 0.7 054 -t0.8 6(745 1=t/08
/f 06.9 - a.5 07.1 -0o.7
/2 ®S. 40.7 07.8 - 1.4
13 06.2 +0.2. 06.5 +0.9 _____
/ 4 031 +3.3 06.8 0o.4
/S 05.6 +0.8 06o.8 -o.4
07 0.
I 534E SE R ED Az M U H
8 25 0 6.43 t 0.19
OWNA ARERRATIONi_____ + 00.32______
ELEVAI/oN of MARK (/20 Fr.) .60.00 _____
ECCENATRIC T/ ES 03 o2R. 93 _____
FINAL ASRO NOMiCAL AZIl UT/-I 8 28 09.68 0./,9
R ATIN __ __ _ __ __ _ __ _
0 RE. EC.TI0NS____________
Po8A O ER oR of A IGE0o ER VT loN = t 1.08
TABULATED BY DAT [HCKD BY DT
-I/ms 2ocT. 43 -4m5l 12 feC'6.3
6571962 GPO 90547U. S. CHOVZRNRICT PSnIMD OFWE: 1957 0 - 4211a
Figure 9. Summary of azimuth observations.
DIAGRAM PROJECT LOCATION AZIMUTH BY HOUR ANGLE METHODTRUE Sth Tes uW v______(TM _____-237)_
Ae St ORGANIZATION LATITUDE (") LONGITUDE (X) STATION
mO 7.I I 914418
LMSMARK INSTRUMENT (Number and type) STANDARD TIME (Meridian)
k ERVER CELESTIAL BODY(S) WATCH V . SLOW (+) WATCH COMPARED (Time)
Atc kec.________ _
DATE OBSERVERWETR
SET NRI SET NR 2 SET NR 3
.. TIME Hos. ANGLED TIME 11oa. ANGLE TIME Boa. ANGLE
Has M:;;.;; EC. a ' $Has. MIx. SEC. c 1 HRs. MIN'. SEC. 0 P U
Mean / 2. . 19SET NR 1 SET NR 2 SET NR 3
HaS. MIN. SEC. HRs. MIN. SEC. HRs. MIN. SEC.
1 Mean time of observation j o HL. I.1
21 Watchecorrection .0+ __ Zr ~ _70
3 Time Zone Correction (TZC) f - 1- - 61
4 UT of observation (I+2+3) 7 3.1. 1(SUN 5 Oh Gnih EQT[r Sid.T.
OSERVATION renic-orI. 21 . 2 . .L
For star 1 6 UT X var. EQT per hour [or .table III Eph.] f j 4--[observation,
use factors I7 (5+6) correct EQT [or G. Sid. T. (4+5+6)1 t .1 .1.M 6 Ienclosed 8-47 A o A
in bracketsi .S17GA~rA ~ j...9 GHA in time (GAT-12h) [or (7-8)1 2 2 a.~ ~Q~
10 GHA in arc Q/~50
11 Longitude, West (-), ~~~~) f L4 2 ~~Q~ 4
12 LHA (10+11)=t (or 360-LHA -- t) 1-871L I. SET NRI4ETN 2 SE N
t 0 o U 0 1' Mean true azimuth to Mark
Lat. (*) f Grid correctionf
Dec. (8) t 03 o Grid azimuth to Mark
Sin t f Magnetic azimuth to Mark
Cos t 1 0624 *5f2 Magnetic declination E(-), W(+)
Sin* f t.. q5~ 6617 M "
Cos f in t4%
9179302 79_32 Z717 -Tn Acos 0 tan 8-sin 0 cos tTan a
-Tan A . ,90 - ubo a o ~If LHA is greater than 1800, sb
A (E .LIIL .Ltract from 3l60° and reverse sign.
Azimuth of 8 Obtain a from Ephemeris.~L IL L. I 49i I..Check signs and quadrants by use
L, Mark to 8 1?4 of sketch.
Tr. Az. to Mark 2 0, 2'l /COMPUTED BY DATE CHECKED BY DATE
id FEB457/ eb.
FORM 0
Q Natural functions, DA Form 1905
Figure 10. Computation of azimuth using a circumpolar star at any hour angle (angle method).
DIAGRAM
TRUE NORTH
A 4r St'br
BSERVER
PROJECT LOCATION..r IAZIMUTH BY HOUR ANGLE METHODI(Logarithmic) (TM 5-237)
ORGAN IZATION LATITUDE LONGITUDE fSTATION
MARK
LENOX Az/MUM iflarkCELESTIAL BODY(S)
DATE
5S Wnv, .5.3OBSERVER
INSTRUMENT (Number and type)
WATCH 110 - SLOW W+
7 _<w
N. . s.
STANDARD TIME (Meridian)
000WATCH COMPARED (Time)
WEATHER
MSd 11aSET NR I SET NR 2 SET NR 3
.... TIME HOR. ANGLE TIME HOR. ANGLE TIME HoR. ANGLE
HRS. MIN. SEC. 0 , FHRS. MIN.I SEC. 0 j r HRS.~ MIN.~ SEC. 0 j
Mean Z /01g07 ,730.I 11,1 112.5_ 1 .
SUN
OBSERVATION[For starobservation,
use factorsenclosed in
bracketsj
SET NR I1 SE's NR 2 SET NR 3
I Mean time of observation HRS. MIN. SEC. HRs. MIN. SEC. HRS. MIN. SEC.
:2 Watch correction f -f
3 Time Zone Correction (TZC) :1f - -
4 UT of observation '(1+2+13) 2~1. ~ L ~ . 65 Oh Greelwich EQT for SidT.] f !11_1f 96 UT'X var. per hour [or table III Eph.] - 49. 3.i -9 O
7 (5+}6) correct E.QT [or G. Sid. T. (4+5+6)1 f 2 L Q 2. 1 X 2_ 6 UL8 (4 +7) GAT [or RA] jJ ~ ~ ~ ~9 GHlAin time (GAT-12h)[for (7-8)J IQ.L .Q.Z...Q2 &
10 GHA in arc IA 3Z 1 5 0/ 2401 SJ_411, Longitude, West (-),' ~ f -9j144 j" 91 44 1 L 4 L&Q
S= 89°03'047iS 1121 LHA (10+l-1)-t (or 3600-LHA=-t) -87121 la'.5I-%142 L~oLA5id4 [~M
SET NR 1 SET NR.2 SET NR 3--.
Log sin # Mean true azimuth to Mark ,
Log cos tGrid correction
(Sum) log A ~ a~~zzGrid azimuth to :Mark
A 0.2 0 2il& 120531 Magnetic azimuth to Mark
Log cos 4 Magnetic declination E(-), W(+I)
LogTa A=---_S7809nt
B TnA Cos 0 tan a-sin * cos t
B-A If LHA is greater than 1800, su~btract
Log Sill t _ 9999858 from 3600 and reverse sign.- iiii~~Obtain a from Ephemeris.
Log (. -2. A) Check signs and quadrants by use of sketch.
A (E eg a ilJ I * O1JL92 1"
Azmt5fS 1J. I 9 J A 2 COMPUTED BY~ ,~ DATE
ICHECKED BY DATETr. Az. to Markj 8 2813 41 7j . ""'o 6 f SS.
DA I FORM510
GLogarithmic functions, DA Form 1906
Figure 10-Continued.
west stars can be observed at any latitude by the
equal altitude method; i.e., a pair consisting of
an east star and a west star should be observed
at the same altitude. The stars should be near
the prime vertical, or when the latitude is near
300, the declination of the stars should be ap-
proximately twice the latitude. For any type
of observation on east-west stars, the accuracyof the observers latitude is critical.
b. The Hour Angle Method. Either the Direc-
tion or the Hour Angle Method may be used as
outlined in paragraph 14. Because of the rapid
movement of the stars as compared to circum-
polar stars, recording of time is more critical
and chronographic recording of time is recom-
mended. Since the stars' azimuth will be between
450 and 90 ° from the meridian, DA Forms 1903and 1905 should be modified to a cotangentformat (fig. 11) such as:
-cot A cos 4 tan S sin 4sin t tan t
(1) Usually, it is easier to compute the stars'azimuth for each direct and each re-verse pointing on a star and then deter-mine an azimuth for each such pair.This eliminates the necessity for com-puting the curvature correction. If themean time of the direct and reversepointings is used, the curvature correc-tion is computed by the followingformula:
Curvature correctionsin A cos 4 sec2 h (cos h sin -2 cos A cos 4) (t)2 sin 1"
8
where t is the time interval between the
direct and reverse pointings expressed'
in seconds of arc.
(2) The diurnal aberration correction is
applied to the azimuth of the mark reck-oned clockwise from the observer's me-
ridian. It is positive if the star is ob-
served north of the observer's latitude
and negative if observed south of the
observer's latitude.
c. The Altitude Method. The formulas are:
cos A-sin -sin h sin 4cos h cos 4
1 /cos s cos (s-p)cos A= cos 4 cos h
tan1 A= sin (s-4) sin (s-h)2 V cos s cos (s-p)
in which s= 2(4+h+p), p being the polar dis-tance of the star.
(1) The second and third formulas are pre-
ferred for logarithmic computations. DA
Form 1907 (Azimuth by Altitude Meth-
od) is used for the computations by the
first formula (fig. 12), and DA Form1908 (Azimuth by Altitude Method,Logarithmic) is used for the computation
by the second formula (fig. 12).
(2) The procedure is as follows:
(a) Arrange the observations by positionsin the case of the direction method,
and by sets if by repetition. Find the
mean horizontal circle readings and
the corresponding mean vertical anglesfor each position or set.
(b) Correct the mean observed vertical
angles for refraction.
(c) Obtain 3 for the date from an ephemeris
and subtract from 900 to obtain p
when the formula requires it.
(d) Compute by the applicable formula.
(e) Subtract angle, mark to star.
(f) Determine the mean of the azimuths
determined by an east star and by a
west star. When more than one
east and one west star are observed,the final azimuth is the mean of the
average value obtained from all east
stars, and from all west stars.
(g) Apply diurnal aberration, etc., if neces-
sary for the required accuracy.
16. Observations on the Sun
a. Altitude Method. The altitude method (fig.
13) is preferred when the time of the observation
is not precisely known. It is subject to con-
siderable error on account of inaccuracies in the
altitude, declination, or latitude. Hence, these
should be taken from the record book and tables
with care. The altitude method is not suitable
for observations on the sun when it is near the
meridian.
PROJECT LOCATION AZIMUTH BY DIRECTION METHODS OL 0/A10 (TM 5-337)
ORGANIZATION MARK LATITUDE ( ) .. LONGITUDE W~ .STATION
vsAAIS T7? S 06/ 223 /h1 /( RAILCHRON.-NR. INST. (NR.) LEVEL VALUE (d) ECC.' (INST.) (SIGNAL) JOBSERVER G. CIVIL DAY
2E 1000/ 73 26.593 6.3/1 ____ J. DOE 6.404 JAN. 63
Datei19 63 postion 3(D)) 3C(R_____
Chronometer reading 02 48 21.8 02 49 00.5Chronometer correction + 1.1 . / _ ________________
Sidereal time 02 48 22.9 02 49 01.6
RA(a) of 30 8 (star) ' 08 05 58.8. 08 05 58.8 ______ _____
HA(t) of star (time) /8 42 24.1 /8 43 02.8_______
t of star (arc) 280 36o 01.5 280 45 42.0 _____ _____
Deal. (a) of star -. 24 i1 48.53 -24 1/ 48.53 _____ _____
81n 0 Cos ~ Tan5a Cos 0TIMna
Constantsafor star. -. 091 60957 +. 995= 79500 -. 449 3505'8 -.44 7 46146Sint -982 93402 - .982 4/240'
rAN ~ -5.34 32378 - S.26 /3053
e~54 Cos t .1 83 95850 +.1./86 72408
0 CoS f' AN S91144 sunlt .455 23041 +,.455 47212_____
2~ SINj O
C1sataN u~ ot +.0 /7 14495 +. 01741195
S+438 08546 +. 438 06617:
A (Az. of star from -. )t - 66 20 32.9 - 66 20 373._____M. S. M0. S. M0. S. m. S.
Diif. in time between D. &R.
Curvature correction
Altitude of star (hi) /48'/ 29. 73 S7 5 /92/0
~tan h(level factor) -330 .334Inclination 9- 0. 5 _______
Level correction .* 0.2
Circle reading on star /97 27 28.4 ______ ____________
Corr. reading on star 197 27 28.6CORRECTED
Circle reading on Mark 22 01 20. 9 ________ _ ________________
Duff. (Mark minus star) /84 33 52.3________MEANGm. Az. of star, from t*t 5 -66 20 35.1I
l8~ .9 .°9 X890 00' 09 . 1800 00' 00".0 180° 00' 00'.0
Azimuth of P(clockwise from south) 1/18 /3 /72 ________________
To the mean result from the above computation must be applied corrections for diurnal aberration, elevation of mark,and eccentricity (if any) of station and mark. Carry times and angles to tenths of seconds only.* Give volume and page of record for eccentricity, if any. t Minus, if west of north.
COMPUTED BY DAECHECKED BY DATE
G .T r - A4S MA-AY '(03 IJ. a.a, -i- - AM5 I MAY '63:
D A ,FES 71903GPO 908848
U. S. 6OVSRNMSNT PRINTING OFFICE : 1657 0-0010
Figure 11. Computation of azimuth using east-west stars.
DIAGRAM PROJECT LOCATION AZIMUTH BY ALTITUDE METHODTRUE NORTH 12-4(TM 5-237)
$Sfar ORGANIZATION *LA rUDE LONGITUDE STATION
Inc.pi e 0 2'0 0 *
MARK INSTRUMENT (Number and type) STANDARD TIME (Meridian)
CELESTIAL B0 Y(S) WATCH FAST (-) SLOW (+) WATCH COMPARED (Time)
DAEOBSERVER WEATHER
INRSETN1R 2 SETN2R 3
Ro. ARE, VE . ANLE HOR. ANGLE VERT ANL oa NLE V f. NL
Mean time 0 4&O
Weatc rrction-
4 40 4 6 __
MI SEnh +S +N Rs.5/90 ~ cmue* SiE+C.~___ _ _ __ _ _ _
o I! 0 o II 0 I y
Tu Az to ark r
Men tru azmt
Si eCompto he eaecmuted sprtl
Mag aziut to MakCsA insnhsn
taine fro TM526¢pl acorcint
AziAstr ooi azmt ato eto ot.I o Ai -,Ai ewe 0 n 8
DrAz.t MOrk 19070ea e iut
Q~ Natural function solution, DA Fornm 1907
Figure 12. Computation o f azimuth using the altitude method (star).
DIAGRAM PROJECT LOCATION AZIMUTH BY ALTITUDE METHODTRUE NORMh /72r (Logarithmic) (TMS-237)
$ ORGANIZATION LT DE LONGITUDE STATION
44$/f A>denc ' e pgMARK INSTRUMENT (Number and type) STANDARD TIME (Meridian)
OVS B RVEIR CELESTIAL BdgY(S) WATCH FAST (-) SLOW (+) WATCH COMPARED (Time)
DAEOBSERVER WEATHER
01171 'SET NR 1 SET NR 2 SET N& 3
HORt. ANGLE VERT. ANGLE HoRt. ANGLE VERT. ANGLE Bait. ANGLE VERT. ANGLE
O~. Mix SEC Ba MIN SEC. is MN. SEC. N
Mean tm . ___ ___
Wac crerection
TZC
Universal time (UT) .21... L6ii'&at O'UT / N e / N
UT X avar. per hr
p_ 19 '53 1 Logarithms p, Logarithms p,__ Logarithms
cSec c
hhe
ZLL2 72 2
oaCos Coss 266 e5
p__ _ _ p __ _ _ p_ _ _ _
CsCos
os 9 -s-p s_ f-p__ _____
1. (sum) /f 02 a2. (sum) 3. (sum) ________
Sum-.-2=log cos A/2 9,g ASum-+-2e=log cos A/2 Sum-+-2=log cos A/2
:. 4..NR 1. NRJC: 2i{:i NR"ii; 3s:i:}iv};;T; } A.?v........ v:ivi:'Gl ' i:i::'.:y ii:::{
A/2 Mean true azimuth to Mark
A, (E orW) 7~ 33 .__ Grid correction ±
Asimuth of S Grid azimuth to Mark
Angle, Mark to S - 33 L 3fo - Magnetic azimuth to Mark
True azimuth to Mark 3QQ -- - .- -Magnetic declination E(-), W(+})____
Computations:-'three sets are computed separately for check. a Correction =UT X variation per day -: 24Refraction and parallax from TM 5-236. If a is (+{),p == 90 - a; if bis (-),p = 900 +} a
(s - p) = arithmetical difference, always positive._TZC=Time zone correction to universal time. Fo~l ics A_/cos s cos (s-p)
s='x (O+h+p) Foml scs V cas~ co hA =astronomic azimuth east or west of north.
COMPUTED BY '"I/DATE, Ms CHECKED BY DATE~
® A I ORM X A7® Logarith mic functions, DA Farm 1908
Figure 12-Continued.
TRENR--DIAGRAM PROJECT LOCATION AZIMUTH BY ALTITUDE METHODTRE ORH ./. 5'55 (TM 5-237)
ORGANIZATION LATI UDE LONGITUDE STATION
38 43 Inc . 3394 77 O09' 00Y ~14'N Ez~foi~tt MARK INSTRUMENT (Number and type) STANDARD TIME (Meridian)
S RVER USTN_____________ 5
- CELESTIAL BODY(S) WATCH FAST (-) SLOW () WATCH COMPARED (Time)
-s DATE OBSERVER WEATHER
Sfv.J5Dt5 T F Smith . _Clear t Wain,SET NR 2 SET NR 2 SET rR 3
oR. ANG. VRT. GEAGLE HoR. ANGLE VERT. ANGLE H R. ANGR ES. ANGLE
Mean
Parallax + 0 *0Mean refraction 4h (sum) Hs _
HRS. SEC. HR MN. SEC. HRS. MIN. S
Mean time J ~44 ____ 0 15E __ L. 14 ___
Watch correction
TZC 4.±. ____ +___..
Universal time (UT) JjBat Oh UT f ~ .J. 2 2. 1L .. .& l.UT X a var. per hr. f .- J1L ... it.. - .=z... ~ A
*. 27 e
b
2 2 Q6 4,. 2.. Q .A 2. 6 2
h .. 9 Jz 40 Z 394. 5 5 71
*8- 9.i 3f. AE 2&. 3E AE4 38E .3E 49Sine d O.036g2471 .38$qO0742Sin h + + .77 /60 73%0 .7 5 q9Sin # + # f .6,247-4687 0. 624749 0.6247467Coe hi f1 ~f p* 65261283 0.a 43,36220 0. 6/8492221CoBs # . p. 7802735f 0. 7B= 3 Q 2731
Cos A - 0. R564 740~ ~ ~ __
Azimuth of $ 4 5L 2.i5 2. l4.. 1 IiX AAngle, Mark to S 07 __ Jj~24 J2 A S l8_True Az. to Mark JL I....E /4 .&JLI. L..AMean true azimuth0
to Mark 14L. 5 Cos A sin B-sin h sin 0
Grid correction cos h cos 0Computation = Three sets are computed separately
Grid azimuth for check, refraction and parallax corrections are ob-tained from TM 5-236. Apply watch correction to
Mag. azimuth to Mark _____ observed mean time. TZC=time zone correction to
Mag. eclintionuniversal time.
B=declination, (+) if north, (-) if south. h=altitude. -0=latitude, (+) if north, (-) if south.A=Astronomic azimuth east or west of north. If cos A is (-), A is between 900 and 1800.
COMPUTED BY / 0 /jDATE 1 CHECKED BY ~, ~DATES
FORM 10DAIFEB 571 0
Q Natural functions, DA Form 1907
Figure 13. Computation of azimuth using the altitude method- (sun).
DIAGRAM PROJECT LOCATION AIMUTH BY ALTITUDE METHODTRUE NORTH (Logarithmic) (TIW5-237)
ORGAN IZATI ON Lii'ruoE p LON.G ITUDE STATION
MARKS INSTRUMENT (Number and 'type) STANDARD TIME (Meridian)
4*.6 t;e G/1uo/__ ______ 5s.t C RVER CELESTIAL BODY(S) WATCH FAST (-) SLOW (+) WATCH COMPARED (Time)
DAEOBSERVER WEATHER
.StIp 5gpt52 I~a it Sm/f": .; ,;i' ,SET NR 1 SET NR 2 SET NRl 3
i:+at:',':'."\r" 3O.ANGLE VER. ANGLE Has. ANGLE VEST. ANGLE Has. ANGLE VET ANGLE
__ _ _ ___ _ 49L4 _L4L8
H; 2c:.}}>: i:'.: s:25:s;; .?:;: as. MIN. SEC. HaRs. MIN. SEC. HiS.. MIN. SEt.
Mean time io j4 59 ilL ___
Universal time (UT) j~J ~ __ 16. 14
isat Oh UT 1 o f 2 2L LL 2 22 11L 2 22 ilL.UT X _a var. per hr 1 __ ~-~- ~ 4
p Logarithms P ~ Logarithms P, Logarithms
he Sece
LL X2 2 2/
2s.-p 0 1 4 5q99997 5j) a 042 15- f966 s 1 5-p O
1. (sum) 2. m) 3.~2~Id4 (sum)
Sum-o-2'=log cos A/2 92 47 Sum-.-2=lbg coon A/2 J4 , Su-2=gcoAf
A/2 2 I q Mean true azimuth to :Mark ..4-7LS711_2 W-UM iA, (E or W) g 1522Q 47Grid correction
Azimuth of S 4 523 Z- 224 04 06 4Z Grid azimuth to Mark
A n l ,t a k t a n t c a i u h t a k_
9_8True azim uth to M ark A 1 3 5 159 1 1 1 3 5 j5 j M agnetic declination E (- ), W (+{)
Computations:-Three sets are computed separately for check. 8 Correction = UT X variation per day -I- 24Refraction and parallax from TM 5-236. If b is (+), p =900 - a; if bis (-), p 900 + S
TZC=imezonecorecton t unversl tme.(s - p) = arithmetical difference,. always positive.TZCTim zoe Crretio touniersl tme.Formulla is cos ,s A - /cos s cos (s-p)
S(O+h+p) Vcos 0coshA = astronomic azimuth east or west of north.
COMPUTED BY DAT Ca... ~ n~fHECKED BY /)/ DAsE4 , 4
®A ~E 71908
QLogarithmic functions, DA Form 1908
Figure 13-Continued.
(1) The formulas for logarithmic compu-
tations are:
1 A cos s cos (s-p)
s2 A cos 4 cos h
tan1 A sin (s-h) sin (s-c-)2 cos s cos (s-p)
in which
h=observed altitude corrected
for refraction
O=latitude of station
p= polar distance of sun (900--5)
s= 32(h++p).
This first formula is simpler, while the
second is slightly more precise.
(2) The formula for machine computation is:
cos A=sin a-sin h sin 4cos h cos 0
(3) The procedure of computation is as
follows :
(a) Compute each position by the direc-
tion method, or each set by the repe-
tition method, separately. The ob-
servation may consist of a series of
positions (D&R observations) or a
single pointing on the mark followed
by a series of direct pointings on the
sun, and an equal number of re-
versed pointings on the sun, followed
by a final pointing on the mark.
(b) For each position or set, determine the
mean of the recorded times and the
horizontal and the vertical circle read-
ings. For each position or set, the
angle from the mark to the star is the
difference between the mean circle
reading on the mark and the mean
reading on the star regardless of the
number of repetitions.
(c) Each mean vertical angle must be cor-
rected for parallax and refraction.
Tables of parallax (table VIII) and
refraction (table V) are included in
appendix III. For lower order obser-
vations the parallax may be neglected
since it is never greater than 9".
(d) Convert the local time of observation
to Ephemeris time as explained in
paragraph 10d.
(e) Extract, from the AE&NA, the declina-tion of the sun for the Ephemeris timeof the observation.
(f) The value of p is determined by sub-tracting the declination of the sunfrom 90 .. If the sun is south of theequator the declination is consideredto be negative. The quantity (s-p)is always considered positive.
(g) Solve for the value of "A" by use of theformula.
(h) Subtract the horizontal angle, mark tosun, from the computed azimuth of thesun to determine the azimuth from
station to the mark. The final azimuth
is the mean of all the azimuth values.
(i) Either DA Form 1.907 or 1908 may beused for this computation.
b. Hour Angle Method. The hour angle method(fig. 14) is preferred when the time of the observa-tion is accurately known. It is not greatly affected
by errors in the latitude and declination, and
should always be used when necessary to observethe sun near the meridian.
(1) The formulas are:
1 sin (€-- ) 1tan 1 (A-q)= s ( ) cot ! t2 cos (++) 21 ( cos (-) 1
tan (A q)=Sin (-) cot 1 t2 sin z(4+) 2
sin ttan A== -
cos 0 tan -sin 0 cos t
When A exceeds 450 from the meridian,
the last may be stated as follows:
cos tan sin-- cot A= sin t tan t
The algebraic signs of the functions may
be maintained, or the following rules
followed in the case of the first formula:
(a) t is taken less than 1 2h and positive.
If over 12 h, subtract from 2 4h .
(b) If t is less than 12h, the azimuth is to
the west of the meridian. If greater
than 1 2h, it is to the east.
(c) (A-q) and a(A+q) are taken out as
less than 90 ° .
(d) When (4-6) is positive, add numeri-
cally the values of (A- q) and (A q)
to obtain the azimuth angle between
757-381 0 - 65 - 3
DIAGRAM
TRUE NORTH
:..5
S&nPERVE
PROJECT LOCATION
I V, i inIAZIMUTH IT HOUR ANGLE METHOD1
I (TM 5-237)
ORGAN IZATION LATI UDE () LONGITUDE (1)) STATION
_AiS 382' I 77 O 1 . 054IMARK 2CELESTIAL BODY(S)
DATE
INSTRUMENT (Number and type)
WATCH F49--) SLOW (+)
7.1 -IcfOBSERVER
SET NR 1 I SET NR 2
STANDARD TIME (Meridian)
WATCH COMPARED (Tion)
WEATHER
r1-.1
I -SET NB 3
Ti E HoRn. ANGLZ TIuNE Horn. ANGLE TINE j Hon. ANGLE
*Hos. MIN. SEC. a " Hits. Mir. SEC. 0 " Hag Is . SEC. "
egan I42.81s/A/4343/
SET NR1I SET NR2 8ET NR3Has. MIN. SEC. HRs. MIN. SEC. Has. MIN. SEC.
1 Mean time of observation _15 _4Z ISE SL_ .i" at. ua2 Watch correction -f +. - -
3 Time Zone Correction (TZC) - -3 - #
4 UT of observation (1 +2+3) 2o. 142 .5. . 21 a5 Oh Greenwich EQT 0jas;:q~ f I&L1 4 _8 248
6 UT X var. EQT per hour in -t -28 a 30 _
7 (5+6) correct EQT al - lfhuj
8 (4+7) GATacA fL & =2L.Q
9 GHA in time (GAT-12h)4a6] .
10 GHA in arc
11~ Longitude, West (-), ~aT~s f z'. oi~.~ i .
12! LHA (10+11)-t 0001& .) I 5s]3~ I4J 7I4Rl~J &21 I~ I/AS_________________________________________________ -- j a~ a ~. & -- _______ --
t ~ NENR .0 NR SETNRS Mean true azimuth to Mark
Lat. ( ) fGrid correction
be._a_______ 105-,2410l Gi Man:zimuth to Mark
Sec. (8 Mgi azimuth to Mark
Cost i .Magnticdeclination E(-), W(+)
Ssin t
cos* tan a-sin *cos t
1 If LHA is greater than 1800, sub-A 4me W) X-2 asJf1Z ~ ~ - tract from 3600 and reverse sign.
Azimuth of 8 Obtaiin a from Ephemeris.
Check signs and quadrants by useL Mark to 8S AX v- of sketch.
Tr. Az. to Mark /4 a O _____________
COMPUTED BY DATE ss CHECKED BY &.CnDT1g4,& c.I. ID
DAI 7O 1905
O Natural functions, 1)A Form 1905
Figure 14. Computation of azimuth using the sun at any hour angle
SUNOBSERVATION
For star
use factors[enclosedin brackets
At
02
u-I
TrRUE NORTH
i
fJp ~
4:A* Mk
PR~OJECT
14w-2-3LOCATI ON
Vrai n/aAZIMUTH BY HOUR ANGLE METHOD
-(Logarithmic) (TM 5-337)
ORGAN IZATI ON LATI'U DE LONGITUDE STATION
AOS L9842'3 OA/ 7 0846#J5MARK
CELESTIAL BODY(S)
&n,DATE
A D .52
WATCH .) SLOW (+)
OBSERVER
C _ . I R~rnnvet
STANDARD TIME (Meridian)
750
WATCH COMPARED (Time)
WEATHER
C, e leSET NR 1 SET NR 2 SET NR 3
TIME HOlt ANGLE TIME HoR. ANGLE T~IME. OR. ANGLE
Has. MIN. SEC. 0 P M HRS. MIN. SEC, 0 f I HR.MI SEC. 0 f i
I SET NR I SETNR 2 SETNR 3
SUNOB3SERVATION
For star1ubseratios
[oservactonsenclosed in
brackets
S
1. -22°47' O'
2. -.22-47 07
3- 2247 10
1 Mean time of observation HiRS. MIN. SEC. Hal. I MIN. SEC. Has. MIN. SEC.
2 Watch correction 4 _ 1.~I j __1.
3Time Zone Correction (TZC) - - - -
4 UT of observation (1I+2+}3) W 4 .2 20- -5t 2L QL5 O h G r e e n w ic h E Q T [e -8 + dl - j f
,1 -8 L 2 4 & 1 4 .2 6 U T. X1
4.p r±or [ ip & H p h .
7 (5 +6) correct EQT (.~ f.Od .4----~ 44 152± l8 (4+7) GAT.~p* JQ£ 120 LQ9 G11A in time (GAT-12h)-4.wo*-43J .. j~f..* 2
10 GHA in arc 139 7-5~ ~ 22. 111 Longitude, West () l) f 77 dA 7 O2 a U
12, LHA (10+11)-t (1i.: .SS.3044i. 5714 [39 _I913 IRA______ SET NR 1 SET NR2 SET NR3 -
Log sin 4S I 76 Mean true azimuth to Mark /41 Q4. LQ.Log cos t Grid correction
(S u n ) lo g A G rid a z im u th to M a rk
5 , 2 4 5 4 9 8 1 AM g e i a z m t t o a r
Log cos. 0 ________ _______ Magnetic declination E(-), W(-l-)
Log tan E - 6232 963126UW
(Sum) log B - 9 5l725,571 Z § 07I 2,5t% 0/40-TnA -- slt
B-A7 Q270 o a 8sl o
- 6383B60 B- If LHA is greater than 1800, subtract
Log sinl t *from 3600 and reverse sign. 9 M1_2 .8(jObtain d from Ephemeris.
Log (B -W A) -= 9JZIJ 9 Check signs and quadrants by use of sketch.
L~o(- tan A) (diff.) A-83540,1.07%624 013129.iA .'= W) Loi59 a12 0-521 24 L& 20Azimuth of S 230 2&Q 0 2 22 0 COMPUTED BY, e DATE
CHECKD BYDATE
Tr. Az. to Mark
169 5710
(0 Log functions, DA Form 1906
Figure 14-Continued.
I NSTRU MENT (Number and type)
1 CSOlft'D l
/hn r
the meridian and the sun. If negative,subtract numerically. q is the paral-
lactic angle, which cancels in the
solution.
(2) The second is the standard azimuth
formula.
(3) The procedure is as follows:
(a) Compute each position or set sepa-
rately, using the means of the times of
pointings with the corresponding
means of the horizontal circle readings.
(b) Convert the local standard time of the
observation to GAT.
(c) Subtract 12h from GAT to obtain GHA,and convert GHA to units of arc.
(d) Subtract the longitude of the station
from GHA to obtain t, the local hour
angle of the sun. West longitudes areplus, east longitudes minus. If using
rules to disregard signs, subtract t from
3600 when it is over 1800.
(e) Take declination of sun from an ephem-eris for date and UT of observation.
(f) Apply formula.
(g) Subtract angle, mark to sun.
(4) All pointings on the sun refer to its
center. The method of pointing on the
sun should always be recorded by thefield party. The computer should in-
spect this record and apply any correc-
tions for semidiameter or other correc-
tions that may be required. Ordinarily,opposite tangents will be observed so
that a mean of the readings will refer tothe center.
Section III. DETERMINATION OF LATITUDE
17. Relation of Latitude to Zenith DistanceNearly all observations for latitude, other than
those using the astrolabe or zenith camera, consistof measuring the altitude of a celestial body whenit is on or near the meridian. The latitude of aplace may be defined as the altitude of the pole orthe declination of the zenith at the place. Eithercan be obtained from the meridian altitude orzenith distance of a body of known declination.The equation is:
where is the meridian zenith distance and 8 isthe body's declination. is positive when thebody is toward the equator from the zenith,negative when toward the pole. 0 and 6 are posi-tive north of the equator, and negative south of it.
18. Latitude by the Altitude of a Circumpolar
Star at Culmination
a. Formula. Latitude, by this method, is de-temined by use of the following formula:
4=h±p
Where: h=the corrected altitude of the star.p= the polar distance (900--6). p is
negative if the star is between thezenith and the pole but is positive ifbeyond the pole.
b. Procedure of Computation. The computationis as follows:
(1) Apply the correction for refraction to the
observed altitude.
(2) Extract, from the Apparent Places of the
Fundamental Stars, the declination of
the star for the date and time of the
observation.
(3) If the star was observed at upper culmi-nation, subtract the value of p from the
corrected altitude (90°-). If the star
was observed at lower culmination, thevalue of p must be added to the corrected
altitude (fig. 15).
(4) If the position of the star, during obser-
vation, was not recorded, the following
may help in determining the position:
(a) If the local sidereal time and the right
ascension of the star are equal, the
position is upper culmination.
(b) If the local sidereal time is equal to
RA+ 12h, the position is lower culmi-
nation.
19. Latitude From the Zenith Distance of a
Close Circumpolar Star at Any Hour Anglea. The Formula. Latitude, by this method, is
determined by use of the following formula:
O=h-p cos t+-p 2 sin2 t cot " sin 1"
Sp3 cos t sin2 t sin2 1"
+ p4 sin4 t cot 3 " sin 3 1".
Latidue Ay Altitude JPoo/arn: ato/rrnot ion
Local Dk le'/~oyS4 Eastern Sfondord ime ofoMar& 10:40PMries ohserved of lower culrrnoa OI'servedo/itudr of str 42°/7' 35
Temperature 45-F Barometric Pressure 30.4 inches
. Neron re,%ction. Tale V AppendiZT(use zenifi distance ofgA; as arumentJ 0- 0/' 04'
2.Correction to refracian for Toperahirsamew Ta/eosl) t 0/
3.Correction to refraction for Pressure, (some Tohk as 2) t OL
4 Corrected rerhiorn (if2,3) 0' 0/ 06
S. Observeda/flt/oe ofstar 42 17 35
6. Corrected alitue of star (-4) 42 /6 25
1 Declnatin ofstar (for ate) 8? 02 58
8 Poor Dislonce, p, of star (?o°-Deciroti 40 57 02
Q Latitude (6t8) 430 /3' 3/U
Figure 15. Computation of latitude by altitude of a circum-
polar star at culmination.
Where: h= corrected altitude
p=polar distance (in seconds of arc)
b. Procedure of Computation. The computation
(fig. 16) is as follows:
(1) Abstract, from the field records, the
zenith distance and the sidereal time of
observation. Correct the time for chro-
nometer error. Apply the refraction, level,and collimation corrections to the ob-
served zenith distance.
(2) Abstract, -from the Apparent Places of
the Fundamental Stars, the declination
and the right ascension of the star at
the time of observation. Determine the
local hour angle (t) by use of the formula
t=LST-RA of star and the polar
distance by use of the formula p=90°-.
Reduce the polar distance to seconds of
are.
(3) From the appropriate function tables,obtain sin t, cos t, sin 1" and cot .
(4) Apply the formula given above, the termsof which are the star's elevation aboveor below the pole. DA Form 2839,Latitude From Zenith Distance of Polaris,is designed for a logarithmic solution ofthis method, and can be used for otherclose circumpolar stars.
20. Latitude From the Meridian Zenith Dis-tance of Any Star
a. The Formula. Latitude is determined fromthe meridian zenith distance of any star by useof the formula:
Where: b= declination of star; positive if north ofequator, negative if south.
1=meridian zenith distance of star cor-rected for refraction. r is positive
when the star is on the opposite sideof the zenith from the pole; negativewhen the star is between the zenith
and the pole.
b. Procedure of Computation. The computation(fig. 17) is accomplished as follows:
(1) Determine the refraction correction andapply it to the observed zenith distanceto determine the corrected zenith dis-tance.
(2) Extract, from the Apparent Places of theFundamental Stars, the declination ofthe star for the date and time ofobservation.
(3) Should there be any confusion in alge-
braic signs, a diagram together with aroughly known value of the latitude willindicate the proper procedure. Thisconfusion might occur in the case of asubpolar star observed at a very highlatitude.
(4) Substitute known values in the formulaand solve for the latitude.
21. Latitude by Use of Table II, AE&NA
a. The Formula. The following formula isused in determining latitude by use of table II,AE&NA:
q ==h+ (ao-f-a,± al2).
Where: h=observed altitude, corrected for re-fraction.
ao, a, a2=factors in table II, AE&NA.
b. Procedure of Computation-. The computation(fig. 18) is as follows:
(1) Extract, from the field records, the meanobserved altitude and the date and timeof observation. Reduce the time to
local sidereal time if necessary.
(2) Determine the correction for refractionand apply it to the observed altitude todetermine the corrected altitude.
(3) Interpolate in table II, AE&NA, for thevalues of a0, at, and a2.
(4) Substitute known values in the formulaand solve for the latitude.
PROJECT I LATITUDE FROM ZENITH DISTANCE OF POLARIS(211 5-237)
OBSERVER UNIVERSAL DATE STATION
1.S.12.1e8 Jul '63 NORTHASST. OBSERVER RECORDER TEMP. BARD.
FE. W I R.A. G. 314 OF 2980 "IN ST. LEVEL VALUE IdREFRACTION FACTOR POSITION NO.
WILD T-4 No. 3744( L d/2 = 04977 /.0239 /
h In a G "
CHRON. TIME 17 32 /7. 9 to .28 22 /6.95CHRON.-CORR. 0. LEVEL CORR. +g 00. 98LOCAL SID. TIME /7 32 /7. 9 REFRACTION CORR + 32. 01R.A. - 01 S57 SP2/ COLLIMATION + .. 70HOUR ANGLE (t) iS 34 /8.8 r-28 23 01. 64
t 2330° 34' 42.0o"900 001 00 00'"
LOGp 3.16 4957 a - 89 o5 1530LOG COS t 9. 773 584 0 - /0 p ° 5 ' 44.70 "
LOG 1 3 290 0797 ______ 3284.70
LOG 0.5 9. 698 9700 - 10 LOG 1/3 9. 522 88 - 10
2 LOG p 7 032 9914 3 LOG p /0.549 492__LOGSINt 9.8112348 - 0 LOG COS t 9 7735s8 -10
LOG COTr 0.26733)0 2 LOG SIN t 9,811 23-1/0
LOG SIN 1 '4. 6855749 -10 2LOSI1"93715-0
LOG II 1. 46 110/ LOG III 9 o28 33 - /0
_____ _ + /950 .20 ~ LOG 1/8 9. 0%691 -10
I3.34 4 LOG p 14.665 98111 +. 00 / 4 LOG SIN t 624
IV + 00. 00 " SLOG COT o 02osum + 1 98/ .6(5 S LOGOSIN i' 4 52-2
sum 0 330. . LOG IV 7644/0 -10h(90- r)61 34 58.36 ______________
#A_____ 62 /0__00.0/ ____ ___________
A= h - I + II - III + IV
SIGNS: I and III. are PLUS in the first and fourth quadrants.
ii and IV are always PLUS.
- S OTE 6 CHECKED BY - MS Juy 6COMPU~~~
AM Y'63
A M JC Dju '(
DA FORM 2839, 1 OCT 64
Figure 16. Computation of latitude by altitude of a circumpolar star at any hour angle, DA Form 2839.
LoafA~e from a feridi/n Zenith Dt's/nce COMPUTATION OF LATITUDE BY ALTITUDE
OF POLARIS (TABLE 11) AEA NA)
Stfo/n : I-of,'
Ce/estial Body: aune (GCope/o)
Date : 20 Feb 55
Lon9,itde: 75*30'W
Terre Zone leridian;
Local Standard Time
Time Zone Correction
Universal 7me
A /ttde
fWraction
7S W
/8" /6"S /0S
236 As' le
440 /6' 23"
4 1 59
44 15 2
Zenith Dstance (,) - 5a 441 36il
Declination hi) 45 57 27
£cditude 0f) /2' 5/"
Figure 17. Computation of latitude from the meridian
zenith distance of a star.
22. Latitude by Circummeridian Altitudes ofa Star
This is a more accurate method for observing
latitude by means of the instruments ordinarily
available. It is an extension of the method of
meridian distance and the accuracy is increased
by taking a greater number of pointings. A series
of altitudes or zenith distances is observed on astar at recorded times during a period of a few
minutes before and after transit. From the hour
angle of the star at the time of each observation,
the observed altitudes are reduced to the equiva-
lent meridian altitude. The observing period
should not exceed from 10 minutes before to 10
minutes after transit. The procedure is as fol-
lows:
a. Reduce each observed altitude (or zenith dis-
tance) to the meridian (fig. 19).
(1) Compute the LST of the pointing by
correcting the recorded time for chro-
nometer error. The numerical difference
DATE: /1 APRIL /63
LOCAL SIDEREAL TIME:
ALTITUDE OF POLARIS:
ALTITUDE (h)
TABLE Hf AE NA
+t
az =
LATITUDE OF STATION
09h 596 /95
380 28 59
380 28 590
28 //.9
06.9
00.0
38 57 04.0
Figure 18. Computation of latitude by altitude of Polaris
(table II, AE&NA).
between the LST and the star's right
ascension is the hour angle (t).
4)s cos e5
(2) Let A= cos , 'in which 0 is a closelysin -
scaled map latitude, or is a trial value
computed from an altitude taken near
the meridian. should be the value nearest
2 sin2 (t)the meridian. Also, let m s
' ~ sin (t)
in which t is in seconds of arc (table IX.).
(3) Then hmrh+Am or m,= -Am; h orbeing the observed values, corrected for
refraction, for the pointing being re-
duced.
b. After all observations are reduced, take the
mean of all consistent values.
c. Apply formula: 0= +b.
d. If for any reason it is necessary to reduce
observations taken more than 10 minutes from
the meridian, the formula,
sin hm=sin h+ cos 0 cos 2 sin2 (t)should be used.
e. DA Form 2840, Field Computation of Lati-
tude-Circummeridian Zenith Distances, is de-
signed for a logarithmic solution of this method.
23. Precise Latitude by Circummeridian Alti-tudes of Stars
a. Circummeridian altitudes can be used for
the determination of precise latitude at latitudes
FIELD COMPUTATION OF LAIUE- CIRCUMMERIDIAN ZENITH DISTANCES(TM 5-237)
PROJECT ASR _jSTATION
LOCATIOND ASRORGNZT /SQ//GUA (CC.LOAIO RGNZAINDATE IELEVATION
0/SKO /S. ---- US'4MS 121 AU6.6WPMINSTMMPN TCHRONOMETER TEM AROM.
WILD -:4 No/0_37446 2E /200/ - o./ c 26.81 _OBSERVER I ASST. OBSER VER RECORDER
R. SLVERMO SER _ _I -J.M.LEW/SIG(0EFAPPROX. POSITION .,h ~ SECCENTRICITY
:N 6r°46"55.5 a: f 03"28 /0.0 ODIRECTION FROM. STAR NO. feNi.0!
TO: 17______ (SOUth)
POSITION NO. 1/ 2 13 /4
CHRON. RAN_ 00 35 .06.7 00 35 42.2 100 36 ./.5 oo 36. 43.7,
CHRON. CORRECTION - /9 ~ 19 +o. -0.
SIDEREAL TIME OQ 3S 08.6 00 35 44.1 00 36 /3.4 00 36 456
R. A. OF STAR 00 341.0 00 34 51.0 00 34 51. 0 00 34 S1.0o
HOR NLE() 7.6 S3.1 0/ 22.4 o 0 54.6
M I 0.17 1.5S4 3.7/ 7/6LOG. COS O~A 95862_
LOG. Coss 9.7 72 5056 -
-LOG. SIN JTI 9.442 9473
SUM LOG. A[98821
A_ _ _ _ _ _ 0..73811
ZENITH DITNC /6 05 _42. 0 / 6 05-42.9 /6 0S 44.0 1 /6 05O 46.9
RE FRA CT ION .,L/5.5 , f15.6. t /S."5 +/5.5
CORRECTED~ 0' I 0 o I' , "" O----- 16 oS 575 /6 05S58.4 /6 05 S595 _._6 0 6 02.4
A MI -00. - 01.1 02.7 0 _5.3__ /6 05 574k /6_ S 05673 /6 05 1 ~t~05
DECLINATION 8 o ft' "t
5-- 3 -40 -S92 6 3 40. S5?2S ¢ S392- 53.:4o05.2LAIUD"
C' 6.9 46o 66t94 ,-.5 6?9 46 S6.0 69 4656.3REMARKS
COMPUTED BY DAECHECKED BY JDATE
DA FORM 2840, 1 OCT 64
Figure 19. Computation of latitude from circummeridian altitudes of a star, DA1 Form ;!840.
between 600 and 900, where other precise methods
are impractical. Observations should be made
with a broken telescope theodolite.
(1) This method requires a number of zenith
distances observations at recorded times
of selected fundamental catalogue(APFS) stars, during the period extend-
ing from 5 minutes before to 5 minutes
after transit. The stars are observed
in pairs one north and one south of the
observer with the zenith distance of
each being less than 300 to minimize ir-
regular refraction conditions. For the
pair, the difference in their zenith dis-
tances should not exceed 30 and pref-
erably should be less than 10. Ten to
twelve pairs are required and the time
difference between observing the stars
of any one pair should not exceed 1
hour. If a close circumpolar star, at any
hour angle, is used as the north star of a
pair, see paragraph 19 for computation.
(2) When a pair of stars are close to the
zenith (i.e. their azimuth factors (A) are
less than 0.15 as determined from the
formula, A=sin see 6), they will move
fast with respect to azimuth and the
instrument should be clamped in the
meridian. The star is bisected in rapid
succession with the horizontal crosshair,
noting in the recording where the star
crosses the vertical hair. A meridian cor-
rection can be computed and applied to
the hour angle before reducing the
zenith distance to the meridian if the
telescope was not properly alined.
b. The procedures followed in the computation
are:
(1) Abstract the field data from the field
books (figs. 20 and 21).
(a) Chronometer times and date of ob-
served zenith distances.
(b) Observed zenith distance corrected for
the bubble correction. This inclina-
tion correction is computed by the
formula zenith distance= observedd
zenith distance+~ (L-R) where d is
the vertical circle level value in seconds
of are and (L-R) is the difference
between the left and right level
readings.
(c) The radio time signals for computingthe chronometer correction.
(d) The temperature and barometric pres-sure readings.
(2) When the computations are to be com-pleted using normal observations.
(a) The rigorous equation for the reductionof the observed circummeridian zenithdistances of a star to the meridian is:
(cos 4 cos ) (2 sin2 t
' sin _\ sin 1 '
±(cos 4 cos b)2 (2 cot sin4 it)
sin sin 1"
Substituting:
2 sin2 tsin 1"
cos 4 cos S
sin j
2 sin 4 Itsin 1"
B=A2 cot (1
We have: rl=r-Am+-Bn, and
1I=-f+Am+Bn, for subpolarstars.
In the above equation:
'= Observed zenith distance cor-
rected for inclination, refraction,and for index error of the vertical
circle (collimation and zenith
point error).
'1= Zenith distance of star on the
meridian.
4= astronomic latitude of the station.
t= hour angle of the star at the in-
stant of observation.
S= declination of the star at transit.
In the above equation, the third term was
neglected. This third term is:
+4/3 (1+3 cot2 1) Asin 1t
sin 1" 2
(b) After finding the mean zenith distance
of the two stars of the pair, the latitude
for the pair can be computed as follows.
s+0n= the latitude for the pair,2
where
'~,=-6+- i, for stars south of the
zenith, and
~,= -- n= , for stars north of the
zenith.
STATION]S//V ft' ECC DATE~./A9dU I9 6/
POS. CHRON TIME STOP TEL CIRCLE
RE. OBJECT OBSERVED H' N S WATCH D / R
___ Souz -y, 06
3 6.'ee p/ ' __3 jn4- - _
3/ / 6.
3s2~ -6 __
3-6 42(7 -5
22s
/ -2. Os-
RECORDER 67 O4ffee 'e WEATHER 7G'4 1V / '
MICRO. / VERN. MEN ~iDR/N EVL
IST/A(') 20B' /' REMARKS
# Zz Y19.610 44, */
o~ ' /, o~ __ _ _____
,f 17/ 107./
Figure 20. Field observations-Star 17.
STATJON -S/ G E CC. -. DATE-'-' ' 96/I
-POS. I -I-IRON TIME * TL CIRCLE
OBJECT OBSERVED H F WATCH DI/R -____I_ - -- T S
-V /V ,/er/7' 55 -- > -
71,c W4 1= __ __z
_ - V__
Ii STRUNTAN !P 0_ NSRUhEi7T T' 37A
RECORDER G Ofr E___ WEATHER,-: tTG.~I i/Z
MICR0. /VERN - -- 7 EVELSl M MEAN MEAN DIR /JAN REMARKS
IST/A(' "T)/B! D/R , ___ W E
7Jse "p7 _, 1-
.3.5 / S_;,5 _ - - y~a -3 t,~ ( -
Figure 21. Field observations-Star Na.
(c) Southern latitude and declinations areto be considered negative, the sign ofthe cosine being plus (+), and the
sign of the tangent, cotangent, andsine being minus (-).
(d) The factors m and n are functions of
the hour angle t. Values of m may befound in table IX, appendix III.
Values of n are tabulated in table X,appendix III.
(e) Constants A and B are computed
for each star. Since closely ap-proximated values of 0 and 1 are
required, it will be necessary to re-
compute the meridian reductions if the
values used vary by more than 5"from the true value.
(f) The approximate values for 4 and (1may be determined as follows:
Select the observation nearest to
the transit (minimum zenith distance)
for the north and south star of a pair.Correct these two zenith distances for
refraction and level error. Compute
the latitude for the north and the southstar. The difference between the tworesults is equal to the double collima-
tion error (index error) of the verticalcircle. The mean of the two results isequal to the approximate latitude. If
the computed latitude for the north
star is higher than the one for the south
star, then half of the double collimation
error will be added to the zenithdistance of the north star and will besubtracted from the zenith distance
of the south star to obtain the approxi-
mate corrected meridian zenith
distances.
(g) Having found the arithmetical means
of the zenith distances of the two
stars comprising one pair, we have the
latitude:
n-=8-(- 1, for the north star, and
-,=-8 +[, for the south star
Then the latitude derived from thepair is:
2 ,2
(h) The arithmetic mean of the latituderesults of all acceptable pairs deter-mines the observed astronomic lati-tude of the station.
(i) No further adjustment is necessary ifthe hour angles are distributed sym-metrically before and after transit; ifthe observer has bisected the staralways at the center of the cross wires;and if the recorded time in realitycorresponds with the exact local side-real time of the bisections. An ap-proximate method of adjustment is
employed in order to determine theerror in t; to correct all individualmeridian distances; and to obtain amore accurate final zenith distance ofthe star.
(3) The computations of observations, whenthe instrument is clamped in the merid-ian, are as follows-
(a) The reduction of the observations tothe meridian are made by the followingformula:
, sint tm sin Xsin 26, where
sin 1"
t=the true hour angle of the star atobservation.
m'= the correction to reduce the meas-ured zenith distance to what itwould have been if observed attransit.
Table IX may be utilized for thiscomputation. Using t as the argu-ment, the m values from the tables
sin 28must be multiplied by 2 'which is
a constant for all observations of thestar.
(b) Since the horizontal center wire isseldom exactly perpendicular to theobserver's meridian, it becomes neces-sary to determine the inclination of thecross wire in order to reduce all ob-seivations as if observed along aperfectly adjusted cross wire, that is,in a plane perpendicular to the ob-
server's. The adjustment of the re-duced zenith distances due to inclinationof the cross wire is shown in figure 22.(1) Explanation of the terms:
PROJCT HYPO HE TCA ITABULATION OF GEODETIC DATA(TM 5-237)
LOCATION o a.' ORGANIZATION
Oc0
=SSTATIONtZDo VO Itt [~O] Z I V
0500o - 300 35.05 1453 ____ -/5-00 26.06 -. 6204 /2 -252 2.6o -12.08J -/2.60 20.00 +-.o303 25 - 205 o.2 - 9.76 ______-10.25 20.0o3 .0o
02 /01-/30 26.90 -6.38 _ ____-6.50 20.40 - .37
0/ 30 - 90 24-$g -3.78 _ ____-4.5-0 19.80+.23
00 /0 - /0 20.09 +o.43 _ ____-o-50 /.959 +.44
00.53 + 53 /7.65 +2.87 +____ 2.5 20.30 -..27o/ 47 +107 /4.70 4.5S82 ¢ #535 20.05S -. 02
025S/ , 17/ 11.41 +9.11 +" +8.55 ,5'9 91. #07
04 04 + 244 07.92 + 12.6a ~ + 12.20 20.12 -. 0901503 # 303 04.85 +IS.67 +1/515 20. oo +.03
y, CN
-109 20.523 -03 +' + 20.027 +.03
__ _ _ _ /lI, -10.9 -0.03
___________4-9.90909 +0.00237
______________+ 415 473.0000 - 20 724.3300
________- 1____ -080.0908 - .2 976
+414,392,9092 - 20 724.6276
__________~~~~~ __ _ _ _ _0 0.0500/202 -_ _ _ _ _
AZD + .00237
_________- .49557 -0 4) + .909)
ZD 20:S230 ____
TABULATED BY DATE / 2 CHECKED BY DATE
D FORM 16DAI ES 5716GPO 92196f U. S. GOVERNMENT PRINTING OFFICE : 1957 0 - 421182
Figure 22. Adjustment of zenith distance for inclination of cross wire.
t=the hour angle of the star
ZDo= the observed zenith distance
corrected for curvature and
refraction.
v=the residual from the mean
AZD= correction applied to ZDo
a=inclination of the cross wire, or
AZD per unit of t.
The normal equations are:
n(AZD) - [t]a+ [v] = 0
- [t] (AZD) + [tt]a- [tv]= O
(2) After solving for a and AZD the
individual AZD's may be com-
puted as follows:
at=AZD; then the final ZD is
ZD=ZDo+AZD
(c) There remains another source of error
of the observed zenith distance if
the line of sight (collimation axis of the
telescope) does not lie in the plane of
the meridian. The equation for the
correction is:
A01 = -Am, where
cos 0 cos a 2 sin2 't
sin -1 sin 1"
A- ==the correction to be applied tothe reduced mean zenith distance
of the star, and
t= the meridian error of the instru-
ment expressed as an hour angle(local sidereal time at transitminus right ascension of the star).The correction is applied so thatit will numerically decrease thezenith distance.
c. Following is a completed example of the fieldnotes and computations for one set of stars using
this method. This pair of stars was observed at
station Test on 21 August 1961, P.M. date. The
longitude of the station is 03h 28" 10.08 W. The
chronometer correction for both stars is -1.9".
The constant for the vertical circle level is 2.845"
per div. The stars observed are APFS stars 17 and Na(1) The Greenwich Civil Day and the short
period terms are computed for each star
as follows:
Local sidereal Time
Longitude
Sid. Time at Oh
Universal Time
Greenw. Civil Day
Greenw. Civil Day
Star 17 Star N.
00h
34m 518 01h
03m 258
03 28 10 03 28 10
- 22 00 14 -22 00 14
06 02 47 06 31 21
(22 Aug 61) (22 Aug 61)
362.7833m 391.35
1444 1444
22.251 Aug 61
d¢= -0.042
dE= 0.077
22.271 Aug 61
not applicable
(2) The right Ascension and Declination for
each star are then completed.
Star 17
Longitude 0. 145 day west of Greenwich
n 0. 8145
0. 8145 (-0. 1855)B" -- 0.03784
Star Na
n 0.145
RA for 14 Aug UC OOh 34m 50'. 745 RA for 22 Aug UC 01h
03m 25. 328
0. 8145 (+0. 294) - . 2395 0. 145 (0. 24) -0. 03
-0. 0378 (-0. 099) + .0036
(0. 067) (-0. 042) + (-0. 090) (0. 077) + .0048
Right ascension 00 34 50. 99 01 03 25. 35
Note. The FK4-FK3 corrections are to be applied for 1962 and 1963 only.
Declination 14 Aug UC 530 40' 56. 85" Declination 22 860 02' 49. 47"
Aug UC
(0. 8145) (20. 95) + 2. 403 (0. 145) (0. 33) + 0. 048
-0. 0378 (0. 29) - .011
(0. 39) (-0. 042)+(0. 15) (-0.077) - .028
Declination 530 40' 59. 21" 860 02' 49. 52"
Note. The FK4-FK3 corrections are to be applied for 1962 and 1963 only.
(3) Approximate Meridian Zenith Distancesand preliminary Latitude are then de-termined as follows:Star 17 (South star) Star Na (North star)
Observed Zenith Distance
(position 11) 16005'42/'
(pos. 9) 16015'40"
Refraction - + 16" + 16"
Corrected Zenith + 16005'58" - 16°15'56"Distance
Declination
Latitude (g)
Mean Latitude
(preliminary)
+53°40'59"
+-69046'57"
f86O02'50'
+ 69°46'54"69046'55"'5
Zenith Distance (p') of star 17: 16°05'58"-1.5"= 16°05'56.5"'
Zenith Distance ('1) of star Na: 16015'56"-1.5" = 16015"54.5/ '
(4) For the computation of the CorrectedLocal Sidereal Time, refraction, andcorrected observed zenith distance (figs.23 and 24), Roman numerals wereassigned to the columns for clarity in theexplana.tion.
(a) Column I. Number of position.(b) Column II. The corrected local side-
real time is equal to the observed(chronometer) time plus or minus thechronometer correction.
(c) Column III. The observed zenith dis-tances as taken from the field records.
(d) Column IV. Level correction.
=- R) (2.845").
(e) Column V. The refraction correctionis always plus, increasing the zenithdistances numerically. The formulafor the correction is: (rm) (CB) (CT)(table V).
(f) Column VI. The corrected observedzenith distance is equal to the observedzenith distance+level correction+re-fraction correction.
(5) In the computation of the final zenithdistances (figs. 25 and 26), the Romannumerals are continued.
(a) Column VII. The hour angle t is equalto the corrected local sidereal time ofthe observation minus the right ascen-sion of the star (II-R.A.).
(b) Column VIII. Factor m to be foundin table IX.
(c) Column IX. Factor n, to be found intable X.
cos € cos 6(d) Column X. Am, where A=
sin 1
(e) Column XI. Bn, where B=A2 cot f'.(f) Column XII. ,, the meridian zenith
distance (VI+X+XI).
(6) The adjustment of the reduced zenithdistance is based on the assumption thatthere is a constant error in the correctedhour angles at the instant of each bi-section. There are a number of sourcesproducing this error, some of which arelisted below:
(a) The star is not observed at the inter-section of the cross wires.
(b) Delay in time from the instant of bi-section to the instant of reading thechronometer.
(c) Personal error of the recorder in read-
ing the chronometer.(d) Horizontal collimation error of the
instrument.
(7) The adjustment of the zenith distances(values listed in column XII of thesample computation) is carried out bysolving two simultaneous equations:
Equation I: n (ZD)+[t] a--[j]=0
Equation II: n (ZD)+[t] a--[ l]=0
(a) In equation I, n denotes the numberof observations before transit; ZD isthe final adjusted mean zenith distance
of the star reduced to the meridian;[t] is the algebraic sum of the hour
angles before transit (in seconds oftime); a is the correction for the zenithdistance per second of the hour anglet; and [1] is the sum of the zenithdistanc'es before transit.
(b) In equation II, the explanations for
equation I holds, except that "aftertransit" should be substituted for
"before transit."
(c) After the unknown a is computed,column XIII may be computed bythe formula: XIII=(a) (t), in which
t is to be taken from Column VII (inseconds of time). Apply Column XIII
to Column XII to obtain the adjustedzenith distances.
STAR G.C.D. =22.25/ AUG. /96/
#/7(SouA) BARO. =:26.8/ TEMP: _-0./C
S. 3 40" S92/1 Cos 0 0.34S 59/7 Chrotn. Corr : +1-S9RA o0/734" 509 Cog 0.592 2507 C =CB 4 0.928If, /6< 05" 5"65 51I , 0.277 2984 ___________
0 69 ° 46' 555 S COT f, 3464 8020
Po_____Fs. Corr Loc. S31W. 0,6Os. Z.D. Level Refr. Corr Obs. Z.D.
/ 0 29 24.8 /6 06 23.55 - .28 + /5 53 /6 06 38.80
_____2 30 01J.7 14.45 -14 f 1-S2298
_____3 30 3559 06.35 - .14 + 1/.62 2/-73
4 3/ /1.5 00.20 .00 4- /552 /5.72
_____ 31 48.4 /6, o5 54.55 ,oo t- /552 /0.07
6 .32 23.7 49.6o 00 4- /552 0S.12
7 .32 53.5 46.10 oo 1- /552 01.62
_____8 33 21.0 43.50 00o f /S651 16 05 59.0/
_____9 33 53.o 42.00 *oo + /5.5/ S________
/0 34 24.7 42.05 +.28 *f /6.5/ 5784
1/ 3S 08.6 41.90 +. l.4 + 15.51 57.55/2 35 44.1 42.80 +14 4 15.51 58.45
1.3 36 13.4 43.90 ,.4 -5/59.55
14 36 456 46.80 4-. 14 +-.1/552 /6 06 o2.46
1S 37 /,54 4 9.40 -. 14 + /552 04.78
_____ 6 37 48.8 53. 95 -. /4 + 16.52 09.33/7 289 24.9 .59. 75 -14 t- 15.52 i.S. 1.3
18 .39 04.9 /6 06 07/0 1. 4 f /552 22.481___ 9 39 38.8 /4.85 .o0o+/.52. 30..37
20 00 40 /4.7 23.20 .00 + 1S.53 38.73
TABULATED BY DATE/2 CHECKED BY RQ DATEjl
DA 1 FW 571962 GPO 921961 U. S. GOVERNMENT PRITIG OFFICE : 1957 0 - 421162
Figure 23. Correction to observed zenith distance-Star 17.
GREELAN A rR TABULATION OF GEODETIC DATA(TM 5-237)
bSiA4et ORGAN IZATI ONSta.- /SL/NG(UA Ec c. USA M11S , 0,hs. R9. S.
S TAR 6. C.t D:-22.2 7/ AUG./1961Nq~ (orth) BARO.2 ~8 /g TEP +02 C
6 :802'4952 COS 0 0 . .6 5917 Chron-. Corrc+15
RA 0/ h03 '5. 3 5 COS (6 = 0. 068 9366~ C =CB CT 0.92716*15'S4'5 S/Ny 0i=. 280 0827
05 = 46'5.. COT f, 3. 427 4733
____ II .ZifF1 "
____Pos. Cor. Loc. Sicl T Obs. Z.D. Level Refr Corr. Obs. Z.D.
____ / 00 .57 32.3 16 15 45.75 0.00 + 15.68 /6 /6 01.43
____ 2 58 /2.8 44.10 +o.43 + 1668 00.2/
3 68 53.5 42.40 +0.43 + /5.68 /5 58.514 59 35.6 40.95 4#0.85 f /568 57.485 o/ 00 /6.8 41.00 +0.85 + 15-68 57563
6 0/ 01.3 39.00 i-.42 + /5.68 6 .10
7 0/ 50./ 39.35 # 1.42 +'-15.68 56.45
8 02 28.2 38.85 ,j56 +#1568 66.099 03 25.3 38.80 t 1.85 +15-68 6'6.33
/0 04 04.3 38.65 + 1.99 4- /6568 56.32
1104 4.5.9 38.50 +2.28 +-15.68 66.46
/2 0 24.9 39.36 +1.56 +-16568 5-6.59
13 06 03.2 39.30 + /.56 - 1668 66.5S4
14 07 /1.7 42.80 -1.28 t 15.68 6720
1___ 5 07 54.3 44.50 -1/.0 oo -/5.6o8 59.1/8
/6 08 22.7 45.00 -1.00 -15.68 59.68
1___ 7 09 /1.6 45 90 -0.43 +/568 /6 16 01./6
/8
TABULATED BY
0/ 09 ..f6.o 48.00 -0.281+ /568
D ,FORM 16®AEI FEGPO 921961 U. S. GOVERNMENT PRINTING OFFICE : 1957 - 421182
Figure 24. Correction to observed zenith distance-Star Na.
757-381 0 - 65 - 4
03.4048.00
PROJECT GARENLAND SIR TABULATION OF GEODETIC DATAGREELAND,,~Jw I(TM 5-237)
60GAT O, TORGANIZATION
Sta. /SL'AGUA Ecc. USAMS
sTTe'COS60COS S~ 2STAR 1/7 A= si 0.73811 S= AcotL 1.88765
AdSj.Fs. t M A m 8r f7 . atv
/ -. 5- 2652 58.03 .0082 -42.84 + .02 55.98 4-58 56.56 .00
2 4 49.3 45.65 .005/ -33.69 ,,.0/ 56.15 f-. 5/ 56.66 - . 10
.3 4 151 35.49 .00.3/ - 26.20 +.0/ 5554, # 45 55'99 +.5S7
4 3 395 26.28 .00/7 -/1940 .00 .56..32 4-.39 56.71 -. 1
51 3 '026 /8./P - -13.43 - 56.64 f.32 S6. 96 - .q0
6 2 273 /18.4 - 8.74 56.38 + .26 56.64 -. 087 / 575 753 ___- 5.56 S6. o6 #-. 2 / 56.27 t .29
8 / 30.0 442 -3.26 65.75 + /6 655.9/ *.66
9 0 6'8.0 1.83 ___- 135 56.16 f . /0 56.26 -30
'o0 0 26.3 0.36 __ -0.28 ___57.66 -OS 5676/ -1.05-
/4- i 0 176 o.17 -0.13 _ __5742 - .03 57.39 - .83
/2 0 S3.1 1.5S4 - 1.14 ____573/ - .09 6722 - .66
/3 / 22.4 3.7/ - 2.74 56.81 - .165 56.66 - .10
/4 / 6S4.6 7/6 6__ -28 57. 18 - .20 56.98 - .42
,'S 2 24.4 /137 - 8.3 9 _ __56.39 -. 26 S6.13 +1.43
/6, 2 678 /7.24 -12.73 56.60 - .32 .56.28 7-*.28
17 -3 33.9 24.96 -/8.42 - 56.7/ -. 38 6.33 -'. 23
18 4 /3.9 23'16 .0030 - 25.95 .oi/ 56.54 - .45 56.0,9 f .47
/'9 4 47.8 4518 00oS0 -33.35 .0/ 5703 - .5/ 56.52 ,-.o4
201 S 23.7 57/14 .00 79 -42-18 *"0/ 66.56 -. 58 55S.98 '. s8
56.554 __ 56.558 + #05
A= +.0241 /0ZD - 1711.8a - 562.54 = O
t= +2.~2 /OZ0 -/669.2 a. -%68. 55 0
___ ____-3381.0 a+ 6.0/ :0 _ _ __ _ _
IPf9 . .t32 a__ -- o. 00j,77
/P . . = f . 6 /OZD: 568.5 - 2.?67 =56$5 3 Z'D =56. 6-
TABUATE BYCorr 4For error in~ It, .002TAUAE YDATE CHECKED BY DATE
V/62 _______/______
DA FEe a71 962 GPO 921961 U. S. GOVERNMENT- PRINTING OFFICE : 1957 0 - 421182
figure 2?5. Computation of final zenith distance-Star~ 17.
STi? 4 'Na A=C 0osC_ 0.08506 3 = As cot f, 0.02480Adj .
Pos tfm l A m Bn f, a , v
/ -5 53.1 68.00 - 5.78 55.65' -.0 9 5574 -. 05
2 5 /2.6 53.29 - 4.53 ___5561 8 .02 5. 76 -. 07
3 4 31.9 40.32 -3.43 55S08 f'. o7 55.15 t . 54
4 3 49~8 28.80 -2.45 5 03 +.06 55.09 9- 60
s 3 08.6 /2940 -__ -/66 5588 t . 5 55.93 -. 24
6 2 24.1 /133 -o. 96 5. 4 .0o4 55/8 t. 51
7 / 353 4.95 _ _-0.42 _ _56.03 + .02 56o.05 -36
8 0 572 1.78 ___-0.15 55.94 +-0/ 55.95 -. 26
9 0 00.! 0.00 ___0.00 ___56.33 .00 56.33 -. 64./0 f0 38.9 0.83 -_ -o.0o7 56.25 - .0/ ,5-6.24 -55
(I / 2o.5 3.54 ___- 0.30 'i. 56. i6 - .o2 56.14 -. 45
/2 / 59.5 778 -_ -0.6 6 C 5.93 -. 03 55.90 -. 2/
/3 2 378 /3.S8 ___ -//6538 - .04 55.34 +.35S
/4 3 46.3 27.3 ____ -2.38 54.82 - .06 5-4.76 -. 93
15 4 28.9 3943 ___-3.35 2 55.83 -. 07 576 -. 07
/6 S 04.3 50.5 So _ -4.3o 56 38 -. 08 55 30 +- .:39
/7 S 46.2 6o537 ___-656 S_ .59 -. 09 555 so />
/8 6 36.6 83.2/1 __ -. 7.08 __ 56.32 - .1/0 56.22 -. 53
_____ __55.69 5568 7-.0
92D /-6352. 7at - _5_0.76 =0___ _ __
92D 1933.Oc'.- 5/.-66 = _______
__ _ _ _-358 6.7 a +1 .90 0 ____02
5 0.0 251
f= :2 S7 RZD = + 00.? 76Z 55.686 __
Olt *
P~s /. =± .3/ + -4/5 ____ 4-.00H'
. 07 __ _ 5 50/ /7 _ __ 17S____ _ _ _
Rej. L. = /.'Q __ __ * Cor. 4ro r e rro in "it
TABULATED BY pDATE CHECKED BY Rq&DATE'/62 86,VOIK COM.PU/TER 7 /62
DA 1 FesR11 96V2 GPO 921961 U. S. GOVERNMENT PRINTING OFFICE : 1957 0 - 421182
Figure 26. Computation of final zenith distance-Star Na.
(d) ZD, the mean of the final adjusted
zenith distances, may be computed
by substituting a in one of the two
equations.(e) In the event that observations before
and after transit are not balanced in
number of observations, it is advis-
able to employ a least squares adjust-
ment of the reduced zenith distances
(values in column XII of the sample
computations). Figure 27 shows the
least squares adjustment of star 17.
The two normal equations are:
n(as)-[t]a+[v]=O
-[t(A 1) +[tt]a-[tv]=O
(8) In the above adjustment, the zenith dis-
tances will be reduced to a meridian
which is, by the amount of dt, off the
astronomical meridian. Normally, not
a large error in t is to be expected if allprecautions in the determination of the
local sidereal time of the bisections were
taken. The correction to be applied to
the mean of the adjusted meridian
zenith distance of the star is:
Correction-A sin2 IdtCorrectionA sin 1 Am, where dt is the
average error in the hour angles.(a) The average error in the hour angles
(dt) may be found in the followingmanner:
Differentiating the formulal'1- =-Am+Bn, and neglecting the secondterm, we can say that
sins 2
tsin2 t
d (A--) ,, sin t". dt(radians) =sin 1"
A • sin t".dt"
(b) Assuming that dt (error in hour angles)
and t (hour angle) are both in error by1 second of time, then:
d (- 1-1-)=A (0.000,072,722) (15)=0.00109"A, which means that thechange in the zenith distance due toan error of 1 second in t is equal to0.00109" (A) . (dt).
This relation holds true since thesine and arc of t are linear, when t issmall.
(c) From this, it follows that the average
error in t may be computed by the
formula: dt 0.0019(A)0.00109(A)
a is
plus, then the chronometer time of thebisections is less than the correct local
sidereal time, and the final adjusted
mean zenith distance will be lower
(numerically) than the zenith dis-
tances from column XII of the sample
computation. If a is minus, the oppo-site would apply.
(9) The latitude from one pair of star (fig. 28)
is equal to 2, where s = -+1-2
for the star south of the zenith and 4=
-i, for the star north of the zenith.
The mean of the summation of the
results of all pairs will give the latitude
of the station. Eccentric reduction, re-
duction to sea level, and the correction
for the variation of the pole must be
applied (fig. 29).
(10) The probable errors (PE) are computed
as follows:
(a) PE of a single pointing on a star:
eo= ±0.6745 (n-v)(n-1i
(b) PE of the arithmetic mean of
duced zenith distances:
the re-
ro= e
(c) PE of a single latitude pair:
e=±0.6745V ([v 1
(d) PE of the arithmetic mean of all
pairs (latitude result);
r=n
In the above equations,v=the residual, and n=number of
observations.
(e) It should be understood that the
probable errors thus derived represent
the probable errors of the observations
only, and do not include constant
LEFIST SQU/ARES ADJUSTMENT
PROJECT I TABULATION OF GEODETIC DATAGREENLAND A S7RO I(TM 5-237)
LOGAT4ON,1 ORGAN IZATION
Sta. /SUNGUA Ecc. U'SA lviS
STAR~ /7
fl 6 V [te[ W11 a6 Ad~j.j V2
/6x05" /6° 05,
/ - 3216.2 568 0.3S7 f-o. 45 56.43 7-o 1
2 - 2893 S6. /S 0.40 4-0.40. S6.55 # 0.0/3- 255/1 55*54 /10/ V-0.35 5 89 *067
4 - 2/95 56.32 4 0.23 -A0.30 56.62 -. 065 - /82.6 56.64 -0O.09 _ *0.25 56.9 -0o.33
- /47.3 56.3.8 9-0.1/7 __ i0.240 56.56 - o.o 2
7 - 1/75 56.06 -1-.49 _ "0./6 56.22 f 0.34
8 - 90.0 557 *-0.80 ,_ #0. 2 55.87 .~0.69
9 - 58.0 S6. /6 +0.3-9 +0O.08 56.24 f+0.32
/0 - 26.3 5756 -/10/ +10.04 57,60 - /.o4
/1 + 76 5742 - 0.87 -0.02 .5740 -o.84
/2 + .53.1 57.3/ -0.76~~ - 0.07 5724-.6
/3 # 82.4 56.8,' -0.26 ^ - __o.// 56.70 -___.__4
/4 +146 5.S~-.6 50 04
/S ~ 144.4 56.39 +0.16 . -0.20 56.1,9 ____o._37
/6 1 77.8 56.60 -0.06 "' -o.25 ,56.35 -f 0.2/
/ 7 42/3.9 56.7/ -0/16 __-0.3,0 56.4/ .1
/3f 253.9 56.54 -A0.0/ -0.35 S6/2 - .3
/9 +~2878 3703 -0.48 - o. -04o 56.63 - 0.07
20 f 323.7 56.56 -0.0/ - o. -04S S6,11___ ___o__4
-42.6 56.55 -0.09 +0O04 56.156 4- 0. 07
AJ, + .0/ [N2]nrz .00 _____
56.56 CHfECK 56.56
__ __ __ 0.
_____ +2o. ooo -A42.6 -0.090
- 2.13 fo. 0045
FE: t 0.o8 4 757 704. 62 f. 052. /7/0
- 90. 738 + .19/7
+757 6/3.882 1/0523627 _____
A9 - ._00/7______
TABULATED BY JDATE CHECKED BY DATE
7/62
DA I RED 71962 GPO 900e47
Fig~ure 27. Adjustment of zenith distance-star 17.
11. S. GOWZEBnm.6 PRDITIN OflCE. 1"1? 0 - 21 is
GPRNOAJEACT? TABULATION OF GEODETI DATAGREELAN A STRO(TM 5-237)
LCATION, ORGANIZATION
Stu. /SIAG A Ccc. "USA MSSCIO4N
STAR ZEN/ITH DIST. DECINI/A TION LA TI/TUDE
(1) (2) (142)
Noa (N') - /6 °/5.569 #86°oO2 49s2 6~9 46'53.83 _____
/7 (S) 4/16o 065'.6-6 t S3 40 592/ 5S.7
______ - 'Nr 2s 69 46 54.80 -oo>
t 07PAI o=o
TABULATED BY DATE CHECKED BY DATE
DA 1 FED 71962 GPO 921961 U. S. GOVERNMENT PRINTING OFFICE :1957 0 - 4Z1182
Figure 28. Observed latitude-single pair' of stars (17 & Na).
SUMt-MARY OF LATITUDEREUT
PROJECT I TABULATION OF GEODETIC DATAGREENLAND ASTRO (TM 5-237)
LOCATION, ORGAN IZATION
Sta. ISUNAGUIA Ecc. UISA MSSTATI ON
PAI/R NW'. 0V 1 V2 v v]
/ 69 46 S4.8 , .1 00
2 53.0 (-o 0 67) R-
-3 654.70 4 0.23 0/15
4 54.87 -/-0.06 -0.02
S 5-5.02 _ 0.0o9 -0.176 54.76 -0./7 'o
7 54.67 -A o. 26 -A0./8
8 55/3 -0.20 - 2
9 54.83 +4.D t0.0 2
Mn. or Su- S.5493 -0/
54.85 -0o.02 (0.6769)0. /736
6,~±1745 .8 -' .20 R~j. Limit =.Zo x 2.84 =+0.17
e2 x'10745 7 O.Rej. Limnit =.t .1/ x2.76. ±0.30
r -± .V 0.04______
MEAN 08 ERVED ASTRONOMIC LA T/T DE 690 46' S485
____ ____ ___ ___ __ _ ____ ___t o o4
TABULATED BY DATE/6 CHECKED BY DATE'42~ 7/6
D ,FORM 16 GPO 921961 U. S. GOVERNMENT PRITING OFFICE : 1957 0 - 4ZI182
Figure 29. Summary of latitude results-circummeridian altitude method.
RESULTS
errors of the instrument and the errors
introduced by inconsistent refraction
conditions.
(11) Chauvenet's rejection rule should be
applied for the rejection of single point-
ings on the star (Column XIV of the
sample computations), and for the re-
jection of latitude pairs. The rejection
factors are found in table XI.
24. Latitude By the Horrebow-Talcott Methoda. The Horrebow-Talcott method of first-
order latitude determination is used in most of
the world today for latitudes up to approximately
600. This method utilizes pairs of stars on
or near the meridian and of approximately equal
zenith distances observed north and south of the
observer. Excellent results are obtained because
the method depends entirely on differential
measurements which can be accurately de-
termined.
b. The basic formula used in computing latitude
from observations made with a Wild T-4 broken
telescope theodolite is:
O=2 (8+ ')+1 R(Mw-ME)
+l (d+di) [(n+nil+s) w-(n+nl+s+s8)E
1 1+I (r-r')+ (m+m').
Where: 8 & 8' are the apparent declination of
the stars of a pair.
Mw & ME are the micrometer readings with
ocular west and east respectively.
n, nl, s, s, are the readings of the north and
south ends of the two levels.
d & dl are the values of one division of
each of the levels.r-r' is the difference in refraction of
the two stars.m+m' is the sum of the meridian dis-
tance corrections.R is the value of one turn of the
latitude micrometer.
Note. If the Wild T-4 is not used, the micrometer and
level sections of the formula (i.e. m, n, s, and d) must
conform to the instrument used.
c. The first step in the computation of theapparent places is to obtain the mean place of eachstar for the epoch of observation from a standardstar catalogue. It should be noted that theapparent places of stars for observations made
subsequent to July 1st must be computed for the
epoch of the following calendar year, since the
Besselian and Independent Day Numbers in the
AE&NA are tabulated on that basis. The best
available star catalogue is the "General Catalogue
of 33342 Stars for the ]poch 1950," by Benjamin
Boss. In the Horrebow-Talcott method the right
ascensions need be accurate only to the nearest
second.(1) This accuracy will usually be attained by
adding algebraically to the catalogue
right ascension the product of the annual
variation and the number of years between
the epoch of observation and the epoch
of the catalogue. If the epoch of obser-
vation is earlier than the epoch of the
catalogue, then the difference in years
must be considered negative.
(2) If we denote by ao, the mean right ascen-
sion for the beginning of the year nearest
the time of observation (to) and by am
the catalogue right ascension, then the
complete formula for use with the Boss
General Catalogue, Epoch 1950, is:
o=am,+ (to-1 9 50) An. Var.
1 1+-± (t-1950)2 0) Sec. Var.
100(to--1950\o)
In the case of circumpolar stars, it may
be necessary to use more than two terms
of the above formula.
(3) If rm represents the catalogue declination
and ro the mean declination for the
beginning of the year (to), then the com-
plete formula for mean declination is:
o=0 8,+(to-1950) An. Var.
+~ (to-1950)2 1 Sec. Var.
(to-1950 3dt.
All the terms of the above formula may
be needed to determine the mean declina-
tions with the accuracy required by the
latitude. Usually the mean declinations
are computed accurately to the nearest
hundredth of a second.
(4) The above formulas for reducing the mean
right ascension and declination of a star
trom the epoch or the catalogue to tile
epoch of observation contain all theelements necessary for the reduction.These elements consist of first- andsecond-order terms of the precession andproper motion. The terms of the formu-las are arranged according to ascending
powers of the time interval from theepoch of the catalogue to the epoch ofobservation in the manner indicated in
these formulas.d. After the mean places of the stars for the
epoch of observation have been obtained, theapparent declination of each star is computed bymeans of Besselian and Independent Day Num-
bers. Using the trigonometrical expressions forBessel's star constants (Formulas for the Reduction
of Stars, AE&NA, any year) the formula forapparent declination customarily expressed interms of Besselian Star Constants and BesselianDay Numbers has been transformed into:
6=6,+rI+t cos 6o+X sin ao+Y cos ,+AbaI-Faa,.
Where: X=-(B+C sin 5,)
Y=D sin o-+A
(1) DA Form 2865 (Reduction, Mean toApparent Declination) has been arrangedfor this computation (fig. 30).
(2) ao and 6, are computed for the "appro-priate year" and entered on the formalong with their natural sines and cosines,each in its designated column. Fourplace tables provide the required accuracyof the trigonometric functions. Be sureto use the correct signs of these functions.
(3) The proper motion in declination, u', istaken from the star catalogue (G.C.).(If the change in proper motion per 100years, namely, 100 A ', is large enough itmust be taken into account in taking thevalue of ' from the catalogue.)
(4) Usually when the period of the observa-tions does not exceed 4 hours, the valuesof A, B, C, D, T, and c can be obtained
with sufficient accuracy by using themean Universal time, commonly abbrevi-ated UT, of the observations. If themean right ascensions of the set of starsare spaced with approximate regularity,the mean of the first and last right ascen-sions will provide a sufficiently accuratemean local sidereal time of the observa-
tions. Kometmes when there is a pro-
nounced break in the regularity, the setshould be divided and the mean epoch
computed for each part.(5) To the mean local sidereal time, add the
longitude if west of Greenwich or sub-
tract if east, thus obtaining the corres-ponding Greenwich sidereal time. _Fromthis Greenwich sidereal time or as
abbreviated, GST, subtract the Green-wich sidereal time of the nearest preced-ing Oh UT. Divide this interval expressed
in minutes by 1444 and the result will bethe fractional part of a civil day fromOh, UT chosen.
(6) After quantities A, B, C, D, r, and c havebeen determined, compute the X and Y
values for all the stars. When these havebeen entered in the proper places on theform, all the necessary information for
computing the apparent declination of astar is on the same horizontal line with
the exception of T and I which are at thebottom of the form. By machine, all thesteps of the computation from mean to
apparent declination may be carriedthrough in one continuous operation.
The signs of the products must be
carefully adhered to.
(7) The corrections Ass and A,5a must beapplied to the apparent declination.
These corrections reduce the valuesobtained from the system of the General
Catalogue to the system of the FK4.The requisite tables may be requestedfrom the Americas Division, Department
of Geodesy, Army Map Service, Wash-
ington, D.C. Distribution of the tables
will be made as soon as they are available.
e. The information which should be entered bythe observing party on DA Form 2842 (LatitudeComputation) (fig. 31) consists of the following:
(1) At the top of the form, the names of thestation, chief of party and observer, the
date of observation, the kind and number
of the instrument used, the number of thechronometer, and the elevation of the
station if available.
(2) In the designated columns for each star,the star catalogue number; the position
of the star with respect to the zenith
N or S; the position, E or W, of the
ocular during the observation on the
AP AONR 2 LCIN SAREDUCTION, MEAN TO APPARENT DECLINATION
ATOMARY? LAND , UA2 MAY1963 PM *1 5237)
CATALOG MEAN MEAN 1 * *1 APPARENTSTRN. Mag. RIGHT ASCENSION DCIAONsin aI C~a 4 I Stu, Y DECLINATION E
/47 4/ 4 23 m4 B 2 _0/_ '578 " , .7882 "6/54 -404 473724 -58 75 -/0.820,- 8092 1..9 It./5 52 -/ - 1 .l
/22i02 /08 4387 .88 o2.52s4 -. "987 00-. 8009 L i9) .,6.i, 26 0/ 04.07/ 9(87j A A. 34 2j3-_4.73 3973 I 9/77 _4.015 +1927' 624 S45' 74 _2 ./6 23 43.1_
/97424 55 37 05 5 4 /o 5709 .585 2 ;.024 4.68 -. 633 - /.132 7716 23 243/8
/9800 6.4 40 /3 12/ /6 62.25 3629 938 01+44 645 47, 7(S .2 .7 -/4.&f
2907/ 6._ _5 20 !S .6/.1636 480 -005S+80451 -. 6604 -/.483'. 7509 *i41 .16 56 46 /4.4420/5/. 7/ 22l_5 20 02 /997' .3427 .9395 -. 070 #1.166' -. 68/5 -4.702! -. 73/8 22; "/8: 20 02 /732
Io IS 6 2 74 9566 ~ 7~ -~ 8462 .529-.025 4 8.1.80! -. 95 I1.6/7' - .7/82 ./4 .7 57 4 490
-. 75- 27 6.949, -. 6940 I.22' .181 47 33.38
2o4211 S6 0S 20 298 22 3 3224746.490 .02 20.9187/5 -032 1.22 . -7353 = 6.730-77 9 .9 29 22 03.68
20489< 6.2 : 2 3Si 3/ 55 30.17 .5288__. .848.7 -027 +~3.7S?9 -. 7448 -7.258 -. 4,672 .20' .18: 3/ S5 2746
2058o6 6.8- r 7 oo0 45 45 13.80~ 7163 .6977 +.004 +6.370, -. 7576 -9.833' - .65289 .2/: .181 45 45 /1.57
-20696: 1/ 21 40 30 25 13.23~ .50o63 L 8623 '-.195 +-3.44 -. 77 -94 -632 9!1830 5 /.1
2074h 690 6 9434 56 47 1/ -5/~.36'.76'.3~#.,-77 /0.070 - . 62 95 .2/' /8' 47 1/ 22.65'
213 4 6 3 02 23 69 70 .51#44 ./32890-._5860 .8.19 40 28 /8.95
2/8 / 3 3 37 38 06.61' .6/ 06 j. 7919 x.03 S434 -83 8.89/0 .5770 .204 .191.37 38 03.042/4 S51 46 06+.62 42 48,21 .8887 ' 4584 -,o6I -A8,772! -. 834/ -/2.200 51 l09_2 4 453
4132 59 5O82 157 204.1 2647 .9643 -1/42 L o~o8o; -. 8441 1 - 3,630 - 5362 .26. 20 /S 20 44.64
5/2 9 71,27' .4923L.84216841 6.6_ /6 0534 3 0958~ _ 7385 i'86 - .0/5 43.348 -. 8660 _-6.852, -. 500/ 191 2 .2029 57 _0795,
_ _0_5_ _6743~ - +66801 -"878(0_-/0/381 .4776 22' 1' 47 36 04.8927/ .51 09 /2 43 54 S6.81~ 6936 .7204 .309 4 054 -8854 49.52/1 -4448 .20; 9 43 S54 51.8/
2/863L 54 i 13 17 33 S7 03.88i .5585 ' 8295 {-086 + 4.172, -8935 -7.-665'1 -. 4490 .2/ 19- 33 66 58. 95
22 0 6 . ' 6 _ 2 1 2 5 S9 08.09 .438 / .8989 f -. 04 4.2.495 '. -.9004 - 6,/2 -4350 1 .19 i .19 1.2 5 59 03 41 / "7 9 28) 52 07 33.65 .7894 .639 -. 04 + 7389~ 9053 -/0.8311 -. 4247 ) /9 .19 52 07 28.23
22094 6.9 ' 3 S9f5 25 06,1~ .8233 .-5676 !-.358 L786/K -9126 -/113021 -. 4088 "/1 .19' 55 25 00.4322216 6.0 29 38 1 2 /6 25.14 .3790 _.9254 ) -004 +1.6724,- 9233 -5200 -3842 .2/ 8 22 /16 2039
2234 6.6 _i /6"_ 2656 2/ 96 7? -0./08 V .9324 - 3.445 -3614 .26 .17 14 32 51.53
232 64 36 32 63 08 48.55 .8922 .4517 o.9/ +8.820 -. 9344 -12,248 -. 3562 ./ /14 63 08 42/16Vb m h m BESSELIAN AND INDEPENDENT DAY NAMBERS The mean epoch should be for a period not exceeding .4 hours Use oil sines and cosines to 4 decimal
MEAN EPOCH OBSERVATION // 22 A +.L 0A p laces; compote X and Y to 3 decimal places. *Reduction from Boss General Catalogue System to nR-. ' .4,System to 2 decimal places.
CORRECTION FOR LONGITUDE 5 08. 5 D ~ + .3.607 -___-____
GREENWICH SIDEREAL TIME 2/ 30.5 13; /3 29 1 Fomls .= BCs,, dl
SIDEREAL TIME Oh O.C T.- -__- -) /4 40.7 4 5- 13+D34- .4 - -... A snd
SIDEREAL INTERVAL I 6 49.8 1 )- 6.041 ____ OE T AECECDNSRT
GREWC CVLDY!MAY 3.284 [' t 0. 3346 a° F -j'.. - AIVS July 63 f.lR. n. do - AMIS DEC. X63
DA FORM 2865, 1 OCT 64
Figure 30. Reduction, mean to apparent declination (DA4 Form 2865)..
STATION~~~ LCTOPRJ T INSTR.IOENT (Type an No.) LATITUDE COMPUTATIONPAEN. N OPGS
MPATO24A LAOW/LD r-4 No. 56095 (TM 5-237) 7ZII75 m~ H/A'. CADDESS R. SAL VERMOSER 2 MAay 196
3 - /2R = 76.5 d +d / 16 .//168 /246o
RO N E MICROMETER LEVEL CHRONOMETER RMENHLESMO CORRECTIONSPAIR BOS OR ON TIME OF RDIS- DECLINATION DECLFNATIONS LATITUDE REMARKSSTAR NO. N N READIN U -RE N S W-EU OBSERVATION DELNTOSTANCE MURRO LEVEL REP EI
t d t d d d d h IM N NS
/9467 N E l/ 45.9 30.1 51~7 1__ 4 24 /14 -/6 52 0/ /6.69 .03 38 S6 TIPPED LATE
1-3.248 /28. .66___o_+6.6 39 0/ /038 04 08.47 +0,71 -. 08 62.63 _______
i 9528 .5 W 8 21.1 S.t 32.0 14 27 07.2 - l 26 0/ 0407 TEMP. ME4ANJ 38 OF
/52.4 /300 BAR. M6EAN = 3
032
Ojn
/9687 5 W /2 93.y 49.8 21.o.1 14 .94 27.4 -2 23 24 35,18 (,.o24)(/.o0.1= 1.034
2 +Z239 1470/24.+ Lo 38 47 46.40 09 /3.78 +0/2 +.17 60,.47/9742 {N E S6700 258 494 14 .97 074 -3 54 /0 .5762
__124.214
198001s 1/EI83.0 32556.1 /4 40 /4.4 -2 21/16 4986 _____
3 ___-3.547 /30.0/52.5 +4.3 39 0/' 32./S 04 31.35 '0,5 -. 08 61..23
1_ 9907 N1 W 8 28.3 573 33. /4 45 275 - 8 56 46 /4.44 ______ 0/ TIPPiD LATE
/535S /3t0 _______
20041 S E 8 876 252 / 4/ SI 5.3 -2 20 o2 /7.32____________________
4 .-. 42 114,1146.8 -0.4 _ ____38 55 03.31 a/ 5796 o o5 +,o4 41.26___
20151 N1 W 10 41.8 48.7 25,0 ,4 S6 23.3 - 2 57 47 49.30___________________
/46,9241______________________ _____ ____
20308 N W /0 62.2 5932 297 /1504 /3.2 - 2 48 17 33.68 _____________ _______
6 +5460/ 1496/126.9 +333 38 49 48.68 07 08.48 3,89 +.12 __61/33 ______
20421 .5 E 5 02.1 20.7 445 ___5. 09 670 -375 29 22 03.68 ____ 16_______-TPPED lArE
/190 141.9
20489 5S E 9 40.4 215 451 /'. /3 05.0 -3o 3/ 55 27.46 ./ TIPPED LATE
6 +S,247 /20.6/43.0 -3,3 38 50 /9.52 +06 41.40 0.39 +,12 60.76
20586 Nd W 1465/1 43.9 20.0 ~/5 /70O.S -24 44 /157
/43,01/20.0 _ ____
_ _ _ _ _ _ _ _ _ _ _ _ _ .SESP SAP4 - A S DATE CHNEDBYNATF
____________________ ______________ July X63 0,RNk - AMS D0EC. (03
DA FORM 2842, 1 OCT 64
Figure 31. Latitude Computation (DA Form 2842).
star; the micrometer reading in turns ofthe screw and in divisions and tenths ofdivision of the micrometer head; therespective readings of the north and south
ends of the two latitude levels; and thechronometer time of observation.
f. In the office, the computer will complete the
computation.(1) If the observing party has not furnished
the approximate elevation of the station,the value of one half turn of the microm-
eter, and the level value, then these data
should be entered at the top of the page.(2) The apparent declinations computed on
DA Form 2865 (Reduction, Mean to
Apparent Declination) should be enteredin the column designated "Declination",and the half-sum of these declinations,for each pair, computed and entered inthe next column.
(3) The algebraic difference of the micrometerreadings for each pair (in the sense ocular
west minus ocular east is positive) is thenplaced in the "Diff. Z.D." column,usually in decimal form. This differenceis then converted to seconds of arc by
multiplying by the value of one half turnof the micrometer and the result placedin the micrometer correction column.
(4) Next, the algebraic difference of the sumof the level readings for the star withocular west minus the sum of the levelreadings with ocular east is set down inthe designated column. This difference
multiplied by the level value, i.e., by
(d+d ) , constitutes the level correction.
(5) The approximate meridian distance iscomputed by the formula ao-(t+At),where ao is the mean right ascension, tthe chronometer time of observation andAt the correction to the chronometertime obtained from a radio time signal.This distance is entered in the propercolumn on DA Form 2842 (LatitudeComputation). If, for any reason, theobserver has not observed the star on themeridian, it should be noted under"Remarks", giving an estimation of thetime of observation before or aftertransit.
(6) If a star is observed off the meridian whilethe line of collimation of the telescope
remains in the meridian, the measured
zenith distance is in error on account ofthe curvature of the apparent path of thestar. Let m be the correction to reducethe measured zenith distance to what itwould have been if the star had beenobserved on the meridian.
Then,
sin2 rm=in sin 2
sin 1"
in which r is the hour-angle of the star.
The signs are such that the correction to
the latitude (2) is always plus for the
stars of positive declination and minusfor star of negative declination (south ofthe equator), regardless of whether thestar is to the northward or to the south-
ward of the zenith. 2 or m is then al-2 2
ways applied as a correction to the latitudewith the sign of the right-hand memberof the above equation. For a subpolar,1800--8 must be substituted for 6,making the correction negative for anorthern subpolar, and positive for asouthern subpolar. Table XII gives thecorrections to the latitude computed fromthe above formula. If both stars of a
pair are observed off the meridian, twosuch corrections must be applied to the
computed latitude.
(7) Although the difference in refraction of a
pair of stars used in the Horrebow-Talcott method is small, it must beapplied as a correction to the latitude.The refraction for each star of a pair is
very nearly proportional to the tangentof the zenith distance, so that the
differential refraction will be very nearlyproportional to the square of the secant
of the mean zenith distance. In addition,the differential refraction depends uponthe pressure and temperature of theatmosphere at the time of the observa-tion. For a mean state of the atmos-sphere (pressure 29.9 inches or 76 cm.and temperature 500 F. or 100 C.), the
correction to be applied to the latitudefor differential refraction will be givenby the formula:
r-r' 57''9 s
2 sm ( -3') sec22 2
where r and are the refraction and
zenith distance, respectively, of the
star observed with ocular East, the
primed letters referring to the star
observed with ocular West. Differential
refraction will, therefore, have the same
sign as half the difference of the zenith
distances as measured by the microm-
eter. The two zenith distances of a
pair of stars used in the Horrebow-
Talcott method are so nearly equal
that either may be used to determine
the sec2 in the formula.
(a) Table XIII has been computed by
means of the above formula with half
the difference of the zenith distances
as measured by the micrometer for
one argument and the mean zenith
distance for the other.
(b) In as much as the refraction obtained
from the above table is only valid for
the assumed mean state of the atmo-
sphere, it will be necessary to apply
to this differential refraction, factors
obtained from the regular refraction
(table V) in order to reduce it to the
differential refraction for the pressure
and temperature at the time of
observation.
(c) When the micrometer, level, refraction,and meridian-distance corrections have
been combined algebraically with the
mean of the declinations of a pair of
stars, the latitude as determined from
observations on that pair of stars is
obtained.
(8) The correction to the mean latitude and
the correction to the value of one-half
turn of the micrometer are computed by
the method of least squares. Separate
computation of each night's observation
is not necessary unless there has been a
distinct change in the value of one-half
turn of the micrometer. Such a change
should have been recorded in the field
records.
(a) DA Form 2843 (Astronomic Latitude
Summary) should be used for the
adjustment (fig. 32). The data in the
first four columns are obtained directly
from DA Form 2842, Latitude Com-
putation, the micrometer differences
being taken to the nearest tenth.
(b) Before proceeding further with the ad-
justment, it is necessary to find out
which, if any, of the results are to be
rejected. An absolute rejection limit
of 3" from the mean of all the latitudes
in column 4, each considered to have
unit weight, is first used. Then a
mean of the remaining latitudes is
taken, and the probable error of a
single observation, e p, is computed.
Any latitude with' a residual, 0O, equal
to or greater than 3Y2 e is automatically
rejected. In addition, other values
may be rejected if the residual is ex-
cessive when compared with all others.
Before final rejection of any value, the
records should be reviewed to deter-
mine whether a star has been misidenti-
fied, level values follow pattern, or the
turns of the micrometer may be a full
turn in error. These are the most
common causes of error. Notes by the
observer may indicate doubtful ob-
servations. Another criterion for re-
jection is table XI. For a small num-
ber of observations, its use may in-
crease the number of rejections.
(c) After all rejections have been made, the
accepted observations remain to be
adjusted in order to determine from
the observations themselves the most
probable value of a turn of the microm-
eter and the most probable latitude
of the station. The information at
the foot of the third column is now
entered. Then a mean 4m is taken
of the unrejected latitudes in column
4 and the difference A¢= -- entered
in the next column and summed
algebraically at the foot of the column.
(d) If p is the number of accepted latitudes,c the amount in seconds by which the
mean latitude deviates from the proba-
ble latitude, and r the amount by
which the preliminary value of one
half-turn of the micrometer is to be
corrected, then there will be p equa-
tions of the form.
c- Mr +A = v.
Inserting the condition that [v2] or
[vv] must be a minimum, the normal
equations to be solved for c and r are
PROJECT I ASTRONOMIC LATITUDE SUMMARYI (TM 5-237)
OBSERVER DATE STATION
A'.A/.CADDESS 2 MAY /963 PM MAP AS7RO 2CHIEF OFPARTY }INSTRUMENT LOCATION.
R?. SALVfiRAVSER WILD r-4 (No. 56o95 IMARY-A N DSTAR NUMBER Mic. Diff. j _ 2 Mr ADJUSTED ___2
BOSS GEN. CATALOG M 381 !7~ i Q,
19467 /9528 -3.2 Io2 63 /44S 1. 3 02.50 -130/9687 1/9742 +72 oo.47- +0.7/ +__~-.30 00.77 +0.43/_9800 /8907 - 3.5 01.23 -0 .0s _ -. 1/4 0/. 09 + 0. /1
20041 20/51 + i.S 01.26 -0.0.8 4.o61.32 -0o.,2__20308 2042/ ,S.6 0/.33 - oS 4_ 4. ?$. 01.66 -0,36 __
_20489 20586 (o +2 00. 76 +o.42 ___ +. 2/ 00.97 +0.2320696 20744 +&6.9 01.54 -0.36 +__ 1. 28 01.82 -0.622/032 2/086 -4.8 00.67 +i0-5S1 __ -. 20 00.47 +0.7321246 2/32/ -3.7 00.68 +0o.50 - . /S 00.53 40.6721534 21684 +8.2 00.40 + o. 78 +__ "34 00.74 +0.462/761 2/86 3 +0.8 00.32 1 +0.86 +__ 03 00,35 +0.8521937 22003 -4.9 02.23 -i/. oS ___-.20 02.03 - 0.8322094 22216 +4.9 R___223,42 22382 +4.9 01.13 +.0 o __o5 +-20 01. 33 -. 1322646 22785 -. 1 0 2.2/ -/.o3 .o0 o2.2/ -1.0o/
22866 22980 -3.9 R23/32 23225 #9.6 0 1.7$ -0.57 +~ .14 01.89 -0.6o923433 23574 -3.3 01.64 -o.46 .- 14 01.50 - 0.30
23 770 2.39/9 -0.2 0. 29 - o. / / _ - .0/ 01.28 -0o.0824003 24 /SS #3.1 b00.764 #0.421 . .13 00.89 +0,3124279 24433 f 0. 5 0o.9 + o. 19 +.o2 01.0/ +0.1924538 24413 -2.7 00.86 +0.32 - .1 00.73 +0.4524699 24816 +2.9 01.41 -0.23 ./12 0/.53 -0.33-24 92o' 25.040 -4.8 00.86 + 0.32 - .20 00.66 + 0.5426290 26358 -2.3 00.49 + 0.69 - .09 00.40 f 0.8026475 26 542 -1.2 R _____26,632 26749 -1.9 004 +.7/ -. 08 00.39 +0.8126996 27047 -6.9 02.21 -1.03 ___-.28 01.93 -0.73
+±SUM 50.3 /2.12 6.48 1__ 4.15 6.54
- SUM 42.3 /7.47 (o.57 __ 5.71 6.65 __
ALGEBRAIC SUM +8.0 29.59 - .09/0.0279 ___29.810 +.08 9.2468MEAN -l- 0.32 01.18 __ _ _ _01.20 _ _
UNADJUSTED ADJUSTED
MICROMETER ONE-HALF TURN VALUJE 76.5 76.54/26(1) MEAN 0 12 PAIRS WITH PLUS MICRO. 01FF. 01.0/ 0.1/8121 MEAN 0 13 PAIRS WITH MINUS MICRO. 0IFF. 0.'. 34 _01. 2/DIFFERENCE 121 - (1) + .33 + 03
NORMAL EQUATIONS
MASERMI LATITUDEFGDEISTIN N 38n 57 (03.240 .0
COMPUTED BY DATE CHECKED BY DATE
-AMS IJU l '(p3 0. R.fli.oka~ - A MS DEC. '(03
DA FORM 2843, 1 OCT 64
Figure 32. Astronomic Latitude Summlary (DA. Form 2843).
pc- [M]r+[A4,] =0
- [M]c+ [MM]r- [MA4] =0
where [ ] is the standard symbolindicating summation in operations
with least squares. In the solution bythe Doolittle method (fig. 33), twocolumns are added to obtain the propercoefficients necessary to determine theprobable errors of the micrometer andthe latitude. The equations take theform:
pc- [M]r+ [A-+ 1.0+0=0
- [M]c+[MM]r- [MAC]+0-+ 1.0=0
(e) Each micrometer difference, M, ofthe accepted latitudes is now multi-plied by the value of r and the resultsplaced in the proper column. Thento each preliminary latitude thereis added algebraically the corre-sponding Mr to produce the correctedlatitude which is entered in the desig-nated column of DA Form 2843(Astronomic Latitude Summary). Amean of these latitudes is now takenand entered at the foot of the column.
In the next column the difference,mean latitude minus individual lati-tude, is set down for each pair ofstars. The sum of the squares of
these new A¢'s is entered in the nextcolumn. The mean of the correctedlatitude is the mean observed astro-
nomical latitude of the latitude sta-
tion, uncorrected, however, for eleva-tion of station above sea level or
variation of the pole.
(.f) The probable errors have been com-puted by means of the following formu-las with data used in the adjustment.
= ± /0.455[A 2]
p-2
V= [M 2 P [M]2
(p-2) ([MM]-
== eP [MM] [M2
(9) The correction to the latitude to reduceit to sea level is given by the followingformula:
A=---0"000171 h sin 20
where 4 is the correction in seconds ofare, h the elevation of the station inmeters, and 0 is the latitude. Thiscorrection can be obtained directly fromtable XIV.
(10) When the x and y of the instantaneousnorth pole are known for a given date,the reduction to be applied to an astro-nomical latitude observed in west longi-tude (X) to reduce it to the mean poleis as follows:
A0=- (x cos X+y sin X),
x and y are in seconds of are.
(11) If the observations have been madefrom an eccentric station, the reductionto the geodetic station is computed bycosine of the azimuth x distance inmeters divided by difference per secondof are in meters. The sign of the correc-tion should be checked on an orientedsketch of the eccentricity.
25. Latitude By Meridian Zenith Distance ofthe Sun
This is a rough method used for convenience inobserving and it will not yield precise results.The computations are similar to those in para-graph 20, except for the necessary determinationof the sun's coordinates.
a. The field observations are usually made byfollowing the sun in altitude near noon andaccepting the highest obtained altitude as themeridian altitude. This ignores the slight differ-ence in the meridian and maximum altitudes dueto the changing declination. In finding the maxi-mum altitude by trial, it is seldom possible tosecure a reversed pointing. Hence, the observedaltitude must be corrected for index error ofthe vertical circle, refraction, semidiameter, andparallax.
b. A more accurate method consists in knowingthe meridian from a previous azimuth observation,or in computing the exact time of the sun's transitfrom a known watch time and the station longi-tude. The vertical circle is then read a fewseconds before transit, the telescope reversed andthe other limb observed. The index and semi-
diameter corrections are thus eliminated.
MAP ASTRO 2 SOWT/oON OF LATITU E ADJUSTMVENT
c r a-R s.25 ooooo - 8.ooooo - 0.0 9000 + 4. 00000 0.00000
C:= +0.32000 +0.00360 -0.04000 0.00000
______+463.52000 -/8.99200 0.00000 0-I0000O
- 2.56000 -0.02880 +0o.32000 0.00000o
+460.96000 -/902080 +0.32000 + 1.00000o
r=+ oao4126 -~00069 -0.00217
r= +. 04126__ ____
c: +. 0/(680 _____
+./0.02790_ _ _ _ _
0. 00032 -0.04000_____
0. 78480 - 0.00022 - 0.002/7
9.2 4 278 .04022 . 00 217
e1- 9.24278,(0. 45495) ±0.42758 ___
______ 23 _ _ _
e 7 2 ±0.08S5.
e = .0021~7 =± 0.01/P92 _____
TABULATED BY* DATE . CECKED BY DATE
Al &. 11MS Jul363 0.1?. a- AM s DETC. 6 3
DA Ifly 61 b GPO 380647U. S. GOVUBUmWa PR99Th63 OFMZ : 1957 - 421132
Figure 33. Solution of latitude adjustment.
c. The computations for a above, are-
(1) Correct the observed altitude (or zenithdistance) for index error (if known),refraction, semidiameter from the Amer-ican Ephemeris or equivalent, and par-
allax from table VIII. Parallax may beneglected in rough work and when theindex error is unknown.
(2) Find the sun's declination as follows:(a) Accepting the observation as having
been made on the meridian, the local
apparent time is 12".
(b) Add the longitude to obtain Greenwich
apparent time (GAT).
(c) Subtract the equation of time from
GAT to obtain UT.
(d) Correct the apparent declination for
the date for the elapsed UT from Oh.
(e) In case the local standard time of the
observation is recorded, the UT is
found at once by adding the time zone
difference.
(3) Apply the formula:
or 4= +(90 0 -h)
26. Latitude By Circummeridian Zenith Dis-
tances of the Sun
This is an extension of the previous method for
greater accuracy. It is similar to the method in
paragraph 22. The observations are made start-ing about 10 minutes before local apparent noonand continued for about the same interval afternoon. Pointings are made in direct and reversedpositions alternately upon the different limbs.The computation procedure is as follows:
a. Take the means of each pair of D and Rpointings and the means of their vertical readings.
b. Determine the hour angles (t) of the meantime of pointings on each pair. These are thedifference between the observed time and the timeof transit.
c. Scale the approximate latitude from a map,or compute a trial latitude using the highestobserved altitude.
d. Apply the formula, finding the correctionsAm for each value of t, and find the equivalentmeridian altitude or zenith distance by the equa-tion:
hm=h+Am or ,'m= -- Am
e. Mean all the consistent values of hm (or tm),
and apply corrections for refraction and parallax.
f. Obtain the sun's meridian declination byfinding the UT of transit and using tables of theephemeris.
g. Apply the following formula:
4= (+fm
4= 8+(90°--hm)
Section IV. DETERMINATION OF LONGITUDE
27. Basic MethodThe longitude (X) of a place is the arc of the
equator between the meridian of the place and theprimary meridian of Greenwich. Since there is adirect relationship between longitude and time,determination of the time at the place with respectto the time at the meridian of Greenwich will
establish the longitude of the place. Present day
radio time signals broadcast by WWV, WWVIH,GBR, JJY, and several other major observatorieshave been synchronized and provide an excellentmeans of obtaining time at the meridian of Green-wich. Time at the place is determined by observa-tions on various stars using several differentmethods and procedures.
28. Determination of Longitude By Star Transits
The most direct method of determining longi-tude is by observing the instant of transit ofknown stars over the observer's meridian. At thatinstant, the observer's hour angle is 0h and the
local sidereal time is equal to the right ascension
of the star. This method is applicable to any
class of observation but is seldom used except for
first or second order work since the preparatory
work of placing some types of instruments in the
meridian (para. 29) will provide a longitude having
the required accuracy for lower order work.
a. The instruments used in this method are
usually large meridian transits or universal type
theodolites with very sensitive levels and imper-
sonal type, automatic recording eyepiece mi-
crometers.
b. The following formulas and identities are
applicable:
AX-Aa+ (a+Aa-t-At)=v
A= sink secb= sin- tanb cos4
B= cos sec 3= cos O+tanb sine
C=secb
k= 0.0213 cos 4 sec6
l=2 (m+s) C
757-381 0 - 65 - 5
b=i(d)/15 (n)
The symbols in the above equations are:
AX= correction to an assumed longitude
A= azimuth factor of the star
a= azimuth of the line of collimation (amount.by which the instrument is off the meridian
assuming the collimation error of the instru-
ment to be negligible)
a=right ascension
Da=the short period terms of right ascension
t=mean time of transit corrected for levels,diurnal aberration, width of contact strips
and lost motion
At = chronograph correction
v=residual of a star in the solution of a star
set (usually six stars)
= zenith distance
= declination
0= astronomic latitude of observer
B = level factor
C = collimation factor
k=diurnal aberration. The sign is minus for
stars observed at upper culmination and
plus for subpolar stars (i.e., observed atlower culmination)
1= correction for width of contact strips and
lost motion
R= equatorial value of one turn of the mi-
crometer in seconds of time
m=lost motion in terms of divisions of one
turn of the micrometer
s= average width of contact strips in terms of
divisions of one turn of the micrometer
b=inclination error in seconds of time
i=mean level value in seconds of are per
division of bubble
d=difference of bubble readings (refer to in-
strument manual to determine sign)
n= number of level bubble readings
c. The following data should be furnished by
the field party:
(1) DA Form 2844 (Longitude Record)
containing the following information
(fig. 34):
(a) date and headings(b) star names and/or numbers
(c) level records(d) time of radio time signal comparisons
(e) remarks as applicable
(2) Chronograph sheets and/or tapes which
contain the record of the star trackings
and radio time signals
(3) Instrument and level constants including
information as to when and howdetermined
(4) All data abstracted on the proper formsincluding the scalings from the chrono-graph sheets and/or tapes
(5) Field computations
d. Scaling of the Favog Chronograph record isfully covered in TM 5-6675-210-15. In scalingother types of chronograph records a suitableglass scaler or a variable scale may be utilized.Figure 35 is an example, using a glass scaler.
(1) In scaling radio time signals on DAForm 2845 (Radio Time Signals) (fig.350), 20 breaks should be adequateif good reception was obtained. Themean epoch of the radio time signals isreduced to the nearest second and themean chronograph time is corrected tothat epoch.
(2) In scaling the star transits (fig. 36), atleast 10 matching pairs of breaks arerequired. If the residuals of the leastsquare solution for the star set appearerratic, it may be desirable to scale allrecorded matching pairs of breaks, makethe obvious rejections, reject others onthe basis of pattern, and then obtain anew mean value for use in the computa-tions. With an experienced observer,it is seldom necessary to scale additionalpairs except under conditions where it
was difficult to track the star.e. DA Form 2847 (Comparison of Chronometer
and Radio Signals) is used for radio time signalcomparison computations. The procedure is asfollows (fig. 37):
(1) Fill in all headings. The latitude andlongitude should be the closest approxi-mation which is available.
(2) Enter the year of observation and the
meridian of the local time which is being
recorded.
(3) For each column, enter the local date,recorded local standard time, the chrono-
graph time of signal, transmitting station,and frequency on which received.
(4) To the local standard time, add if west
(and subtract if east) the meridian of
the local standard time expressed in time
and fill in the appropriate date and
Universal Time (UT).
(5) From the American Ephemeris and Nau-
tical Almanac published for the year of
ROJ, CT LONGITUDE RECORD (Original Trait Level Readings)(TMt 5-237)
LOCATION SAION
MAR~YLAND MAP ASTRO 2ORGANIZATION CHIEF OF PARTY CHRONOMETR DATE ( oca
IUSA MS R. SALVERMOSER 1247i4 /8 JUNvE 063 pmOSERV[R INSTRUMENT P9 0. RECORDER
R. SAL VERAMO1SER WILD T-4 No. S4095 H. N. CADOESSSET NO.__2 _________SET NO:- 3 ________
LEVELS LEVELSSTARS (ta Sidn"i*) STARS (Or Signals)
__ _ _ _ _ _ _ _ _ _ _ _ _ W. E. W. E.
531 (E) A -.3(6 72.0 34.1 572 (E) A +,20 70.2 30.8cc /4k23'.565 33.3 71.1 oC /5 26 /5 33.0 72.6
1A -.36 38.7 37.0 IA +.20 372 41.8
534 (W) A +./6 33.1 71.0 578 (w) A +,23 .33.0 72.5/430 14 72.2 34.3 /5 33 04 65!.4 29.8
IA -. 20 39.1 36.7 IA +.43 36.4 42.7540(E) A .-. /6 72.0~ 33.8 /42 (E) A -. 20 65'9.3 29..8
14 3728 .3/ 9 70.1 /S537 03 33.8 73.3Z A -.36 40.1 .36.3 Pcked la fte FA .+.23 35.5S 43.5
1386 (w) A +.02 31.9 70.3 583 (W) A +.41 33.2 73.0/4 4739 71.1 32.S 15 44 25 71.1 31.3
Di thy&4haze EA -.34 392 378 Picked a late ZA 4.o4 37.9 41.7S51 (E) A +.42 71.0 32.1 /416(EW A -.08 70.3 30.8
14 54 24 32.o 71.0 /S S/ 2/ 33.6 73.1Pike4 u~ ate EA +.08 39.0 '38.9 IA + .5(o 36.7 42.3555(w) A -. o4 32.1 71.1 595(w) A -48 33.8 73.4
/5 00 44 71.3 32.2 /5 56654 691 29.3FA +.04 392 38.9 Pkke4 ~Ite IA +,08 36.3 4441
13 95(E) A -24 713 32.0 __
/5-0410 32.6 71.8Ver di f vt z 1 -.20 38.7 39.8 _______
563 (w) A +.i0 32.0 71.-3/S /368 73.0 33.7
IA -. /0 41.0 376o _______
REMARKS
W WV /S MC
s Er 2 N /4.18 (2) /434 (V) /440 (4.) 144.5 (.s) /5/6s ETr 3 (1) I523 (2) 1541 (;3) /6.59
sint proper sequence with stae to show time of receptiom, include following data for each tine signal received:
Time (Local, standard, etc.), radio-station identification. frequency.
DA FORM 2844, 1 OCT 64
Figure 34. Longitude Record, DA Form 2844.
NOTE: As shown here, it is not uncommon to scalethe chronometer time of the ending of theJJY signal rather than the resumption, 0s02is then added to the scaled chronometer time.
4%
U, _ 1
i I 11111111' * ' 1 l
p Graphic sample
Figure 35. Time signal scalings.
the observations, obtain both the sidereal
time of OhUT (apparent sidereal time,HA of first point of Aries) for the UT
date and the change in nutation.
"Change in Nutation" is the propor-
tional part of the UT day multiplied by
the tabular difference of the Equation of
the Equinoxes for the date. The sign of
the tabular difference is determined by
reversing the sign of the equation of the
Equinox at Oh and adding algebraically
to the equation of the Equinox at 2 4h .
(6) From table IX in the Ephemeris
(AE&NA) determine the correction mean
solar to sidereal time for the UT or multi-
ply the UT expressed in minutes by
0.1642746 which is the rate of change per
minute.
(7) The transmission time between the trans-
mitter and the astronomic station may
be determined by first using the formula;
Cos D= sin 41 sin 4,+cos 01 cos 42cos AX where 01 and 42 are the latitudes
of the respective stations, AX is the dif-ference in longitude, and D is expressed
in degrees of are and decimals thereof.Then the correction for transmission
time becomes AT=0.000401 D, the
constant being based upon a speed of278,000 km/second for short wavereception.
(8) The "correction to signal" is obtained
from "Time Service Bulletins" published
periodically by the Observatories moni-toring the time signal. If the monitor-
ing observatory is not close to the trans-
mitting station, it will be necessary to
correct for the time of transmission
between the two points to obtain thecorrection to signal at the transmitter.
The UTO corrections are currently being
used. If a common pole is adopted in
the future, it may be desirable to convert
to UT1. The correction used should be
identified so that future conversions may
be made if warranted.(9) To obtain the Greenwich sidereal time,
add (3) through (8) above.
(10) From (9) above, subtract the approxi-
mate longitude to obtain the local side-
real time (LST).
(11) The chronometer correction is the differ-
ence between the LST and the chronom-
eter time of signal. The correction is
positive if the chronometer is slow and
negative if fast.
(12) The rate per minute of the chronometer
is determined by dividing the difference
between two chronometer corrections in
seconds by the difference in their
chronometer times in minutes. The rate
is positive if the chronometer is losing
and negative if gaining.
f. Computation of factors A, B, C, k and 1 are
ryO
F,
H I i I I/ //
i Il
IL ( I IIcL p
PROJECT IRADIO TIME SIGNALS(TM 5-237)
OBSERVER DATE STATION
. SAL Vi/?MOS'E /8 &1'vE. /1963 PMq MAP AS TR O 2RECORDER TIME ZONE OF SIGNAL FREQ OF SIG NAL
H. CADDESS R WW' 1/' / MCSLocal Chrono- Correction t Local Chrono-LclCrn- oretno LcaChn-Sending Time of meter Time of Reduce to meter Timo Sedn Time ofLoaCrn- Crecint LclChn-SednIieo meter Time of Reduce to meter Time of Sga
Sina Sinal Mean Epoch Mean Epoch SinlSignal Mean Epoch Mean Epoch
___h. ~ . __ 4_ /h. 12 m. 2/ h. 14 h. /4 h. 42 m.m. S. m. S. S. S. m. S. m. S. s., s.
34 22 /2 08.62 .049 26. 66' o4 /9 42 /0.54 .057 31.59724 /0.64 .044 .684 2/ /255 .052 .6o226 /2.63 .038 .6-68 23 14..56 .047 .6o728 /4.64 .033 .673 25 /(6.56 .04/ .60130 /6.65 .027 .677 27 /8.56 .036 .59631 1765S .025 .675 29 20.57 .030 .60033 /9.65 - 0/9 .669 3/ 22.57 .025 .595
__ 35 21.65 . 0/4 .664 33 24.58 .0/9 .59937 23.66 .008 .668 35 26.59 .014 .60439 26.66 .003 .663 37 28.65 .008 .598
41 2767 .- .003 .667 40 31.58 .000 .58043 2967 - .008 .662 42 33.59 -. OO .8545 3/.68 -. 0/4 .666 44 3655 -. 0// .579
47 33.68 = .019 .b6/ 48 396/ -. 022. .588.9 35.69 -. 025 .665 .50 41.6/ -. 027 .583
S/ 3769 -. 030 .660 S2 43.61 -. 033 .S7753 32969 - .036 .654 54 45.62 -'.038 .582
55 41.70 -. 0o41 .659' 56 47.63 -. 044.- .1586S57 43.7/ -. o47 .663 .58 49.64 -. 049 S9
S8 44.72 -. o49 .67/. 60 51.64 -. 055 .S85
____MN 669 ___________ ___M '5/
MEAN EPOCH 20 34 .40.2 MEAN EPOCH 2( 04 3,92
MENSCLD EDIG .675 MEAN SCALED READINGS .5895CMN TO SID ( 0 .,2 ) OOOS CORR, MN TO SID (.8).0022
ADOPTED MEAN EPOCH 20 34 40ADOPTED MEAN EPOCH 2/ 04 40CORCHO TM/4 /2 26.667 CORR CHRON TIME /4 42 31..59/7
CMUEYDAECHECKED BY DATE
DA FORM 2845, 1 OCT 64
® DA Form 2845 (Radio Time Signals)
Figure 35-Continued.
STATION PROJECT ORGANIZATION SET NO. STAR TRANSIT SCALINGSMAP ASTRO 2 TEST: USAMS 2 (TM 5-237)
LOCATION OBSERVER INSTRUMENT' LOCAL DATE 4 No individual sum shall exceedMARYLAND , USA R. SALVERMOSER WILD T-4 56095 /8Ju.63 the mean by more than ± 0.2
STAR 531 STAR 534 STAR 540 STAR /386h h I i
TR IN 14 22 SUMS TRIN /4 h 9 SUMS TRIN /4 36 SUMS TR IN 14 h 46 m SUMS
I 17.5 37.7 55.2 4 22.5 68.7 91.2 2 /8.5 396 58/ I 39.1 43. 2 82.3
2 /9.4 36.o 55.4 5 23.4 67.5 90.9 R 3 20.1 38.0 58.1 2 40.4 41.9? 82.33 21.0 34.2 55.2 6 24.8 66.4 91.2 4 21.4 36.7 58.1 .3 41.7 40.6 82.35 24.5 30.9 55.4 7 25.9 65.1 91.0 S 23.1 35. 58.2 4 3.o 392 82.2
67 26.0 293 55.3 8 272 63.9 91.1 6 24.3 33.7 .5.0 5 44.3 38.o 82.37 < 277 27.5 552 9 28.4 62.8 91.2 7 25.8 32.3 58 6 454 36.7 82.3
8 292 26.1 553 C 29.6 61.5 91.1 8 27.3 30.8 58.1 7 46.8 355 82.3
9 3/.2 24.4 .56 / 30.8 60.4 91.2 9 28.8 29.5 58.3 8 48.2 34.1 82.3D' 32.6 22.6 56 2 2 31.8 59.2 9/.0 C 30.1 28.0 58.1 9 49.4 32.8 82.2
2 36.3 /9.3 556 3 33.2 579 91.1 31.6 26.6 58.2 C 50.8 31.6 82.4
3 37 7 /7.5 S.52 4 34.3 56.7 91.0 2 33.1 25.1 58.2 / 52.0 30.2 82.2
4 39.3 /6.0 563 5 35.6 55.7 91.3 R 3 34.5 23.8 58.3 2 53.4 28.9 82. 3
5' 41.0 /4.4 55.4 6 36.8 54.4 91.2 4 35.9 22.4 58.3 3 54.8 27.6 82.4
6 42.9 12.7 556 7 379 53.1 91.0 5 37.2 20.8 58.0 4 55.9' 26.3 82.2
71 44.4 11.2 556 8 39.1 52.1 91.2 6 38.9 /9.4 58.3 5 57.2 25.0 82.28 46.1 09.4 555 9 40.3 50.9 91.2 7 40.3 /79 58.2 6 58.6 23.7 82.3
9 47.7 077 55.4 b 41.5 49.6 9/.1 8 41.6 /6.5 58.1 7 59.8 22.4 82.2c 49)4 06.2 55.6 1 42.5 48.3 9
0.8R 9 43.2 /5./ S8.3 8 6/.1 21.2 82.3
/ 51.1 04.6 55.7 R 8 44.5 /3.8 58. 3 9 62.4 /99 82.32 52.8 02.8 55.6 - 46.0 /2.3 58.3 8 63.7 /8.6 82.3
h m EAh m* +MEAN h im *S 8OT4 h 48 m MEANOUT /4 25 55.40 OUT /4 30 91.12 OUT 458.18 OUT 82.28
TRANSIT TIME /4 23 57 700 TRANSIT TIME /4 h30 .5 560 TRANSIT TIME 14 37- 29.090 TRANSITTIME 14 4m411Scaled By DATE Checked By{ . AMS DATE
R..±2V t*1?6s L - AS JUNe /963 July 1963DA FORM 2846. 1 OCT 64
Figure 36. Star Transit Scalings, (DA Form 2846).
made directly on DA Form 2848 (AstronomicLongitude Data) (fig. 38).
(1) Extract tan 5 and sec S from thefundamental catalogue at the bottom ofthe page for each star. If greateraccuracy is required, the a should bedetermined to the nearest second andtan 8 to five decimal places.
(2) Apply appropriate formulas as listed in
b above, and as listed on the form.g. The level correction to the time of transit is
computed as follows:
(1) On DA Form 2844 (Longitude Record)(fig. 34), subtract the west bubble read-ings, the difference is considered positive;then subtract the east bubble readingswith the difference considered negative.The inclination is the algebraic sum ofthese two results, which is entered onDA Form 2848 (fig. 38).
(2) Multiply the result of (1) above, by i/60,where i is the mean level value in seconds
per division of the bubble. This resultis the inclination error in seconds of
time (b).
(3) The total level correction in seconds oftime is the product Bb which is enteredon the appropriate line.
h. The uncorrected transit time is the mean ofthe 10 or more matching pairs of breaks from d(2)above.
i. The mean corrected t then becomes the sumof the scaled transit time + (l) + (k) + (Bb) whichis entered on the t line.
j. In computing a and Aa, it is necessary toproceed as follows (fig. 39):
(1) Determine the mean epoch of each starset (usually six stars). The mean of tfor the first and last stars of 'the setwill be sufficient if the star transits are
evenly spaced.
(2) If the chronometer correction is greaterthan 10 minutes, it will be necessary to
PROJECTCOMPARISON OF CHRONOMETER AND RADIO SIGNALS(TM 5-237)
LOCATION HRNMT STATION
M1ARY[IA N0 D_______ / 24 74 MAP A T RO 2ORGANIZATION LAIUE LONGI TUDE
USA MS T__m____ST m 08 t_29__00_A__
LOCAL DATE Jv)NE /963 /8 JUiNE /8 JUNE /8 JUNE /8 JUNESTANDARD TIME OF SIGNAL H M S H MS
MERDIA 20 34 40 21 04 40 2______ ____3_22_25_J
SIG~NLTE TIEO 14 12 26.667 /4 42 3/-592 I6 17/7 30. 316 /(p 03 34. 858
TRANSMI TTING STATION wr w vW WIVIw WW WVFREQUENCY OF SIGNAL /S Mc /M /5S Mc /SMC
UNIVERSAL DATE JuN. /9 1.19 /9TIME (U. T.) H N S M S M M SOF SIGNAL TIME / 34 40,.000 2 0M4,00 2 39 3.0032M3.0
SIDEREAL TIME OF OhU. T.. /7 4~5 .59.845 /7 45 .59845 /7 45 59845 /7 45S9.845CHANGE IN NUTATION .000 - .00/ } .00/ T .00/CORRECTION -MEAN
SOLAR TO SIDEREAL. TIME ,55120 .480 26.210 33.7S8TRANSMISSION TIME 0 0 00CORRECTION TO SIGNAL - 10
G.S.T. OF SIGNAL /9 20 5.2 95 /9 5/ 00.225 20 25 58.955 2/ /2 03.503LOMGITUDE OF STATION 5 0 8 29 S 08 29 5 08 2? 6 08 29.LOCAL SIDEREAL TIME 1_ 4 /2 26o.29.5 /4 42 .3/. 22.5 /S /7 29.9S5 16 03-34.503CRNMTRTOFSIGNAL 14 12 26.6167 /4 42 31.5S9216' /7 30.3/6 /6 0334.858CHRONOMETER CORRECTION - .372 - .367 -31.3S0 CORRECTION #0005 f .0. 006 *0. 006ACHRQNOMETER M0 8 74m 7
SPre/itninary correct ion- UTO si9 nal).
COMPUTED my IDATE ICHECKED BY DATE
cf- AM s. July3 6'(oca 3 AM 5 July I63DA FORM 2847, 1 OCT 64
Figure 87. Comparison of Chronometer and Radio Signals (DA Form 2847).
PROJECT OBSERVER CHRONOMETER SET NO ASTRONOMIC LONGITUDE DATATEST R. SAL VERMOSER~ 12474 2 (TM105-237)
LOCATION RECORDER INSTRUMENT Ty'pe a.d No. STATION
MARYLAND Hf. CADDESS WILD T-4 56095 MAP ASTRO 2ORGANIZATION LEVEL VALUE (dt) LOCAL DATE GREENWICH DATE 77 dTP de
u5 AM S d/60 = s 0 18 6 8 i8 JuN (3 l_1/9.09/J114.(, 52/ -. 207 -. 0/0LATITUDE (40) 0 93IE0COIE1 LONGITUDE() PAGE PAGES
N 38 .57 0/ INE 6286 COIN *.7776o9 5% 8"'295 Y2 to (+ s)= _05 2 OF
STAR .531 534 540 /38(c 551 555 I 39 .663-
DECLINATION (s) +.520/28+.30 32 06 + 44 33.59 +37 .57 55 + 14 35 45 + 40 32 18 +48 17 46 +.33 27 /4
TAN s + 1.281 + 0. 590 0. 98S +0.780 + 0.260 +0.8.55 + . /22 +0. 66 /
CC 8ec 1425 1.16o/ 1. 404 1.24~8 .033 /.31/6 /. 503 1.1/99INCLINATION + 1.7 + 2.4 + 3.8 +1.4 + 0./ +0.3 -1.1 + .3. 4b(1ncl.xrdt) +.0317(a +.04483 +.07098 +.026/5 .+.00/87 +. 00.660 -. 02055 +.0635/
8 15S83 1/ 49 1.397 1.268 0.941 1.315 1.483 1/ 93
Scld 14 14 14 14 /4 h 5h/Transit ~ I s Im s mo s m. S m. s m a m s in aTime 2.3 .57.700 0 /5.560 3 29.090 47 41.140 54 .20.06500 34. 10 04 14.020 /4 02.300
t +) .139 .099 .120 .108 .088 .112 .128 ./O2
K H -. 0o27 - .019 -. 023 - .021 - .0/7 -. 022 -.. 025 -. 20
Bb +..050 +. 052 +.099 +.033 +.002 + .007 - .030 +. 076
t /4h 23 57.862 30 15.692 37 29.28(4 47 41.260 54' 3o. 138 00 34.907 04 14.093 /4 02.458A t -. 370 -. 369 -. 368 -. 36(a -. 365 -. 344 -. 3(o3 -. 3(02
a 141 23 575 42 30 /S.3/5 37 28.945 47 40.877 54 29.682 00 34.525 04 13.808 /4 0o2.o63Da -_ 009' -. 0// -. 0,0 -. 0/0 -. 012 - .0/0 -. 009 - .0/0
a+IAa-(t+-At) .F. 041 - . 019 +. 0/7 -. 027 -. 1/03 -. 028 +. 069 -. 043
A -. 348 +. /70 -. 137 .022 +. 4260 -. 0340 -. 244 +.//sA = sinsec= G iD 0 - tan 8 CoG I/ l =Y2. -' (In+ s) sec S A= da (tA) d qi+:da (E)".df -. 6B =-COs sec s COG + tan 8 sin 9S l IS ALWAYS POSITIVE. 8 6
C +seS g01 ~ 0seSCMPUTED BY GATE
DFOR STARS OBSERVED AT LOWER ______jmA____-____MS___ July_____&3_
(W- E)d/60 CULMINATION K IS POSITIVE. CHEKE BY .~L. M DEc
DA FORM 2848, 1 OCT 64
Figure 38. Astronomic Longitude Data (DA Form 2848).
apply the chronometer correction to themean epoch.
(3) Add albebraically the X (+ if west, - ifeast) to the Mean Epoch.
(4) Determine the time interval in siderealunits by subtracting the value of thesidereal time of oh UT for the nearestpreceding date.
(5) Convert the time intervals to minutesand divide by 1444 to obtain the decimal
part of a day. This is the Greenwichcivil date or the UT date.
(6) Use the UT date to interpolate for d1, andd& in table 1 of the Fundamental Cata-logue.
(7) Interpolate for a in the Fundamental
Catalogue including the second differenceinterpolation as explained in Chapter 2.
Since the interpolation factor 77 is equalto the X expressed as a decimal of a dayfor one day stars, then for 10 day starsit is equal to '10A plus 1/10 the number
of days between the date of tabulationand the date of observation. As acheck, subtract the UT date which islisted to one decimal for the date of
tabulation from the UT date of obser-vation computed in (5) above and divideby 10. The first two decimals of thisapproximate 77 should agree with theprecise 77.
(8) Compute Aa by formula [da N,) X d4,+f-da (e) X dE] and enter on form.
(9) If a and LAa are not in the FK,4 system andcorrections to that system are available,it will be necessary to apply these cor-
rections to either a or Aa.
k. The chronometer correction may
mined from. the following formula:
zAT=Co+ (t-t 0 )r
Wherenearest
t is the
be deter-
Co is the chronometer correction of the
preceding chronometer/radio comparison,
chronometer time of the star observation,
PROJECT TABULATION OF GEODETIC DATA(TM 5-237)
LOCATION, MAYADORGANIZATION
ARYLAP4U USA MSSTATION
MAP ASTRO 2 INTERPOLAT/ON FACTORS
se r S6T'2 Ss T3 SET 4
-t +f (14-) /2 43 2S /4 2 38 /,526 20 /6/68 40
t+ At LAST) /3 /6 54 /S14 02 /S 657 17 2647
MEAN /2 59 40 /4 49 00 /6 41 38 /65S2 /4
APPROX. S 08 29 S 08 29 S 08 29 S 08 29G. S. T. /8 08 09 /965729 2060o07 2200o43
S.Tooh U.T /6 07 26 /746 00 /7 46 00 /7 46 00
SID. INTERVAL 2 00 43 21/l 29 3 04 07 4 14 43G.C. D. MY26.084 JUNE /9.0 9/ JUNE /9.1/28 JUNE /9.176
d d + o7 o082 -.2o7 -. olo -. 205 -.0/1 -.203 -. o/3
2L-.005 .02/ -.062 .52/ -.062 .52/ -0162. ,52/
G.C. D 25.1 /9.1 _ _ _ _ _
TABULATION DATE 24.9 /3.9 ______
APPROX.A .02 .52 ____
ACTUAL 17.021 .521 _ _ _ _ _ _ _ _
C'oMPurATON 4EOF' 24 0. 2/4 FOR /0 DAY
TABULATE YDT HCE YDT
°. TIM BY -A,4M sI u '6o3 c QR.T~. - AMS I Oc. 6o3 BYGO984DA FOR 11w51962
GPO *G6647U. I. GOVERN ,1 ParrTah OFWWZ: 1957 - 4211!2
Figure 39. Computation of interpolation factors.
to is the chronometer time of the signal from whichCo was obtained, and r is the rate per minute bywhich the chronometer fails to keep sidereal timewithin the interval containing the star observa-tion. The algebraic signs of Co and r will deter-mine the sign of the correction.
1. Compute (a+Aa-t-At) for each star.m. Abstract on DA Form 2849 (Least Squares
Adjustment of Longitude) (fig. 40) A and (a+Aa-t-At) for each star, set up and solve simultane-ously the two normal equations:
nAX- [A]a+ [(a+Aa-t-At)] =0
- [A]AX + [AA]a- [A (a +- a-t-- At)]= 0
Then use a to determine each stars residual bythe following formula:
AX-Aa+ (a+Aa-t-At) =v
n. In the above least squares solution of thestar sets, it is not anticipated that good observa-tions with present day equipment should result inany v being greater than 0.080 second. If any vis greater than this amount, an examination of thatstar should be made to determine whether anyerrors in scaling, level reading, or other factorsmay be the cause. If the value is not consistentwith the remainder of the star set, it may berejected. Any star with a v of more than 0.2second of time is always rejected.
o. The summary of results is computed on DAForm 1962 (fig. 41).
(1) In the "Summary of Results" observa-tions by a competent observer usinggood equipment should be such that thedifference between the mean observedAX and any individual AX should notexceed 0.04 second. Any star set havingsuch a large residual should be reviewedto determine whether any errors havebeen made or whether one of the stars ofthe set might have an "A" factor whichwas distorting the AX of the set. In afinal examination, any star set having alarge residual not balanced by one, equalto or only slightly smaller, with an op-posite sign is to be rejected.
(2) The probable error of the longitude iscomputed from the standard probableerror formula:
PE=0.6745/(_ 1
In(n-1)
where v=the deviation from the meanand n=the number of values
(3) The correction to the geodetic station,if applicable, is computed by the formulaC=S sin A.H.
C = correction in seconds of arc.S=distance in meters.H= Reciprocal of the meters per second for
longitude using the 4 of the station asthe argument in the extraction from theappropriate tables of meridianal are fromthe Spheroid used.
A= azimuth to geodetic station.An oriented sketch of the eccentricitywill provide a means of verifying thesign of the reduction.
(4) When the x and y of the instantaneousnorth pole are known for a given date,the correction to be applied to an ob-served astronomic longitude to referenceit to the mean pole is:
AX"(arc)= (x sin X-y cos X) tan 4
West longitude is considered positive.
29. Longitude Using Time By Transits of Pairsof Stars Over a Great Circle Approximat-ing the Meridian
a. This method (fig. 42) is suitable for use witha small theodolite or transit, and is the methodused for placing an instrument accurately in themeridian. It consists of selecting two stars, onenorth and one south of the zenith, which willtransit at a convenient time and interval, say 2 to5 minutes apart. These stars must have highazimuth factors, that is, at least 0.75 and prefer-ably higher. The azimuth factor (A) is computedby the formula:
A=sin r/cos 6
A value of the latitude is necessary, preferablywithin a few minutes. The instrument is care-
fully leveled and pointed as nearly in the meridian
as possible by a ground azimuth, corrected pointing
on Polaris, or even by magnetic compass. The
zenith distance of the first star is computed by
r=4-6, and set on the vertical circle. When
the star appears in the field, it is placed on the
horizontal wire, and the time recorded when it
crosses the vertical wire. Without unclamping
the horizon motion, the telescope is pointed to the
zenith distance of the second star, whose transit is
recorded in the same manner. If the telescope
SET NO.LEAST SQUARES ADJSTME NT OF LONGITUDE2
PROJECT LOCATION STATION
MARYLAND MAP ASTRO 2OBSERVER INSTRUMENT (T p andNo.) CHRONOMETER DATE
R. . __ __WILD T-4 No. s6oOPS /24 74 /5'.09' JUNVE 63
STAR A a+Acz-t-tit AA Adat+Aa A. A____ ____ _ ________-t -eta ,_____ _____ ____
S31 -_ -. 368 1..o41 +073 4.o32 -. 0/9-
534 +./70 - 0/5' -. 034 - oi5 j..o028
540 - .1/37 +.0/7 +.o27 +. 0/0 t,.o03
1386 +.o22 -.027 -. 004 +. o23 -. /0
S5/ +.426 -. 103 -. 084 #.019 -. 006
6S55 -. 036 -.028 +~. oo7 0,.3S -.022
1395 -. 244 +.06' ___ __ +.048 -. 02/ fo34
6S63 +./5S -. 043 _ __ ___-.o2-3 +.o20 -. 007
____-.05S2 -.093 +*4391/ -. 0858 9 .013 +00/
+ 8.00000 +.05200 -09300
o - -. 006.50 +. 01162
+.43911 +.08589'
- , ooo34 .-. ooo60
+.438 77 408649
: -.-197/2
NORMAL EQUATIONS
nmx- [A] a+[ (a+ea-t-eAt) 1.=0
-[A] ea+[MA] a- [A (a.&a-tset) 1.=0
COMPUTEDOBY .DATE CHECKED BY DAjITE
- A MS IJu Iy' 63 D.&R. n oka - A MS Iy16DA FORM 2849, 1 OCT 64
Figure 40. Least squares solution of star set.
MAP ASTRO 2 ASTRONOMIC LONGITUDE SLIMMAARY
_____ G. C.D. adaX V
5 ET 1 25.084 MAY w3 +o./86 +0. 024 .- 0.0/2
2 9 o l J N -'3 - .1 74 0 033 9.128
" 1 + 0 .14 0 +0.027 4-.70o
30-o o .0 / 6
S5 /9.248 " -0-222 +0.018 - 0,006
6 .28.5 - -0.,297 -0.004 +0.0/6
______. N.A~ +0.012 Zv2 =. 000918e = ± 50037(or) ± + '0(o
ASSUMED LO GITLJDE A +______I 0.5" 08'' 29.000
MEAN A A + 00.012
MEAN 0SERY ED LoNGiTUDE (rimrI) + 05 08 2 ?.0 12 t S0 0 4
(ARtC) W 77 0 07 /5/ 8 + 7'06
ASTRO. LoNG I UDE MAP AST 0 2) W 770 07' 1/.! f 0
RenucTioAI To GEODETIC STArbON ~ E 02.54
ASTRO. LoNGI UOE NP(AMS 958)) W 77° 07 /2. (4 + .06
TABULATED BY DATE CECED BY DATE
o° - AM'S July 63 D .R. ,ika. - AMS July (03
DA 1 7 962 GPO 921961 U. S. GOVERNMENT PRINTING OFFICE : 1957 0 - 4z11SZ
Figure 41. Summary of results.
LONG/T(/iUD BY TIME OF TRANSIT OF PAIRS OF STARS OVER A
GREAT CIRCLE APPROXIMATING THEF MERIDIAN
STATION: CASS
STARS
22 H CAML
SCANIS MAJORIS
OBSER VAT/ON
S TA S S
22 N CAML.
$ CAMS MAJORIS
SOLUTION OF 4T
dT -1.47Q -/8.4
4T + /.07a + 05.1
/ 074lT - 1.51290
47AT +. 512 90
,2.54 4T
4T =4 80
DATE : /5 FEB. 56
R/G7- ASCENSION (og)
6h 14
6 18
t
6 /3 47.1
6 /8 44.6
-19.688
+ 7497
/2.191
05.5
39.5
T/E: 20h40FS. T.
S
-369 0
- 30 03
(a-t )
+ /8.4
-51
0
0
LATITUD(): 38 08,
9
-314
68
RADIO SIGNAL
/5 FEB. 1956
TIME ZONE CORR.
U. T, /6 FEB.
SID. T. AT 01U.T.
CORR. FOR SID. GAIN
G. S. T.
CIIROA'. T OF SiGNAL
CORR. (4T)
L. S. T OF SIGNAL
G.S. T OF RADIO 51G.
L.S. T. OF RADIO SIG.
LONGITUDE (rjME)
(ARC)
A
-/47
+/.07
2/ 00 00.0
+ 5
2 00 00.0
9 39 53.6
+ o /9.7
ii 40 /3.3
Ch f h S6h 3)154.2
+ 04.8
6 3/ 59.0
1 40 /3.3
-6 3/ 5920
5 0 8 143
77 03 34.5
Figure 42. Computation of longitude using time by transits of pairs of stars over a great circle approximating the meridian.
axis is not accurately leveled, it will be necessary
to obtain bubble readings and apply corrections.
b. The computation procedure is as follows:
(1) Determine the right ascension and de-
clination of the stars from the ephemeris.To accomplish this, the approximatelocal standard time must be converted to
UT by the difference for the time zone.(2) Determine the azimuth factors of the
stars.
(3) Apply level corrections to observed timeof transit. The correction to the time of
transit for the level error is: BXiXd/60,where B, a factor of the star=cosg/cos,d is the value of 1 division of the bubbletube in seconds of arc, and i is theinclination factor as explained under
computation of azimuth.
(4) Write an equation for each star:
AT+Aa- (a-t)=0
Where T is the desired chronometer
correction; A is the azimuth factor; a is
the angle between the meridian and the
line of collimation in seconds of time; a is
the star's right ascension; and t is the
recorded chronometer time. The values
of A and (a -t) are known.
(5) Solve the two equations for AT.
(6) AT+t=the required local sidereal time.
There remains an error due to the
collimation error of the telescope. This
should be made a minimum by careful
adjustment. The effect can be reduced
by observing a second pair of stars with
the telescope in the reversed position,and meaning the values of AT.
(7) Obtain the chronometer time of a radio
time signal by comparisons. Add the
chronometer correction (mean AT) toobtain the LST of signal.
(8) Compute GST of signal.(9) Subtract LST from GST to obtain
longitude.
30. Longitude Using Time By the Altitudes ofStars Near the Prime Vertical
a. This method (fig. 43) is most commonlyused by surveyors. It is frequently used to
determine the LST for azimuth observations inthe absence of radio equipment or of an accurate
longitude. An approximate value of the latitudeis required.
b. The observations are taken on a star near
the prime vertical and having an altitude of from
300 to 500. A number of sets, each consisting of
one pointing in the direct and one in the reversedpositions, are ordinarily taken. The accuracy is
greatly increased by observing pairs of stars at
nearly the same altitudes and relationship to theprime vertical to the east and west. When this
latter system is used, the reversal of the telescope
between consecutive pointings can be dispensed
with and all pointings can be made with the tele-scope in the same position. DA Form 1909,Longitude by the Altitude of Stars Near the Prime
Vertical is used for the computation.
c. The computation procedure is as follows:
(1) Correct the mean altitude (or zenith dis-tance) of each star, for refraction.
(2) Determine the mean time of observationcorresponding to the above.
(3) Obtain the declination and right ascen-sion of the star for the date and UT ofobservation from the ephemeris. Anapproximate local standard time con-verted to UT is sufficient for this purpose.
(4) Apply the following formula:
1 1sin 2 [1-+( -8)] sin 2 ['-(-8)]1 2 2
sin' t-2 cos € cos 8
where t is the required hour angle, 8 the
declination, and 0 the latitude. The
1value of 2 t will be positive for a west star,
negative for an east star.(5) Add the hour angle t (in time) to the right
ascension to obtain the LST.
(6) Subtract the chronometer reading fromLST to obtain the chronometer correc-tion.
(7) After comparing the chronometer and theradio time signal, that is, obtaining thechronometer reading for the time signal,add to this reading the chronometercorrection found in (6) above, to obtain
the corrected chronometer time of thesignal. This is the LST of the signal.
(8) Compute the GST of the signal by therules for conversion of time.
(9) The longitude is the difference betweenthe GST and the LST.
31. Longitude Using Time By the Altitude ofthe Sun
This is, in principle, the same as the method ofparagraph 30. The differences lie only in thecalculation of the sun's coordinates and time con-version. The computation procedure is asfollows:
a. Extract the mean vertical circle readingsand times of each pair of pointings, D and R.
b. Correct the vertical angles for refraction andparallax.
c. Obtain the sun's declination and right ascen-sion for the date and time of observation.
d. Apply formula, preferably using DA Form1909. In this case, t is the local apparent time(LAT).
e. Convert LAT to local mean time (LMT) bysubtracting the equation of time.
J. Subtract the chronometer time of observationfrom the LMT to obtain chronometer correction.
g. After comparing chronometer and time sig-nal, add the chronometer correction to thechronometer time of the signal to obtain the LMTof the signal.
h. Subtract LMT from UT of signal to obtainlongitude.
LONGITUDE BY THE ALTITUDE OF STARS NEAR THE PRIME VERTICAL (TM 5-237)
PROJECT STATION
LOCATION ORGANIZATION DATE
Ohi Aisfi., Dc.43 -INSTRUMENT (Type and number) CHRONOMETER APPROXIMATE ANGLE BETWEEN STAR AND POLARIS
0
OBSERVER CHRONOMETER TIME OF ANGLE READING
COMPUTATION OF TIME ____ _________
STAR{Eas } h~fSTAR { ih. in. a. 0 1 h. an. a. o I
Chron. Reading Zenith Dist. 18 -f6 582 45 o 2.20 l 1 01 27.0 4 0 8Refraction _________ + +__________
Corrected Z. D.=- 45f0JL. ________ 44 1 24-log cos~ # * S8Z932 4 AL 1 .234log cos a a 8 (335 23 115
log cos #+log coB 8=1og D, #-a 9. 6413230 IL46,4L .Z14330. L 46 4Llog sin [f+(-)J, I9+(-8 1L6842.3 -3L 23 9. 91/L 30 02.log sin i it-(m-anJ +} [r-(m-a)1 2J.iZ2Q.Q7a.Z L 37~b IS 9 871 13....2- ....Sum two log sines=.log N ._____0__
log N-log D=log sin' j t 59.2472&2 9229 1L4.
h. m. a. h. an. a.
t (time) t t(arc) -74 4/ 04.3 3l 0,2 0 4J .3/k3LL.2 22ZORight ascension of star 2 4 0.Sidereal time _______________3L
Chronometer reading _____________ .9 0L..2Z0Q. ______
Chronometer correction _________________ QdR
The chronometer correction is plus if the chronometer is slow, and minus if fast. Carry all angles to
seconds only, all time to tenths of seconds, and all logarithms to seven decimal places.
COMPUTATION OF LONGITUDETIME F RAIO SGNALTRANSMITTING STATION
Chronometer reading (Sid. T.) 9 S Std. timel71 mer.18 2 47
Chronometer correction TZC -+ -1 8I .5
LST_________f UT 23 52 479
TZC= time zone correction Sid. T. at Oh UT 4 43 47.804Longitude (X)= GST -LST Corr. (table III) 3 £37
GST 2 0 3-7
L.ST -23 13 45-?ooLongitude (a) (arc) 17""14 Longitude (X)
COMPUTED BY DATE CHECKED BY DATE
DA, F EB 571 909
Figure 43. Computation of Longitude By the Altitudes of Stars Near the Prime Vertical (DA Form 1909).
Section V. COMPUTATION OF LATITUDE AND LONGITUDE FROM OBSERVATIONS
MADE WITH THE ASTROLABE
32. Basic Proceduresa. The astrolabe is an instrument used to obtain
latitude and longitude by the observation of the
times at which stars cross a circle of fixed altitude.
These stars are distributed in the observing pro-cedure, to each quadrant of azimuth, to minimizethe effect of refraction.
b. The solution of the observation set requires
an approximate or assumed geographic position
of the observing station. This position should be
near the true position in order that the intercepts
in the graphic or analytic solution will be small.
c. Two procedures for reduction of an astrolabe
observation set to a latitude and longitude are
discussed. The graphic solution is the established
method for computing third and fourth order
astronomic position and the analytical solution is
used for the reduction of first and second order
observations using impersonal timing equipment.
The number of significant figures used in the
computation is dependent upon the order of the
work.
d. The Field Record, DA Form 1910 (Observa-
tions, Astro-Fix) (fig. 44), contains the names ofthe stars, approximate azimuths of the stars ob-served, observed chronometer times, and chronom-
eter corrections or stop watch readings. Whenan electronic method of timing is used (as in theexample shown), the scaled results of the timecomparisons and corrected chronometer times ofstar transits are recorded in column (d) of DAForm 1910. This form is used both for an observ-ing program and for a record of the field observa-tions. While sidereal time equipment is preferred,mean time equipment may be used. If more thanone transit wire is used, it is customary for loworders of work to mean all transit readings on theseveral wires, but for higher orders of precision asingle wire transit is required.
e. Computations for observations up to 700latitude may be made by the so-called sine-cosineformulas using DA Form 1911, Altitude andAzimuth (Sin-Cos) (fig. 45 ,()), or DA Form 1912,Altitude and Azimuth (Sin-Cos) (Logarithmic),(fig. 45 ®). Astrolabe observations at higherlatitudes are of little value. The sine-cosineformulas are as follows:
sin H,=sin 0& sin +cos a, cos cos t
sin Z=cos sin t
cos H
where H, refers to the computed altitude of thestar distinct from the observed altitude; 0 is theassumed latitude; and t is the hour angle of thestar in the shortest direction from the meridian.
f. The values of H, are computed for each star.Each value is then subtracted from an assumedapproximate altitude of the fixed circle to obtaindifferences called intercepts, which are thenadjusted to reduce the assumed position to thecomputed position. The azimuths of the stars areused in this adjustment.
33. Computation Procedure
a. From the recorded time, compute the UTfor the purpose of finding the declinations andright ascensions of the stars. An average timemay be taken by inspecting the record, or if theobservations are at fairly well distributed intervals,the mean of the times of the first and last stars issufficient.
b. Enter all the stars and their declinations andright ascensions on the form. Declinations shouldbe to the nearest second of are and Right Ascen-sions (RA) to the nearest OSl of time, for thirdorder work. If impersonal timing equipment isused, and higher orders of precisions are desired,the declination should be determined to the nearest0.01 second of are.
c. Correct the recorded time for the chronometererror found by comparison with radio time signal,and covert fromn GST to LST by subtracting theassumed longitude.
d. Compute the numerical difference betweenLST and RA of each star in such a manner as tomake the difference always less than 1 2h
. One
method is to apply the equation, t=LST-RA,and if t exceeds 12
h , subtract t from 2 4h . Dis-
regard algebraic signs.
e. Solve equation for H,. Complete the com-putation for all stars up to this point, and inspectthe values of He for uniformity. Reject anyoutstanding values after checking to eliminateerrors.
PRJCT SA OBSERVATIONS, ASTRO-FIX',(A(TM 5-237)
LOCATION ORGANIZATION DATE
R YUK Y!/ 45* 4C,,4MS 22.,43/% O . 69WATCH FAST (-) SLOW (+) ASSUMED LAT. (LA) ASSUMED LONG. (XA) STATION
300 27 03.2 08' 4o"52,r FgqaINSTRUMENT (Number and type) OBSERVER As #0. / RECORDER"
sTR~LAE/AToMATc TIMER R.L. 1ONES se = 0.00 W. A: AIMSDESCRIPTION OF STATION; REFERENCES, CROSS BEARINGS. SKETCHES, ETC.
For an astrolabe, put L. Sid. T. in column (e); for a transit, put vertical angle in column (e)
CORRECTED LOCAL SID. 7NR STAR WATCH TIME STOPWATCH TIME AZIMUTH (a) REMARKS
(a) (b) ____ ___(C) ___ (d) (e) ____ (f) (&)
H M S Mf S H M 5 0
752 rSAGITTAE __ ______ 8 17 /0.403 22 00 130.051 255 37_______
1602 4~ P/ScluM _________/8 2/ 2.215 05 12.635 150 46
768 6 DEL PHI NJ _ ______ /8 27 00.692 /a 21957 235 /9
892 C PISCwM~ THESE TWO COL MAIS /8 44 /26/5 27 43.724 /43 07_______
33 A(ANDROMEDAE NO A PLIC SL IF /8 49 50308 33 15.323 65 17 _____
902 w. P/SC/elM TI E CAL D FROM /8 56 04 45 39 3.699 /38 37 ______
765 2'cY6NI CH ON GfPA N. /_ 8 .57 L70O- - 41 03.356 298 29
42 4
ANDROMEDAE RECORDS. 19 02 18.254 45 45.316 7/ 0/_______
17 9CASSIOPEiAE~ ____ 19 06 07.308 49 34.997 3/ 44 _______
777 a( CYGNI (DENEB) 1__ 9 /0 2.862 54 11.305 308 43 _______
45 I P/SC/A 1_ 9,/7 18.562 23 00 48.089 88 0/ ______
78 CYGNI ___/9 22 32.838 06 03.226 285 S/L ______
2 I8
CASSIOPE/AE __ __/9 28 17500 14.8.831 14 24 _______
804 1 PEGAS / 1_ _ _ 9 " 40 45.769 24 /9149 2 56 /0._ ____
52 SI ANDROMEDAE ____19 45 23.119 28 57258 44 /2 ______
64 a rRIANGL/I ___9 8~ 22. 104 31 56.733 83 02 ______
So '? P/,Sclum 1_ ______ 9 53 10.327 36 45.745 /14 23_______
5 i Vcya~ __ 19 54 20.7/5 -_37 56.327 287 43 ______
73 4ANDROMEDAE P __ 19 59 8.227 42 59667 57 28_______
66 '8
RIET/S _ 20 02 39885 46 16.863 /0/ 39_______
1 7r2 CYtN/ _ ____ __20 06 12.249 49 49829 3/7 24 ______
74 a ARIET's_______ 20 to 11.769 _ 53 49984 25' 30_______
878 a' P/SCW __ __U_ 20 23 0 192 24 06 48&533 206 33 ______
COMPUTED BY 7DATE , CHECKED BY DATE
NvOV.59 M . Nov.65
DAIFEB571.1 U. S. GOVERNMENT PRINTING OFFICE: 1957 0-420636
Figure 44. Observation schedule and field data.
757-381 0 - 65 - 6
PROJECT SPAN ALTITUDE AND AZIMUTH (SIN-COS)(TM 5-237)
LOCATION ORGANIZATION DATE
R YUIK YU /IS. 0151A MS 2.4Nov 59STATION ASSUMED LAT. (LA) ASSUMED LONG. (lXA) WATCH FAST (-) SLOW (+)
ERA SU30 27 #o3.2 1089 40 52.'0701INSTRUMENT (Number and type) OBSERVER
ASTROLABE/AUTomATic TIMER R.IL. JONES S4 f ~0. /0 SF =0.0012 3 4
Star 7 2 /602 748 &9'2
Declination /a 9°2 3.267 03 ° 36' 21.079 //1 /0' /2.077 05 024' 38.074
Watch
Corr. slow, fast- 1 ___
UT -.9 0917 /0.403 -09I 2 522/5 09 27~ 00.692 09 44 196/5G. Sid. T
2 2 4 3 doh UT j4g0 6.049
Mean time itrato sid. time (corr.)1 0/ 31.529 01 32.30/- 0/ 33.146 __0/ 35990G. Sid.'T. /3 /9 37981 /3 24 20.565 /3- 29 29887 /3 46 5/.654Long. (NA) E+, W- f 089 40 52.070__
Local Sid. T. 22 00 3005/ 22 05 /2635 22 /0 21.957 22 27 4.724
R. A. /9 56~ 57615 23 0/ 50.370 20 3/ /72/9 23 37 53.454
M. A. +02 03 32.436 00 56 37735 +0/ 39 04.740 01 /0. 0930
M. A.(are) t 300 I306..54 345 50 33.98~ 240 46 //V34 2 27 3.0o5
Sin LA 0 506 799'63 _____
Sin a 0. 33/ 94 742 0.0628925/ 0.19372/07 0.0 9429208A (product) 0 /6823083 0.03187390 0.09817777 0.047787/9Cos LA 0.86206388 _____
CoSa 0.94329789 0.9980203/ v.98/06665 0.99554458COS t +0O858/9798 +o. 94962821 +0.9079988/ '+0.95349955B (product) 0.69787204 0. 83422667. 0.7679250/ 0.8/83/.527A o.16823083 0. 03/873 90 0.09'8 /7777 0-047787/9Sin Hc* 0. 8(o0/0287 0.866/005'7 0.866/0278 0.86o6/0246
H0 60 00 31.9628 .607 joi 0124760 700j319256 60j 00 31.j7934
HO 60 00 33.5800 60 00 33.5800 604 00 3.5800 60 00 L3.5800intercet ne pt "To") H°+ 01.6/72 + 0.576 0.64 fJ 017866.Cos a 0.94329789 0.9980203/ 0.98/0.5665 v.99554458Sin t 0.5/33/884 0.24458361/ 0.4189775 0.30/39443C (product) 0.4842/258 0.2440 994/ 0.411036 00 0.30005/5 9COS HC 0.49986580 0.49986 972 0.49986.59/ 0.499866.51Sin Z (C-. Cos Hc) 0.968 68516 0.48832599 0.8222 9252- 0.60026344
Z 75 37 2 1 05kl 6 S
Azimuth ZN 255 37 /SO 46 235 /93 02~ L7*When L, and a have same sign: Sin Hc=A±B if M. A.<90' and A-B if M. A.>90
0
When L and a have opposite sign: Sin Ho=A-B if Mv. A.<900
and A+B if M. A.>900
COMPUTED BY DATE , CHECKED BY DATE
JAN. 60 7. JAN. 00
DA. FORM '301 U. S. GOVERNMENT PRINTING OFFICE :1957 0-20550
( Altitude and Azimuth (Sin-Cos) (DA Form 1911)
Figure 45. Computation of intercept and azimuth.
PROJECT SPNALTITUDE ADAZIMUTH (SIN-COS)(LGRTMCSPAN AN (TM 5-237) (LGRTMC
LOCATION RYKY I.OGNZTO(44A,1DATERYLKYU/~ RGAIZAIONC/SMS 2.31 N1ov x59
STATION ASSUMED LAT. (LAI ASSUMED LONG. ") WATCH FAST (-) SLOW (+)E/RABU 300 27' 03.'2 08 40' 52.070
INSTRUMENT (Number and type) OBSERVER
ASTRo*ABE WIrth AUTO2MATIC TIMER R. L. ,JONES cry u:*O./D dE 0.00Star /f21 2 6 42
Declination (6) ± x'19 23 /3.267 +03 36 2/. 079 /0 /2.07 4-05_24_38.07
Watch
Corr. slow, fast- ±
UT -9 h 0.9 /7 /0 403 09 2/ 522/5 09 27 00.692 09 44 /96/5G. Sid. T
2 2.4
3dph UIT 04 00 56.049
Mean time intervaltosd ie(or)0/ 31.S259 01 32. 30/ 0/ 33.146 0/ 35 990
G.Sid. T. 13 /9 37 98/ /3 24 20.65 /3 29 29887 /3 46 51644Long. (XA)E+, W- t 08 40 S2.070_____________
Local Sid. T. 22 00 34.05/ 22 05 /2.635 22 /0 2/. 967 22 27 43724R. A. /9 56 6 76/ 23 0/ SO 370 20 3/ /72/7 23 376S36 54
M. A. "02 03 32436 OD 56 3Z 7365 0/ 39 04.740 O/ /0 09930
M. A. (arc) t 30 63 06.54 34550S 33.958 24 46 /. /0 342 27 3/ o
Lg sin LA 9 704 836 29Lg sin a 952/ 06930 8.79859894 9287/7684 8.97447522
Log A (sum) 92250590559 8.50343523 8.9 920/3/3 8.67931//I
A 0. /6823083 0.03/87390 0.098/7777 0.047787/9
Log cos LA 9. 93553 945______
Log cos a 997464886 9.999/3 938 9 99/6940 9 9.9 9805072
Log cos t 9.93358749 998660524 995808527 9.97932049LogB (sum) 984377580 9 92128407 9.8853/881 99/292066B 0. 69787204 o. 83422667 0.76792501 0.8183/527
Sin Hc* 0.866/0287 0.8665/0067 0.86/0278 0.86/0246
Log sin H0
H0 60 00 31.9628 0 00 3/0124 60 00 319256 60 00 3/.7934
Ho 603 00 33.5800 60 00 33.5800 60 00 33.5800 60 00336800Intercet necpt (To>)H0 = - 0/.6172 4# 02.5676 # 0/65544 * 0/ 7866
Log cos a 997464886 9999/3938 9.99/6 9409' 9.998 06072Log sin t 97/038720 9.38842736 9622/8578 9479/3522
LogOC (sum) .68503606 93876674 9 6/387987 9 477/9594Log cos.Hc 95988.5342 9 69885688 9' 698853.55 969885403Log sin Z: (diff.) ~9986/8264 9568870986 99/502632 977834191Z 75°37" 2901/4 55a/ 3653Azimnuth ZN 255" 37' ISO D46' 235 0 / 9 " ,43 °07'
*When L and a have same sign: Sin Hc=A+B if M. A.<900
and A-B if M. A.>900
When L and a have opposite sign: Sin H 0=A-B if M. A.<90° and A+B if M. A.>90°
COMPUTED BY -_P DATE S9 CHECKED BY .DATE
FXoV. '5 J -k ~/5
D FORMD 1 7191U. S. GOVERNMENT PRINTING OFFICE :1957 0-420844
® Altitude and Azimuth (Sin-Cos) (Logarithmic) (DA Form 1912)
Figure 465-Continued.
f. Select an arbitrary value for Ho to an evenminute or second of are slightly greater than thehighest retained value of He. The quantityHo-H,, the intercept, will then always be positive.
g. Solve formula for the azimuth, A, of eachstar using a single value of He.
h. On DA Forms 1911 and 1912, the firstquadrant angle is represented by the letter Z.To determine the quadrant the azimuth anglefalls in when measured from north (ZN) one of thefollowing methods can be used.
(1) There is no satisfactory way to show thequadrant of the azimuth angle by usingsigns and therefore they are omitted onthe computation forms DA 1911 andDA 1912.
(a) A North star with a negative meridianangle (MA) would be in the firstquadrant.
(b) A South star with a negative meridianangle would be in the second quadrant.
(c) A South star with a positive meridianangle would be in the third quadrant.
(d) A North star with a positive meridianangle would be in the fourth quadrant.
(2) When the assumed latitude and a star'sdeclination are nearly the same, de-termination as to the star being a northor south star can be determined fromDA Form 1910, Observation, Astro-fix,where the approximate azimuths of theobservations have been computed.
34. Graphical Solution
DA Form 1913 may be used for plotting theintercepts derived above (fig. 46). This formdoes not require a protractor. The solution is asfollows :
a. Plot the intercept of each star to a suitablescale from the central point of the plotting circlein the star's azimuth direction if the intercept ispositive, or in the opposite direction if the inter-cept is negative. Small variations are displayedbest by adding a constant to all intercepts beforeplotting. In the example given a constant of 50seconds (50") has been added to each intercept.b. Draw lines through the points so laid off,
at right angles to their azimuth lines. These arethe line of position.
c. By trial, draw a circle as nearly tangent aspossible to all the lines of position. A set ofconcentric circles drawn on a piece of transparentplastic will facilitate this operation (fig. 47). Due
to errors in the observations, the circle can neverbe drawn truly tangent to all lines. The dis-
crepancies should be equalized in amount anddirection.
d. Mark the center of this circle. This repre-sents the true position of the station.
e. Measure the seconds and fraction of a second
by the scale of the graph in the direction parallelto the N-S axis of the plot between the assumedand true positions. This amount applied to theassumed latitude gives the observed latitude.
f. Measure similarly the E-W difference be-tween the assumed and true positions. Dividethis value in seconds of time by the cosine of thelatitude, and apply the result to the assumedlongitude to obtain the observed longitude.
g. The algebraic signs of the differences areapparent from the plot.
h. Should the assumed position be in error asmuch as 10 minutes of arc, the scale of the plotwill be too small for the required accuracy. Inthis case, plot four stars in different quadrants toobtain a close approximation of the position.Recompute with this as the assumed position.
35. Analytic Solution
a. When the circummeridian intercepts andazimuths from North are known, the observationequation can be written in the following form:
v=cos0o(AX+AT){sin AN}+ A¢(cos AN})
{-1} AH- (Ho-HC)
Where: {}, braced quantities indicate deter-mined coefficients;
AX, is the change in longitude from
assumed longitude;A¢, is the change in latitude from
assumed latitude;AH, is the adjusted intercept from ad-
justed position,AT, is the correction to the radio time
signal;
AN, is the azimuth of the star from
North;(Ho-H,), is the circummeridian intercept
based on assumed position of station;
v, is the residual.The international sign convention is used. Thisconvention specifies that longitudes east ofGreenwich, hour angles west of the meridian, andlatitudes and declinations north of the equatorare positive. Corrections are to be added al-
PROJECT SP N ASTROLABE PLOTTING(TM 5-237)
LOCATION ORGANIZATION DATE
RYUKYLS /SLAND5 USA MS 22.43i ivovX59
ASSUMED POSITION CORRECTIONS PLOTTED POSITION
LATITUDE (0) 30027I 032 Al om O52 S LATITUDE (~0
) 27 " 0, 0 I\J
LONGITUDE () 3 0 0/' E Da Q."6 E LONGITUDE (a)1)30"c ' f07 £COMPUTED BY IDATE CHECKED BY DATE
___6_____c__ 63
DA TFORM 71913
LIU. .GOVERNMENT PRINTING OFFICE :1957 0-420713
Figure 46. Graphical solution.
Figure 47. Concentric circle overlay.
gebraically. See sample formation of observation
equations (fig. 48).
b. The normal equations are obtained by re-
quiring that the sum of the squares of the residuals
be a minimum. This requirement is obtained by
finding the sum of the partial derivatives with
respect to each variable in the squared observa-
tion equations. The symmetric normal equations
are namely:
I. [(sin AN) (sin AN)] cos 0 (AX+ aT)
± [(sin AN) (cos AN)1AO
+ [(sin AN) (-1)]H- [(sin AN) (Ho-H,)]
=0
II. [(sin AN) (cos AN)] cos 40 (AX+ AT)
+ [(cos AN) (cos AN)1A4
+ [(cos AN) (-1)AH-H[(cos AN) (Ho-H)I=0
III. [(-1)(sin AN)] cos Ok(aX±AT)
± [(-1) (cos AN)IA4-1-AH+ [(Ho-Ho)]=0
The brackets indicate a finite summation of the
set. These equations are solved by the Doolittle
Method. See sample formation of normal equa-
tions in figure 49.
c. Substitution of solved values back into the
observation equations determines the residuals
(v's) for the star intercept observations. See
figure 50 for sample forward solution and back
solution and figure 51 for residuals and probable
PROJET ~ P Al TABULATION OF GEODETIC DATA%J .V I(TM 5-237)
LOCATION, U Y S ORGANIZATION USA4
STATION ' JSET/
RAI5U.
L SIN AN COS AN AH-(H 0 --/c) :0i/ - 0.96087 - 0. 2483 -/.0000 - /.6/72 - 3. 8342
2 + 0.4883 - 0.8727 - 2.5676 -3,95203 -o.8223 - 0.5691 -1.6 S44 -,~4.04584Re
ct d ly O s r r
5 +oi.6003 -0.998 - /. 766 - 2. 98 6/
6 1+0.9079 . o. 4192 -2.680.4 -2.35337 #0.6613 - 0.7602 ______ -2.0852 - 3.17418 -0.8788 +0.4772 -/1.7728 - .17449 +0.9454 +,6.3260 - 2.9S/8 - 2.6 804.
'o .i o.6249 4+ 0.865/2 -1. 4232 - 1.0o4 7/
1i - 0.779' + 0.6264 -0.1357 -1.2888/2 +0.,9993 4-0.0369 -__3.____ -365461 3.50 99
/13 -0o.9620 +#0.2732 ______ -2.7266 --4.415414 f o.2465 + 0.9692 -____ -2. 4/65 -2.2008/5 .- o.97/1 - 0.2388 -1.46,96 -3.8795/6 + 0.666 40.717S -____ -2. 5742 -2.160/17, t o.9928. + 0.120/ ______ -137. 37/2 - I. 2583
/8 +.0.91/0 -0.4125 _____ -3.6839 - 4.1854
19- -. 9528 +0.3036 -1.894/ -3.5433
20 + o. 842'7 + o.6384 ______ -2.2689 - 1.88 78
21 f 0.9795 -0.20/6 _ ____ -2.2625 - 2.4846
Z2 -0.6767 +0.7362 ______ -1.4412 -2.38/723 +o.9948 -0.10/7 _ ____ -2.3882 -2.495/
24 -o.447o -0.8945 -2.1511 -4.4926
____ ____ ___ ___ __ _ ____ ___- 67.4307
TABULATED BY DATE CE D BY DT
-JI.Qa AN. 60j . 7 I JAN.
DA t FOR&..1962 GPO 908847n1. S. OOv3nxMDff PRIDIUIM OFICE : 19ST 0 - 4111
Figure 48. Formation of observation equations.
PROJECT TABUTION OF GEODETIC DATASPAN (TM 5-239)
LOCATION, ORGAN IZATION
RYUKYC/ ISLAND USA MSSTATION E
ERASU FORM4ATION *OF NORMAL. EQUATIONS 6
2coLes 2 x2 .tZCOLS 2 x3 +X1coLS 2 x4 +FcoL's 2 x 5 + Fcois 2 x
Z sNAN sN + Z INAN"cos As + ~SN N .&H + Ls1N AN "(Mo' c A +25IN A N4'c10
:+(+054776)t (-0.2103 + (-3.3324) + -/4.25310) +1 (-2.3/87)
+ Zcois 3x3 + hCoLes 3x4 + ScoLs .3x5 + Icotss 3X6+ FcosAm ecosAN + EcosAN * aH + Icos AN(-c +ZcosAN.(i4a
-+ (75232) + -1.3059) + (-/./969') +-(+4.8/00)
+ZGCOL's 4 x4 + ZCOL's 4 x5 + Fcol's 4 x6
_____+LaH~aH + a-jo-O+ H Z.
+ (± 23.oooa) + (+49.06290)+ (+ 674307)
COEfFFICIENTS ro NORMAL E L'ATIONS _____
I + 15.4776, -0.2/03 - 3.3324 - /4.2536 - 2.3/87Ir +-7.5232 - 1.3059 - .1096 9 + 4.8/00
lrE + 23.0000 +- 49.0690 + 674307
TABULATED BY DATE CHECKD BY DATE
0. 6(9. N/~eo - '4MS JAN .00 G.T.T-&.)nA - AM-S JAN.600
DA a :?:'e 51962SPO 908847
U. S. GOVERNMENT PEUTD= OFFCE : 1951 - 431162
Figure 49. Formation of normal equations.
PROJET CO NI I TABULATION OF GEODETIC DATA.JU~I' I(TM 5-237)
LOCATION, ORGANIZATION
SAINRUY SNDUASs IERA BO FORWARD SOLUTION AND BACK SOLUrIONV
I + S. 4776 -0.2103 -3.3324 - 14.2536 -2.3/874 L = + 0.0/36 +0.2/53. + o.9209 +0.1498
IZ+ 7.5232 - 1.3059 - 1. / 96 9 + 4.8/00
-____ -0.029 -0.0453 -0.1938 - 0.0315
_____+ 75203 - 1.35/ 2 - 1. 390 7 +.4.7785
0 +.1797 +o.1849 -06354
Ill +23.0000 4 49.0690 +67.4307-0.7/75 - 3.0688 -0.4992
- 0.2428 - 0.24 99 1+ 0. 8587
+22.0397 +45.7503 +67 7900
A H =- 2.0758 - 3. 0758
4H =-2.07 58
4=-2.07.58 (+0,797)+ 0./1849 - 0.1881
aL=- 2.0758 (+0.21.53) -0.1881(+0,0136) + 0.9209 =+0. 47/4
( 4 & + 6) ARC - AL/cos o + 0.547 k X -4(411 +T)-a ±D-at
4AA+ aT) IME =aL/6 cos a )++.036 .08'7 40O, 52:070a__ __ _ +,&T rI E =+ 0. 036 ± 0.009
CORRECTION To SIGN L-oT -Go0.001)DIURNAL. ASBER TION f:= 0.277 Cs 0 (E+ W- -+ 0.0/6
T______ RANSM~ISSION TIME 4t =- o.005
A TIME =084' 4 0 "' 52. ff6 ± 0,009
A Apt, 130° /3" 017 74 t0.13
CRC 300 27' 03.20
o - 00. 19 ±0.17
A~ECG = 00.00 ______
g Ed =) - 00.09 ELEVATION of ERAU~ =6(00.06 30M7 2,2 ±01
TABULATED BY0 DATE CHECKED BY DATE
TA5.N. AcLi - M A.10 .T7Ywu. M A.6
DA IFee 571 962 GPO 921961 U. S. GOVERNMENT PRINTING OFFICE : 1957 0 - 421182
Figure 50. Forward solution and back solution.
PROJECT SPAN TABULATION OF GEODETIC DATA'.JI~I1(TM 5-237)
LOCATION ORGANIZATION
RY/KYU~S /SLAND USAMSSTATION $
ERA&BU RIDU5/QALS AND PROBABL.E ERROR
NO. AV si(AN 4 " cos AN. (H-V
/-0.4566 +0.0467 + 2.0758 1.6659 -0.04872 +0.2302 +0./642 2,4702 I+ 0.09743 -0.3876 +0.1070 1.7952 -0.14084 ______ REJEcrLD 8Y O8SERVER
5 + o. 283o +0,1604 2.5092 -0,722(o6 .,#0.4280 - 0.0789 2.424 9 + o.2555
7 . 0.31/7 +0o.14/1 2.5286~ -... 44348 -0.-4143 -,0898 1.57/7 4 -20/11
9,-+0.4457 -0.06/3 2.46~02 041
/0 4-. 24 74 -0o.1/60/ 2.1631 - 0.739
11 -0.3676 - 0.11/78 ______ 1.5905 - .4-648
12 +0.47/1 -0.006o9 _ ____ 2.5400 41.006113 -0o.4536 -o. 05/4 .5709 +1.155714 +0.1/62 -0.1823 _____ 2.00,97 40.4068
IS - 0.45/8 o. 0449 1.6629' +0.006716 +0.3284 -0.1350 2.2692 0. 3050
/7 +0.4680 - 0.0226 2.52/2 -1.1500/8, +0.4294 + 0.0776 2,5828 f 1.i0/
/9 - 0.44 91 - o.057/ ___.__ S6.$Y96 +0.3245
20 +0.3972 -0./013 2.37/7 -0o.1028
2/, +0.4617 + 0.0371 2.5754 - 0.3/2 9
22 -0.3190 -0.1385 1.6/8-3 -0,1771
23 + o. 4689 1-0. 0/9/ 2,.56 38 -0.175624 -o.2107 +~0.1683 2.0334 +0.1177
__I~~~i~~iU~~iv =±=068 :c'9_ 9.1929
/ =t FV } 23-3 =0 .68
Co ___ - ' o. (2 / - 4 .19 e -!o 5 c~ =+0.13rM2 . 7. 70r oe
0 t±./ /44 e _± , O64 5 am -o. 095
TABULATED BY DATE I CHECKED BY DATE
'§.K - A M5 A. 60 G. T rc, AM$S JA. 60DA FORM 16 GPO 921961 U. S. GOVERNMENT PRINTING OFFICE : 1957 0 - 421182
Figure 51. Residuals and probable error.
error determinations. After the residuals are
obtained, the probable errors are obtained with
the following equations:
P= [vv]
S n-3
x (arc)= P 1- [sin 2 AN] cos 4o
ax (Time)= a x (arc)
-= ±[cost AN]
o'-H
e= ± 0.6745a
The statistical results are given in figure 51.Further rejection may be considered by using
Chauvenet's Criteria, table Xl in appendix III.
d. Considerable savings of computation timeand effort will be realized by rejecting observa-tions before the observation equations are formed.Plot azimuths and intercepts on DA Form 1913as is done in paragraph 34. A transparent over-
lay is placed over the plot of azimuths and inter-cepts. The overlay is inscribed with a numberof concentric circles whose diameters differ by
two times the rejection limit. The rejection liinitis generally five times the probable error that will
be tolerated for the class of work. Thus, for
first-order astronomic work, the probable error
should be less than two tenths of a second (0'2) ;
and therefore, the rejection limit would be set at
1 second (1'.'0) and the diameters of the concentric
circles would differ by 2 seconds. The overlay is
moved until a ring is found into which most
plotted intercepts will fall. Those intercepts
falling without the limiting rings are rejected.
The formation of observation equations proceeds
as in paragraph 34a.
e. The graphic and analytic determinations ofastrolabe astronomic position are concluded withcorrections for elevation, mean position of thepole, correction to signal, transmission time, and
diurnal aberration.
(1) Corrections to latitude.
(a) When the observing station is at someelevation other than sea-level, a cor-rection must be applied. This is a
correction for the curvature of thevertical or, otherwise stated, for thefact that two level surfaces at differentelevations are not parallel but convergeslightly as the poles are approached.This correction can be determined byone of the following equations:
AE=-0.000171 h sin240
AOE=0.000052 h'sin24oin which h is the altitude of the ob-serving station in meters, h' is thealtitude of the observing station infeet, and ,o is the assumed latitude of
the station. The correction is alwayssubtracted when the observing stationis above sea level.
(b) Where the greatest accuracy in theastronomic latitude is required, as infirst and second order astronomicobservations, it is necessary to reducethe observed latitude to the meanposition of the pole as is outlined in
paragraph 24.(2) Corrections to longitude. There are four
corrections to the longitude, two causedby small errors in the time, one causedby a small error in the observed altitudes,and the fourth caused by the variationof the pole from its mean position.
(a) The transmission time correction isgenerally computed at the same timeas the chronometer error is determined.This computation is outlined in para-graph 28e.
(b) The correction to signal is the cor-rection to the time service clock as
transmitted. The radio time service
bulletins contain this correction gener-
ally for OhUT and 1200 h UT for the
date and year of observation. See
example in figure 37 and explanationin paragraph 28e.
(c) Because of the rapid rotation of theearth, a star when observed is ap-
parently displaced. The displace-
ment is in the direction of rotation of
the earth so a star toward the east
will appear to be at a slightly loweraltitude than the true altitude of the
star. In the same way, a star toward
the west will appear to be at a slightlyhigher altitude than the true altitude
of the star. This effect is known as
diurnal aberration. The correctionto the longitude of the observingstation for this effect is found bythe equation:
AXD=0"'2771 cos 40 (ARC)
in which AXD is the diurnal correctionto longitude and €0 is the assumedlatitude. The correction is added tothe east longitude and subtractedfrom the west longitude.
(d) Where the greatest accuracy in theastronomic longitude is required, asin first- and second-order astronomicobservations, it is necessary to reducethe observed longitude to the mean
position of the pole as is outline in
paragraph 28o.
f. The corrected astronomic positions are re-corded on DA Form 2850, (Astronomic Results)(fig. 52). This report card is described in para-graph 36.
Section VI. ASTRONOMIC RESULTS
36. Use of FormThe results of the computations of astronomic
data should be summarily tabulated on DA Form2850 (Astronomic Results). This form was de-signed to provide a final summary of the astronom-ic position, azimuth, and LaPlace azimuthcomputations for a station, and to further pro-vide a convenient reference source for this infor-mation. An explanation of its preparation isoutlined.
37. Tabulation of Resultsa. All entries on the astronomic results form
must be complete in all respects to avoid anyambiguity. This involves an accurate stationdesignation, appropriate hemispheric referencein the latitude and longitude entries, referencepole for the azimuth entry, unit of measurementfor the elevation entries, and careful identificationof personnel, equipment, and dates. The formshould not be considered complete until allentries have been verified by someone other thanthe person who made them.
b. The latitude, longitude, and azimuth entriesare extracted from the respective computations.If cardinal directions are used, as on the exampleform in figure 53, there is no chance of error inthe proper application of reduction data in thelatitude and longitude entries. Care must betaken in indicating the proper sign of the cor-rections applied to the azimuth entries.
c. The present design of the astronomic resultsform does not include provision for certaincorrections which are not habitually computed.Effects of polar motion on latitude, longitude,and azimuth are not included for this reason.Occasionally an astronomic azimuth is observedfrom an eccentric station and the necessaryeccentric reduction may be entered as shown onthe example.
d. The computation of the deflection com-ponents and the LaPlace correction for azimuthcan only be made if the geodetic position of thestation is known. It should be noted that thesign of the prime vertical deflection will be in-fluenced by the consideration of East longitudesas positive, and west longitudes as negativevalues. Also, this consideration will affect theLaPlace azimuth and the note concerning the
application of the correction must be carefullyobserved, i.e., aG=-aN- LaPlace Correction (sub-
tract algebraically). The deflection in meridian
is computed considering the sign of north latitudes
as positive and south latitudes as negative.
e. A sketch of the geodetic connection should
be prepared from data available from field recordsand computations. Reasonable care should be
taken in depicting the relationship of the points
involved. Sufficient data should be provided in
order that eccentric reduction computations can
be made.
ASTRONOMIC RESULTS STATION ERABU(TM 5-237)
PROJECT SPAN LOCATION RYUKYU ISLAND
LATITUDEOBSERVER AND ORGANIZATION INSTRUMENT CHRONOMETER DATE OBSERVED
Astrolabe/R.L. Jones - USAMS Auto Timer 496 22 Nov. 1959
I. 0 ur,ELEVATION MEAN OBSERVED LATITUDEOF STATION 600.06 m N 30 27 03.01 ± 0.17
OBSERVATIONS I 2SEA LEVEL REDUCTIONACCEPTED 23 S 00.09
OBSERVATIONSREJECTD ECCENTRIC REDUCTIONREJECTED -1 00.00
PROBABLE
ERROR (PAIR) + ASTRONOMIC LATITUDE 30.17 IN 30 27 02.92 0.17
REMARKS
COMPUTER Haddox - AMS DATE Jan. 60 CHECKER Tennis - AMS DATE Jan. 60
LONGITUDEOBSERVER AND ORGANIZATION INSTRUMENT CHRONOMETER DATE OBSERVED
R.L. Jones - USAMS Astolaber 496 22 Nov. 1959
H M S SSTAR SETS
ACCEPTED 23 MEAN OBSERVED TIME + 08 40 52.116 ± 0.009
STAR SETS MEAN OBSERVED ARCREJECTED 1 E 130 13 01.74 ± 0.13
REMARKS
ECCENTRIC REDUCTION00.00
ASTRONOMIC LONGITUDEE 130 13 01.74 0,13
COMPUTER Haddox - AMS DATE Jan. 60 ICHECKER Tennis - AlIS DATE Jan. 60
AZIMUTH
OBSERVER AND ORGANIZATION INSTRUMENT CHRONOMETER DATE OBSERVED
MARK - pMEAN OBSERVED AZIMUTH
ELEVATION
OF MARK DIURNAL ABERRATION
OBSERVATIONS ELEVATION CORRECTIONACCEPTED
OBSERVATIONS ASTRONOMIC AZIMUTHREJECTED -(MEASURED FROM ) +
REMARKS
COMPU TER DATE )CHECKER DATE
GEODETIC LATITUDE GEODETIC LONGITUDE
DATUM-.NOTE: NORTH LATITUDES AND
EAST LONGITUDES POSITIVE
IfDEFLECTION IN MERIDIAN (OA-4iG "I
DIFFEPENCE IN LONGITUDE (XA-XG)
PRIME VERTICAL
DEFLECTION
LAPLACE CORRECTION (XA--XG) SIN4G
LAPLACE AZIMUTH
(aG) I
SKETCH OF GEODETIC
CONNECTION
NOTE: aG=ALAPLACE CORR (SUBTRACT ALGEBRAICALLY)
DA FORM 2850, 1 OCT 64
Figure 52. Astronomic results (astrolabe).
ASTRONOMIC RESULTS jSTATION NP (AMS, 1958)(TM 5-237)
PROECT CATION MARYLAND
LATITUDEOBSERVER AND ORGANIZATION INSTRUMENT CHRONOMETER DATE OBSERVED
H.N.C. - AMS Wild T-4 56095 12460 2 May 1963
ELEVATION MEAN OBSERVED LATITUDEOF STATION 75 m N 38 57 01.20 0.09
OBSERVATIONS SEA LEVEL REDUCTIONACCEPTED 25 pair S 00.01
OBSERVATIONS ECCENTRIC REDUCTIONREJECTED 3 pair __ N 02.45
PROBABLEPROBABLE r ASTRONOMIC LATITUDE + 009ERROR (PAIR) + Q?.43 N 38 57 03.640.09
REMARKS
Horrebow - Talcott method.
COMPUTER LES (AMS) DATE 23 July 1963 CHECKER ORN (AMS) DATE 11 Dec 1963
LONGITUDEOBSERVER AND ORGANIZATION INSTRUMENT CHRONOMETER DATE OBSERVED
R.S. - AMS Wild T-4 56095 12474 24 May, 18 Jun 63
STAR SETS H MSTAR SETS MEAN OBSERVED TIMEH M SS
ACCEPTED 6 05 08 29.012 ± 0.004o ,
STAR SETS1 it
SRETSD MEAN OBSERVED ARCREJECTED 0 W 077 07 15.18 ± 0.06
REMARKS
ECCENTRIC REDUCTION
Reduced to UT 0 _______ E 02.54
ASTRONOMIC LONGITUDEW 077 07 12.64 ± 0.06
COMPUTER LES (AMS) DATE 22 July 1963 CHECKER ORN (AMS) DATE 9 Dec 1963
AZIMUTHOBSERVER AND ORGANIZATION INSTRUMENT CHRONOMETER DATE OBSERVED
F.E.W. - AMS Wild T-3 53010 12460 10 Apr, 23 May 63MARK o
MEAN OBSERVED AZIMUTH
MAP (AMS, 1958) 008 25 06.43 ± 0.19ELEVATION
OF MARK 348.78 Ft DIURNAL ABERRATION + 00.32
OBSERVATIONS Eccentric Reduction + 03 02.93ELEVATION CORRECTION
ACCEPTED 32 p0$ 00.00
OBSERVATIONS ASTRONOMIC AZIMUTH
REJECTED i 0 (MEASURED FROM South 008 28 09.68 0.19REMARKS
COMPUTER LES (AMS) DATE 2 OctGEODETIC LATITUDE GEODETIC LONGITUDE
N 380 57' 05'.'159 , W 0070 07' 16'.'394DATUM
NOTE: NORTH LATITUDES AND
1927 NAD EAST LONGITUDES POSITIVE
DEFLECTION IN MERIDIAN (4A--G) i -01.519
DIFFEPENCE IN LONGITUDE (XA-XG) ' +03.754
PRIME VERTICAL (XA-)O) COS -DEFLECTION +02.919
LAPLACE CORRECTION (XAXG)jSING 0+02.360
LAPLACE-AZIMUTH
(aG) 008 28 07.32 NOT E:aG=CAiL4PLACE CORR (SUBTRACT ALGEBRAICALLY)
DA FORM 2850, 1 OCT 64
Figure 53. Astronomic results.
--
CHAPTER 4
DISTANCE MEASUREMENTS
Section I. TAPE MEASUREMENTS
38. Data Requireda. In the establishment and extension of hori-
zontal control, the distance between control pointsis a primary requirement. In triangulation, se-lected sides designated as baselines are obtainedby precise measurement, and these are used with
observed angles of connecting polygons to obtainall other lengths of a triangulation net. In tri-lateration, the sides are measured by electronic
methods, and taped distances are seldom involved.In traverse, the distances between stations aregenerally measured directly. One method ofdetermining these distances is by measurementwith standardized metallic tapes, using specialtapes, with a small coefficient of thermal expan-sion, for baselines, and steel tapes for other
measurements.b. Base line computations should be made on
DA Form 1914, Computation of Base Line (fig.54). The first five columns and the temperaturecolumn are filled in from the data in the field
records. These six columns are used to recordthe section measured, the date of measurement,the direction of measurement (forward or back-ward), the number of the tape used for each
measurement in the section, the number of sup-ports used for the tape for each measurement, andthe average temperature of the forward and rearthermometers (a mean of the temperature can beused for all the full-tape lengths). The column
"Uncorrected Length" is used to record the totalnumber of tape lengths and the correspondingtotal length of each section. Odd tape lengths,such as half- or quarter-tapes, or measurementswith a steel tape are also recorded in this column.Set-up or set-back measurements of less than ameter are entered as corrections in the columnprovided.
c. Computation of other tape measurementsshould be made on DA Form 1939 (Reduction of
Taped Distances) (fig. 55). All field data is
entered on the form in the space provided.
39. Corrections to Tape Measurementsa. The field measurement of any line must be
corrected for temperature, tape and catenary,inclination, and elevation above or below sea-level.
b. The information for the temperature cor-
rection and the tape and catenary corrections is
found on either a tape standardization certificate
furnished by the National Bureau of Standards
(fig. 56) or from results of field comparison of
tapes with previously standardized tapes (para.
41). The NBS certificate contains the coefficient
of thermal expansion of the tape and the standard-
ized length of the tape under several support
conditions. Every tape used in a base measure-
ment should be standardized by the National
Bureau of Standards, Washington, D.C., both
before and after the base is measured. The
average lengths of the two standardizations are
used for the computation of the tape and catenary
correction.
(1) The temperature correction is found by
multiplying the coefficient of thermal
expansion (shown on tape certificate)
by the number of degrees difference
between the temperature at the time of
measurement and the temperature of
standardization, usually 250 C. for invar
tapes and 20 ° C. (680 F.) for steel tapes,
times the measured length. The thermal
expansion as given on the tape certificate
usually states a certain expansion per
tape length per degree Celsius (centi-
grade). For example, a tape may have
a thermal expansion of +0.020 milli-
meters per 50 meters per degree Celsius.
(2) As an illustration of a temperature
correction computation, Tape No. 5123
PROJIECT NAME OF BAE COMPUTATION OF BASE LINE
1-442 'Pis,,kTi5.37- - iMh;-4 2LOCATION ORGANIZTION OROER OF BASE FIELD BOOK NR PAGE NR PAGE NR NR OF PnES
DIR. EN"
TAPE APE EMP.REDUCD LENGTH ADPTED LNGTH (V) P.E.SECION GAE zER. NR SPPOT TATN i[R .G i N. Tr n0 zu euttro S. I-v c (Meter. m) (mm)
_ _ . _ r _r I __ __ _
COMPUTEDDBYDATE HECKE B DATE
DA A 1.. 1914
V
Figure 54. Computation of a Base Line (DA Form 1914).
(fig. 56) has a thermal expansion of
0.020 millimeters per 50 meters per
degree Celsius standardized at 250 C.
Section A of a base line was measured in
the forward direction with this tape.There were 12 full tape lengths and 1half length. The average temperature ofthe front and rear thermometers for the12 full tape lengths was 300 C., and the
temperature for the half tape length
was 290 C. The correction for the 12
full tape lengths is:
12 X 0.020 X (30-25) ± x- 1.2 miliimeters
This correction is plus because the
measurement temperature was higher
than the standardization temperature;
therefore, the tape was actually longer
than recorded. When the measurement
temperature is lower than the standard-
ization temperature, the correction is
minus. The correction for the half-tape
length is-
X X 0.020 X (29-25) = + 0.04 millimeter
Corrections are ordinarily entered only
to Yo millimeter
tation form.
on the base line compu-
c. When the tape is supported in the same
manner as when standardized, the tape and
catenary corrections are combined. If the tape
is supported at different points than it was at
standardization, it is necessary to compute the
tape and catenary corrections separately.
(1) Using Tape No. 5123 as an example and
the same section of the base which has
12 full tape lengths and 1 half tape length,the tape and catenary corrections are
computed using the standardization data.
For the 6 tape lengths with 3 supports,the correction is:
(6) (+ 0.00067)= + 0.0040 meter
For the 6 tape lengths with 4 supports,
the correction is:
(6) (+0.00021)=+0.0013 meter
The half tape length is supported at 2
points, and the correction is:
(1) (+0.00028)=+0.0003 meter
The example illustrates a very simple
case where standardization was available
for all the tape lengths used. A more
PROJECT BOOK NR REDUCTION OF TAPED DISTANCESQ/, 30(TM 5-237)
LOCATION TRVREN.FROM STA. TO STA.
-L I94ORGANIZATION TAPE CAL. LENGTH TEMP. TENSION SUPPORT T
A IS 6602DATE CAL. LENGTH TEMP. TENSION SUPPORT 2
SECTION SUPPORT MEASURED TEMP. TEMP. CALIBRATION DIFFERENCE SLOPE CORRECTEDDISTANCE CORRECTION CORRECTION EVTON CORRECTION DISTANCE
200.000L 4.ol -
L 2 200.000 L1o ___ ___
200.000 ___ Z& Q4 2
1-50.865 -0-03% -0-L48. - -0040 70A4
_____ __ __ _ __ _ 4_ _ -A 0. 05B to
2____ ow t~a / .2 -0. ----
_ _ _ _ 1O/, -0 1Q46 A689
*3
t o a ~ 2 0 0 . 0 0 0 L _ _ _ _ _ _ _ _ _ _A
____~2Oo~oo t
____ ____ -0.065 -0.25 -. - 7L~8
20 0 -0 0 0
_ _2
___ __________ .2 -. 048 to
42.aw-0046 - 0Q30 - -QQ8625
COMPUTED BY -DATE CHECKED BY OATEJ4j). C.A,. vjeS4a9
DAIFR 1939
Figure 55. Computation of Taped Distances (DA Form 1939).
757-381 0 - 65 - 7
orm 1
UNITED STATES DEPARTMENT OF COMMERCE
WASHINGTON
Alationat ureu of 'tatbirbg
(CertficateFr
50-METER IRON-NICKEL ALLOY TAPE(Low Expansion Coefficient)
Maker: Keuffel & Esser Co. NBS No. 7871No. 5123
Submitted by
Inter-American Geodetic Survey Liaison.Officec/o Army Map Service6500 Brooks LaneWashington 25, D. C.
This tape has been compared with the standards of the UnitedStates under a horizontal tension of 15 kilograms. The intervalsindicated have the following lengths at 25° centigrade under theconditions given below:
Interval Points of Support Length(meters) (meters) (meters
0 to 50 0, 25, and 50 50.00067
0 to 50 0, 12 1/2, 37 1/2, and50 with the 12 1/2- and 37 1/2-meter points 6 inches above theplane of the 0 and 50-meterpoints. 50.00021
0 to 12 1/2 0 and 12 1/2 12.50088
0 to 25 0 and 25 25.00028
0 to 37 1/2 0 and 37 1/2 37.49701
0 to 37 1/2 0, 25, and 37 1/2 37.50134
For the interval 0 to 50 meters, thermometers weighing45 grams each were attached at the 1-meter and 49-meter points.One thermometer weighing 45 grams was attached at the 1-meterpoint for all other intervals.
Test No. 2.4/G-15354
Figure 56. Tape standardization certificate.
NBS Certificate continued
These comparisons were made on the section of the lines nearthe end on the edge of the tape marked with small dots near thegra duation.
The weight per meter of this tape, previously determined, is25.8 grams.
The values for the lengths of the intervals 0 to 25 metersand 0 to 50 meters are not in error by more than 1 part in 500,000;the probable error does not exceed 1 part in 1,500,000. The valuesfor the lengths of the intervals 0 to 12 1/2 meters and 0 to 37 1/2meters are not in error by more than 1 part in 250,000; the probableerror does not exceed 1 part in 750,000.
The values for the lengths were obtained from measurements madeat 20.2° centigrade, and in reducing to 25 ° centigrade, the thermalexpansion of +0.020 millimeter per 50 meters per degree centigradewas used.
For the DirectorNational Bureau of Standards
Lewis V JudsonChief, Length SectionOptics and Metrology Division
Test No. 2.4/G-15354Date: September 10, 1954
Figure 56-Continued.
difficult computation is necessary ifstandardization is not available forbroken tape lengths. Under these cir-cumstances it is necessary to make use ofthe formula for correction due to sag(catenary correction) which is:
C- 24 l 3
in which:
n=number of sections into whichthe tape is divided by equidistantsupports
l=the length of a section in metersw=the weight of the tape in grams
per metert=tension in grams
The minus sign in the formula presup-poses that the correction is to be appliedto a tape standardized under conditions
of full support throughout. In otherwords, the length measured by a fullysupported tape is shortened by theeffect of sag.
(2) Since the Bureau of Standards does notordinarily standardize tapes fully sup-ported, it is often necessary to reduce
a tape standardized with 3 or 4 supportsto the value with full support. Thisreduction is made by applying thecorrection for sag to one of the standard-ized lengths. It is generally easiest to
use the standardization with three sup-ports, at 0, 25, and 50 meters. The
computation of the sag correction is
greatly simplified by using tables foundin USC&GS. Sp. Pub. 247 and TM
5-236. Both tables required that t-
15,000 grams. TM 5-236 gives the
catenary corrections for various lengths
- 2 -NBS No. 7871
and weights of tape, while SP 247
gives the value of the quantity
- X1010for each 3o gram from 20
grams to 30 grams.
d. To illustrate the application of the catenary
correction, the standardized tape, for which the
certificate is shown in figure 56, will be reduced to
full support and then various odd lengths and
supports computed.
(1) From the certificate, the length of the
tape interval, from 0 to 50 meters when
supported at 3 points, 0, 25, and 50, is
50.00067. The weight of the tape is 25.8
grams per meter. From TM 5-236, the
catenary correction for a 25-meter inter-
val for that weight tape is 1.93 milli-
meters. For two 25-meter intervals, the
correction is 2 X 1.93= 3.86 millimeters.
The length of the tape when supported
throughout is then-
50.00067 +0.00386=50.00453 meters
Starting with this standard length as
supported throughout, the correction for
an odd distance can be found. Assum-
ing a length was measured as 37 $ meters
with 3 supports at 0, 25, and 37 /2, the
correct measured length is:
% times 50.00453=37.50340 minus
the catenary corrections for 25
meters (1.93 mm) and 12% meters
(0.24 mm) which gives a corrected
measured length of 37.50123 meters.
The catenary correction for 12 /2 meters
could be computed from the formula or
taken as 3g the correction for 25 meters.
The fraction y comes from the fact thatthe correction varies as the cube of the
length.
(2) Comparing the computed value for the
37 2 meter length over 3 supports with
the value on the certificate, a discrepancy
of 0.11 mm is found. This discrepancy
is negligible and occurs because of the
markings or irregular stretching of thetape. By using this method, the correctstandard length can be obtained for any
odd measurement or support arrange-ment.
e. The inclination (or slope) correction reducesall measurements to a horizontal plane. In order
to make this reduction to horizontal, the 'difference
of elevation of the ends of the tape must be known.
Also very important are the elevations of any
intermediate tape supports if they differ greatly
from the grade of the end supports. The required
differences of elevation are usually determined by
spirit leveling. This leveling will be discussed
later in the text.
(1) The inclination correction can be com-
puted from the formula:
or
h 2 h4 h6
CG= 21 813 1615
in which CG= inclination correction
(grade correction), = inclined length,and h=difference in elevation of the ends
of the inclined length.
(2) For short lengths or steep grades, use
the formula:
CG=- (1- 2
For 50-meter lengths and differences of
elevation of less than 7.5 meters,TM 5-236 can be used, or the series
formula:
h2 h4CG 21 813
No more than two terms are necessary.
(3) Differences in elevations as abstracted
from the level books and inclination
corrections as computed or as taken from
tables are recorded on DA Form 1915
(Abstract of Levels and Computation of
Inclination Corrections) (fig. 57). In the
second column of the form, the distances
between the points in the first column
are entered. The mean differences of
elevation between the points are written
in the third column. Be sure to indicate
the units of measure for the differences of
elevation by crossing out either meters or
feet at the top of the column. The
inclination correction is entered in milli-
meters or 0.01 foot for each length. The
sum of the correction is obtained for the
section and recorded at the end of the
column. Elevations are computed for
sufficient supports so that a mean of the
100
ABSTRACT OF LEVELS ANDCOMPUTATION OF INCLINATION CORRECTIONS
(TM 5-237)
PROJECT DATE
- 442 .23 .S6LOCATION ORGANIZATION
U.. 5,9. Adis, 1C.DISTANCE FERENCE OF INL AIN ELEVATION MA
POINT Mtr) MENCORRECTION ELEVATION REMARKS(Meters or9+4 (M or 0.01 I) Me(Mhter or
A 1042.4L
2 1 IA 13.0 104#4IO0
aff so ~ .23g L 4~4
6 -n8 ±L0A 11LE ____
0~ .~~+ 1.16 A3A L046,3
i L 50 4.s. 14 AaA7 Q148.16
B3 25 -0.23 L. 1047g1
COMPUTED BY DATE CHECKED BY DATE
~FRM 11DAIFEB 57195-
Figure 57. Computation of inclination corrections (DA Form 1915).
101
elevations will provide a mean elevation
for the section which is accurate to within
2 meters. Use of this mean elevation in
computing the reduction to sea level of a
section is explained in the followingparagraph. The sum of the inclination
corrections for the section is entered onDA Form 1914. Any odd tape lengths,
such as 12%1 or 37%1 meters, or setups
and setbacks, should show their indi-
vidual inclination corrections merely for
the sake of convenience in checking.
f. Each section of a base line must also be
reduced from the measured length to the length
at sea level. It is for this purpose that the mean
elevation of the section is computed on the
Abstract of Levels form. The formula for reduc-
ing the base to sea level is:
h h2
h3
C= -S r+S -S + ...
in which C = correction to reduce a length, S, to
sea level; h= mean elevation of the section; andr = radius of curvature of the earth's surface for
that section. Only the first term of the formula
is needed, except for first-order base lines at high
altitudes. The value of log r can be found in
table IV, appendix III, using the mean latitude of
the ends of the base and the azimuth of the base
as the arguments.(1) For this example, the mean latitude is
250 N and the azimuth of the line is 67°.
The value of log r from the table is6.80459. The mean elevation of thesection, as found on the Abstract ofLevels form, is 1044.9 meters. Theapproximate length of the section is 625meters. The reduction to sea level iscomputed as follows:
log 625 =
log 1044.9 =
colog r =
2.79588
3.01907
3.19541-10
log C = 9.01036-10
C = - 0.1024 meter
(2) Inclination corrections always shortenthe measured length. Sea-level cor-rections shorten the measured length ifthe base is above mean sea level, andlengthen the measured length if the baseis below mean sea level.
g. As the subject of setups and setbacks issometimes confusing, additional discussion is
needed. Setups and setbacks can take either of
two forms. They can be partial tape lengths, such
as 20 meters, 15 meters, and so on, or they canbe short measurements of the order of 5 or 10
millimeters measured with a scale made to bringthe tape end onto the marking strip. Special
care must be taken in the computation of tempera-
ture, tape, and inclination corrections for setups
and setbacks.
40. Final Length and Probable Error
a. The reduced length of the section is now
obtained by applying the corrections to the
recorded length.
b. The adopted length is the mean of the
reduced length of the forward and backward
measurements of the section, provided the dis-
crepancy between the measurements is less than25 mm V/K (K is the length of the section in
kilometers). The limit of 25 mm K applies
to third-order base measurements. Closer limits
are placed on first and second-order bases.
c. In the sample base computation, the reduced
lengths of the forward and backward measure-
ments are 624.6648 meters and 624.6784 meters,respectively. The allowable difference in measure-
ments is 25 mm /0.625=19.76 mm. Theactual difference is 13.6 mm, which is within the
allowable limits, and the 2 reduced lengths are
meaned to obtain the adopted length of 624.6716meters for the section A to B.
d. The value in the v column is for use in
finding the probable error of the section and thebase. The residual v is the difference between the
reduced length and the adopted length.e. The last column on the form is headed P.E.
and may be used for either v2 (residual squared)
or for the probable error of the section. The
probable error is computed, using the followingformula:
P.E.=0.6745 ;n(n-1)V n(n-1)
in which v is the residual, and n is the number of
acceptable measurements made of the section.
Where a section is measured only twice, the prob-
able error is merely 0.6745 times 1 the difference
between the two measurements.
f. The probable error of the entire base is the
square root of the sum of the squares of the prob-
102
able errors of the individual sections. Theprobable error is also expressed as a fraction witha numerator of 1, such as 1/1,000,000, meaningan error of 1 part in 1 million. For third-orderbase lines, the computed probable error shouldnot exceed 1 part in 250,000. For first- and second-order bases, the computed probable errors shouldnot exceed 1 part in 1,000,000 and 1 part in
500,000 respectively.
g. All results are entered on DA Form 2851(Baseline-Abstract of Results) (fig. 58).
41. Standardization of Field Tapes
When the tape used in the field has been com-pared to a standardized tape, a length correctiontable is computed for use in the reduction oflengths measured with the field tape.
a. In order to have a true comparison of thefield and standardized tapes, certain reductions
are made to the comparison measurements of thetwo tapes.
(1) The standardized tape is reduced to itstrue length for the conditions of measure-ment.
(2) Both tapes are corrected for the differencein temperature of observation and tem-
DA FORM 2851, 1 OCT 64
perature of standardization, unless bothtapes have the same coefficient of expan-sion.
(3) The tension must be the same for bothtapes.
(4) The true lengths of the field tape are thentabulated (fig. 59).
b. The length correction table for the fieldtape (fig. 60) is computed and tabulated asfollows:
(1) The correction is computed for tapesegments which are multiples of 10 feet,or 5 meters depending on the units ofthe tape.
(2) The catenary factor, L3 , is computed.This factor is the cube of the segmentlength divided by the total length of the
tape cubed.
L3=(segment length 3
(3) The tape correction, C,, is considered
directly proportional to the length of thetape. In figure 59, the correction for100 feet is +0.002, therefore the correc-
tion for 10 feet is +0.0002. Between
BASE-LINE ABSTRACT OF RESULTS(TM 5-237)
Figure 58. Baseline-Abstract of Results (DA Form 2851).
103
BASELINE
PISULA REDUCED LENGTH 624.6716 METERS
LOCATION DATE RATIO
USA 17 May 1954 PROBABLE ERROR ±4.59 mm 1/136 094OBSERVER AND ORGANIZATION LOGARITHM OF
C. PISULA - AMS LENGTH (Meters) 2.79565176FROM STATION (Master) ELEVATION
A M LOG ARC-SIN CORR 0
TO STATION (Remote) ELEVATION NO. OF OBSERVATIONS REJECTIONS
B M 2 0TAPE STANDARDIZATION
TYPE MANUFACTURER MFG'R. NO. N. B. S. NO. DATE
INVAR K & E 5123 7871 10 Sept 54
INVAR K & E 5124 16 May 54
ELECTRONIC INSTRUMENT
MASTER (Type and No.) DATE CALIBRATED REMOTE (Type and No.) DATE CALIBRATED
REMARKS MICRO FILM NO.
This baseline does not meet third-order specifications, and should be
used as a check base only. Tape K & E 5123 was standardized after use in
the field.
COMPUTED BY DATE CHECKED BY DATE
AMS - R.A. Smith 25 Jan 56 AMS - W.C. Aumen 25 Jan 56
100 and 200 feet the correction is
+0.0050-0.0020-+0.0030 and the cor-
rection for 110 feet is +0.0023.
(4) The catenary correction, C,, is deter-
mined by multiplying the catenary factor
(L3) by the total catenary correction for
the tape. The total catenary correction
is found by subtracting the suspended
length from the fully supported length of
the tape. In figure 59 this correction is200.005-199.929= 0.076 for the 200 foot
length. The check at the 100 foot mark
shows the computed catenary correction
of -0.0095 is very close to the measured
correction of -0.009. This check should
always be made.
(5) The tape and catenary corrections are
combined to form the Ct+C corrections.
(6) These corrections are applied to eachsegment to give a table of corrections for
use with any lengths measured with thefield tape.
42. Broken Base
a. Description. A broken base is a base con-sisting of more than one horizontal tangent.No portion of the base with considerable lengthshould be inclined at an angle of more than 20 °
to the final projected length of the base and themaximum should be kept down to 120 if possible.
b. Computation. To reduce the broken base toa single horizontal tangent, the law of cosines isused.
a2 = b2+c 2 - 2bc cos A
where a is the single horizontal tangent to bedetermined, b and c are the two measured segmentsof the base, and A is the angle at the intersectionof the two measured segments.
RESULTS OF TAPE COMPARISON
LOCATION: Ellensburg, Washington
TIME: 12:30 p.m.
STANDARD TAPE NO: K&E 8159
OBSERVER: R. C. Campbell
DATE: 17 October 1957
TENSION: 20 lbs.
FIELD TAPE NO: 8161
CHIEF OF PARTY: J. E. Norton
SUPPORTED ON A HORIZONTAL FLAT SURFACE
INTERVAL L
0 to 100 feet 100100 to 200 feet 1000 to 200 feet 200
NGTH
.002
.001
.005
SUPPORTED AT THE ENDS OF THE INTERVALS INDICATED BELOW
INTERVALS IrNGTH
0 to 100 feet 99.993100 to 200 feet 99.992
0 to 200 feet 199.929
Figure 59. Results of field tape comparison.
104
LENGTH CORRECTION TABLE
FIELD TAPE NO: K&E 8161
CATENARY
FACTOR
L3
(2)
.000125
.001000
.003375
.008000
.015625
.027000
.042875
.064000
.091125
.125000
.166375
.216000
.274625
.343000
.421875
.512000
.614125
.729000
.857375
1.000000
LENGTH
(feet)
(1)
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
DATE COMPARED : 17 Oct 1957
COMPUTATIONS
(Ct
(3)
+.0002
+.0004
+.0006
+.0008
+.0010
+.0012
+.0014
+.0016
+.0018
+.0020
+.0023
+.0026
+.0029+.0032
+.0035
+.0038
+.0041
+.0044
+.0047
+.0050
Cs
(4)
-. 0000
-. 0001
-. 0003
-. 0006
-. 0012
-. 0021
-. 0033
-. 0049
-. 0069
-. (0095)
-. 009
-. 0126
-. 0164
-. 0209
-. 0261
-. 0321
-. 0389
-. 0467
-. 0554
-. 0652
-. 0760
Ct+Cs TOTAL CORREC-TION W/TWO- (2)SUPPORTS
(5)
+.0002
+.0003
+.0003
+.0002
-. 0002
-. 0009
-. 0019
-. 0033
-. 0051.
-. (0075)
-. 007
-. 0103
-. 0138
-. 0180
-. 0229
-. 0286
-. 0351
-. 0426
-. 0510
-. 0605
-. 0710
(6)
.000
.000
.000
.000
.000
-. 001
-. 002
-. 003
-. 005
-. 007
-. 010
-. 014
-. 018
-. 023
-. 029
-. 035
-. 043
-. 051
-. 060
-. 071
Figure 60. Length correction table (field tape).
Section II. TACHYMETRY MEASUREMENTS
43. Methods UsedTachymetry is generally construed as the
measurement of distance by optical means.Included within this category are the stadia
methods and the subtense methods. Each methodmakes use of a base of some type. The stadia
methods utilize a vertical base which depends on
the length of rod subtended between crosshairs
within the instrument. The subtense methods
utilize an outside base of fixed length,which horizontal angles are measured.
across
44. Stadia Methods
The stadia methods determine distance by the
length of a vertical rod intercepted between two
fixed crosshairs within the instrument. The
method may be used for lower order mapping
where the corrections applied are very rough.
105
c --- ----
It is used as the means of determining distances
for leveling for which the stadia constant is
determined regularly and with considerable care.The self-reducing tacheometer uses an intercept
on the vertical staff which is always 1/100th of thedistance.
45. Use of the Self Reducing Tacheometera. The RDS tacheometer equipment consists
of a tacheometer and a vertical staff. The
tacheometer is constructed in such a way that thestadia crosswires always intercept that portion
of the staff that is 1/100th of the distance, re-
gardless of the inclination of the telescope.
The telescope also displays the appropriate curve
(of four) and the difference of elevation factor
that is required to determine the difference of
elevation at hand. By use of this method, it is
possible to determine the difference in elevation,direction, and distance from the same observations.
b. The distance is determined by reading therod intercept between the lower and upper stadia
crosshairs, considering the millimeters of the
intercept to be whole numbers and multiplyingby 0.1. The difference in elevation is determined
by multiplying the rod intercept, between thelower stadia crosshair and the cutting point ofelevation curve across rod, by the elevation factor
displayed beside the elevation curve. In somecases, two elevation curves and two elevation
factors appear in the telescope at one time. Insuch cases each rod intercept and its corresponding
elevation factor will be handled as described, but
the differences in elevation for both must be thesame. The signs of the elevation factors must be
altered to conform to the elevations in the direction
of progress.
c. The tacheometer should not be used in cases
where the required tolerance in elevation is lessthan 2 meters.
46. Subtense Method
a. Subtense distances, as the name implies, aremeasured by the angle subtended by a knownlength. The length of a subtense bar is usually 2
meters. The directions for the use of a subtensebar state that the midpoint of the bar must becentered over the station, the bar must be level,and the bar must be perpendicular to the line ofsight of the instrument. Figure 61 illustrates theconditions for a subtense measurement.
b. A and B, figure 61, are stations, d is thesubtense bar of known length, a is the subtendedangle which is measured by the transit or theodo-lite, and D is the required length from A to B.Since the figure is an isosceles triangle, theperpendicular bisector (D) of the base (d) alsobisects the vertex angle (a) and forms two righttriangles in which an acute angle (%a) and a side
(Yd) are known. By plane trigonometry, cotD
%a= Yd or D= (yd) (cot %a), but the subtense
bar is 2 meters long which makes 1 d= 1 meter,and the formula becomes D =cot %a in meters.This formula is useful in emergencies, but veryaccurate tables are required for the cotangentfunctions.
c. The best method for solving subtense dis-tances is by using tables XLII and XLIII in TM
5-236, which have the subtended angle (a) as theargument and give the distance immediately inmeters (table XLII) or feet (table XLIII). Thetables are valid only with a 2-meter subtensebar. The whole subtended angle is used as the
argument in the tables.d. It is possible to obtain third-order accuracy
by the subtense method by using the properequipment and careful observation.
d= subtense .bar
Figure 61. Sketch, subtense measurement.
Section III. MEASUREMENTS USING LIGHT WAVES
47. Geodimeter-Model 2
a. The Model 2 series Geodimeter measuresdistances indirectly by measuring the time re-quired for a light beam to pass from the Geodim-
eter to a reflector and back to the Geodimeter.
The Model 2 Geodimeter is primarily a first
order baseline instrument. Its size and weight
limits its use to drive stations and makes its use
on lower order surveys impractical.
106
b. According to the manufacturer, the Model
2A Geodimeter will determine distances up to 30
miles with an instrumental error of 1 cm = 1 ppm
(part per million). The approximate distance tobe measured must be known within 1,000 meters.
c. As with all of the electronic distance measur-
ing equipment, the atmospheric unknowns will
probably introduce more error into the lines
measured than the inherent accuracy of the in-
strument. To reduce these unknown conditionsto a minimum, only calibrated thermometers,psychrometers, and altimeters should be used.
48. Measurement. Reduction
For the purpose of explaining the computationof the length of a line, it is assumed a Model 2-AGeodimeter was used. A calibration was pre-
viously performed (fig. 62). The field observa-
CALIBRATION CONSTANTS
GEODI1,TER NO.11h
Calibrated by Army Map Service, 9 August 1961
Zero Correction (Z) 1.1526 m
Light Conductor Length (L) .7960 M
Transmitter Mirror Lens Focus 7.0 mm
Receiver Mirror Lens Focus 7.8 mm
Photocell Wave Length 5550 Ao
Refractive Index (00C & 760 mm Hg) 1.0003042
Calibration Temperature 27.000
Unit Length, F1 @ OC & 760 mm Hg 7.4925333 M
Unit Length, F2 @ OOC & 760 mm Hg 7.4553801 N
Unit Length, F3 @ 000 & 760 mm Hg 7.2746155 M
Frequency, F1 10,000,000 CPS
Frequency, F2 l0,049,834 CFS
Frequency, F3 10,299,559 CPS
Frism Eccentricity Correction (19 prism bank) .o480 14
Prism Eccentricity Correction (54 prism bank) .0450 14
Velocity of Light in Vacuum 299,792.5 IKS
Coefficient of Expansion of Aluminum 0.000022 1/00
Figure 62. Calibration constants, geodimeter model 2.
107
tions have been recorded on DA Form 2852
(Geodimeter (Model 2) Observations) (fig. 63),
including the Mirror Constant, Approximate
Length of Line, Transmitter and Receiver Focus,
and Eccentricity of the Mirror and the Geodimeter.
a. Determination of the Fine Delay (LC). For
each frequency, the electrical length is reduced to a
physical length. A space near the bottom of the
observation sheet has been provided for this
computation. The formula may be written:
LC=LCI-( LC,,I, -M-) (LC-LC2)
Where: LC = Fine Delay in Meters
LC1 = First Light Conductor Setting
LC2 =Second Light Conductor Setting
LCIm = Mean of First Light Conductor
Readings
LC2, = Mean of Second Light ConductorReadings
M, = Mean of the two Mean MirrorReadings
Watch the signs carefully. Obviously if the LC2
setting is larger than the LC1 setting, the resulting
LC will be larger than LC1.
b. Light Conductor, Coarse (nL). On the obser-
vation form, under the column headed LC, are
numbers 7-68. These numbers indicate that
seven light conductors were used or more accu-
rately six light conductors and a portion of another.
Since the exact length of the light conductors is
shown on the calibration sheet, multiply the
number of whole light conductors by this length
(6X0.7960=4.7760) to obtain the number of
meters. This value is entered on DA Form 2853
(Geodimeter (Model 2) Computations) (fig. 64).
PROJECT GEODIMETER NO. OBSERVATION N GEODIMETER (Model 2) OBSERVATIONS/4/ (TM 5-237)
LOCATION MIRROR CONSTANT BASELINE APPROXIMATE LENGTH
CALIFORNIA -0.0480 AMBATO /0 Km.ORGANIZATION DATE TRANSMITTER FOCUS RECEIVER FOCUS RECORDER
USAM 21 SFP1? /96/ 0.9 mm 1.2 mm /FFDANL4GEODIMETER STATION GEODIMETER HEIGHT ELEVATION GEODIMETER ECC -OBSERVER
&ANAL , AMS 1%,9 10.705 m 56.030 m 0.0/0 o n(IAs.-P4 WIT TERMIRROR STATION MIRROR HEIGHT ELEVATION MIRROR ECC MIRROR TENDER
JUNCTION, usC GS /934 1 5.508 j7 42.572 r ___ MURPRIY
F - MIRROR LC MIRROR LC F D MIRROR LC MIRROR LC F D MIRROR LC MIRROR LC
R E R E R EEL E E L
7-68 7-64 v 7-72 7-68 8-54 8-so./4 _ _2~ __ __ 4_ _
SIGN + - - SIGN - - - - SIGN f.j - + -
S26 5 26 0 250 24 21 28 0 22 / 28 0 2/1 78 1 78 8 78 2 75?9
_ 29 7 29 /129 0 27 4 2 20 2 26 520 2 24 9 2 868 86 9 85 8 841
29 7 32 0 29 0 29 7 3 22'2 28 4 22 3 27 0 3 85 8 138_ 86 0 87 8
4 .23 2 21 0 22 7 19 O 4 22. 7 176 23 O /55 4 73769573 9 67 4
SUM 109 / 08 / /05 7 /00 3 Sum 93 / 94(0 93 5 88 4 suM 324 4 3250323 9 315 2
MEAN 2728 2702 242 2508 MEAN 93 28 23552 38 22MEAN 8/ /0812580987880GEOD MIRROR GEOD MIRROR GEOD MIRROR GEOD MIRROR vGEOD MIRROR GEOD MIRROR
TIME 2037 2035 2046 2045 TIME 2057 2055 2104 2/03 TIME 21/2 21/1 2120 2720
TEMP 22.4 °C 22.2 c 22.2 °c 22.0C TEMP 22.0 °c 21.8 °c 21.9 ° 21.8 °c TEMP 21.8 c 2/1. c 21.7 -c 214 %PRESS /oo.0 j 90.0 i 990 M 898 m PRESS 98.0 m 88.0 98.0 m 87
00 PRESS 970 m 88.0m 96.0 m 870
H 5885 .0 O55 575 F 575 °-5780°r5H or5755,57F i70 575 sf 570 of
73.0 Of 720 °CF 72.5 ° 72.0 F M 720 F 71.5 °F 71.5 r 75 OF M 71.5 °f 71.0 f 710 OF 7.0 O2202 -26.85 23.6 23.33 ,8.s8o ,
LC = 6800 - 2702 -26.85 x.04 LC=.7200 .65- x .04 LC =.5400 - 8125-8.04 x .042702 -25.08 23.65 - 22.10 8/25 - 78.80
__.6765 n.7/17 I =.5 66
TEMP PRESS W D TEMP PRESS W D EMP PRESS W
MEAN 22.20 cc 751.51157.88 1-172.38a FMEAN 2188 OC 75.6815750 °F 7/1.62 -. MEAN 2/.62 °c 7S1.75 15725 F 71.2 OFNOTE: GiVe units for temperature and pressure; give pressure reduCed for inSt ConStant (300m, 1000 fl)
REMARKS
19 PR/SM BANK USED
PRESSURE UNITS ARE MM Ng-
DA FORM 2852, 1 OCT 64
Figure 63. Geodimeter (model 2) observations (DA Form 2852).
108
~ROJET OBSRVATIN NOGEODMETER (MODEL 2) COMPUTATIONS
/ (TN a-237)LOCATION GEODIMETER NO. DATE BASELINE
CAL.IFORNIA /41 2ISept--61 AMBATOORGANI ATION OBSERVER GEODIMETER STATION MIRROR STATION
U SA/vS WHITTER CANAL AMS /96/ IJUNCrION USC(GS/R934DETERMINATION OP PARTIAL UNIT LENGTH ________
LIGHT CONDUCTOR, COARSE(iiL) 4.7760 4.770 5. 5720
LIGHT CONDUCTOR, FINE MLc) .6765 .7/17 -. 5366
ZERO CORRECTION (Z) 1.12 " 1. /26 1. /526
ECCENTRICITY. GEODIMETER + . 0 10 0 I .0/00 f ./0
ECCENTRICITY, MIRROR
CONSTANT, MIRROR . 0480 - . 0480 - . 0480
FOCUS CORRECTIONS . o27- + .0/27 + . 0/ 27
TEMP CORR., LIGHT CONDUCTOR - .0005 - co0 S - 000 7
SUMOF AGOVE (8) Ki 6.5793 K 2 6.-6145 K3 7. 2-352DETERMINATION OP QUARTER WAVE LENGTH
TEMPERATURE (t) (OC) 22. 20 2/. 88 2/." 62
PRESSURE (P) (mm H) 75/ 5 71.68 75/. 75
HUMIDITY (%) 40 42 42
____________ 1.000 2 7821I 1.000278-58 1.000 27885
HUMIDITY CORRECTION - .00 QO004,1 - .o 0ooo4 2 - , 0ooooo41REFRACTIVE INDEX (Na) 1.00027780 /.000 278 /6 100027844FREQUENCY. (f)' cps fi /0 000 000 f2 /0 049 834 f3 /0 2P9 559
QUARTER WAVE LENGTH (U) lU 1 7.492731019- 72 4555 74 213 U3 7274802954DETERMINATION OP NUMBER OP UNITS
PHASE SIGN, fl PHASE SIGN, fa + PHASE SIGN, f3 ..
APPROX LENGTH (LA) /0 000 m. =NxU)K LAs 9993.3855 M.
.LA -Ki 9993. 4207 N2LA-K2/U2 /3404-
N'izLA-Kl/U1 /333 7 '=NiU)K 9997.08,47 i.
Li=(NjxU1 )+K1 9994.3898 Im. U1 -U2 0.037/56283 m
L: L12. 6 949 AN=1f2 -L;i/U1-U 2 725,Nl +N/405 /2=2+A 4 /2 N 3 =N 1 (1.03) 1447
DETERMINATION OP SLOPE LENGTH Q
UNIT LENGTH (UxN) /0 527. 287/ /0 527. 2708 /0526. 6399SLOPE LENGTH (UxN)+K /0533 8664 /0533. 8853 iO533. 875/
NQ 1 r087.4 316.288 ) + gk )4. +10 i- 7 N.AL i+ t .)76
= Wave lenth of light in micron.. Hum. Corr. - 5. x 108 0 U = 2997925001 + t/273 4 (f) (Ns)
-OPTE7 A M4/4p 3IC 1'*o -A S JA N. '64DA FORM 2853, 1 OCT 64
Figure 64. G eodimeter (model 2) computations (D. Form 285).
109
c. Light Conductor, Fine (LC). See a above,for determination of this value.
d. Zero Correction. This value is sometimescalled the calibration constant and is determinedduring calibration. The value is the distancefrom the back centering point of the Geodimeterto the electrical center of the Geodimeter and canonly be determined by calibration procedure.This value will change for each instrument andwill change if any major components are replacedor repaired.
e. Mirror Constant. The reflex used with theGeodimeter must have its constant computed.This value is furnished on the sheet of Calibrationconstants or may be computed. Since the refrac-tive index of glass is about 1.57 as compared toair, the thickness of the prisms from the front tothe apex at the back must be multiplied by 0.57to get the Mirror constant. The correction isnegative as the light travels farther when goingthrough the glass due to the index of refractionbeing 1.57 for glass as compared with 1.00 for air.
f. Focus Corrections. The focus correction isthe algebraic sum of the Transmitter and ReceiverFocus as calibrated, minus the sum of the Trans-mitter and Receiver Focus as determined duringobservations. Watch the sign. The formula maybe written:
FC= (Tf +R f,) - (Tfo+- Rfo)
Where: FC=Focus CorrectionTf,=Transmitter Focus Calibrated ValueRf,=Receiver Focus Calibrated ValueTfo= Transmitter Focus Observed ValueRfo=Receiver Focus Observed Value
g. Temperature Correction of Light ConductorThe difference in observation temperature andthe calibration temperature will cause an ex-pansion or contraction of the light conductortubes. This correction can be determined bythe formula:
nL(0.000022) (to- tc)
Where:
nL=Light Conductor, Course (b above)0.000022 =Coefficient of Expansion of Aluminum
(meters per degree Celsius)to=Observation Temperature (Celsius)tc=Calibration Temperature (Celsius)
(to-t,) determines the sign of the correction.
h. Temperature (t) (°C.). The mean tempera-ture (OC.) of the air for each frequency is broughtforward from the observation sheet. If tem-perature is in °F., convert to OC. using table XV,appendix III.
i. Pressure (P) (mm Hgq). The mean pressureis brought forward from the observation sheetand if necessary converted from altimeter read-ings to millimeters of mercury. Conversiontables are furnished in table XVI, appendix III,or computed from the following formula:
P,(288-0.0065h 5.25 6
Po\ 288 ,
Where: P 2=Barometric PressurePo=Barometric Pressure
(760 mm Hg)h=Altitude (Meters)
at sea level
j. Relative Humidity. The relative humiditymay be determined from tables supplied with
the psychrometer or altimeter or by a HumiditySlide Rule (Short and Mason) using the wet and
dry bulb temperatures as arguments.k. Refractive index (Na). Before proceeding
further it is necessary to compute the refractiveindex so that the length of a quarter wavelengthat each frequency can be determined. Theformulas are:
P) 5.5X10- 8 e
760 1+ t1+273.2273.2
and:
1 2 (16.288 ) (0.1361)Ng=l+ 2876.4+3 \82 )+5\ X 4 _ 10-7
Where: N =Refractive IndexNg=Refractive Index for group velocity
t=Temperature in degrees Celsius
P -Pressure in mm Hge= Humidity in mm HgX=Wavelength of light in microns
(10- 6m)
(1) Ng values are furnished in table XVII,appendix III, for a series of values of X.The photocell wavelength (X) must be
furnished on the calibration data and will
be different for each instrument and/oreach photocell.
110
N 1+273.2
(2) Na is determined from the first portionof the Na formula
NQ=1+ N--1 P
1+ t 760)
Na is the refractive index without the
humidity correction.
(3) Humidity Correction-The humidity cor-
rection is taken from the correction for
humidity graph (chart 4, app. II)using Temperature Celsius and Relative
Humidity as the arguments. This cor-
rection is subtracted from Na to produce
the refractive index (Na).
(4) Frequency (f)-The frequencies for fJ,f2, and f3 are determined during cali-
bration and are given in cycles per
second. These values are assumed to
remain constant until a new calibration
is performed.
(5) Quarter Wavelength (U)-The quarter
wavelength is determined by the follow-
ing formula:
U_299,792,5004(f)(Na)
Where: 299,792,500±400 meters per second
is the velocity of light in a vacuum as adopted
by the International Union of Geodesy and
Geophysics and the International Scientific
Radio Union at their international meeting at
Toronto, Canada in 1957.
1. Determination of Number of Units. With
the value for a quarter wavelength at each
frequency of modulation corrected for the temper-
ature, pressure, and humidity at the time of
observation, the number of whole quarter-wave-
lengths at each frequency in the line being meas-
ured can be determined.
(1) From the observation sheet determine
the phase signs from the four signsof each frequency at the top of the page.
If the four signs for the frequency are
alike, the phase sign is positive; if they
are unlike, the sign is negative.(2) From the observation sheet, enter the
approximate distance on the computation
form. This length must be within 1,000meters of the correct length or an in-
correct result will be obtained. If the
approximate distance cannot be de-termined by any other method, a shortbase with all angles turned should beobserved by the field party to providedata for an approximate length.
(3) The sum of the corrections for frequency1 and frequency 2 (KI and K 2) aresubtracted from the approximate length.
LA-K1 and LA-K 2
(4) Nl and NZ are determined from theformula:
N LA-KI LA-K 2-
SLA-K and N LAU1 U2
NI and NZ are rounded to agree withthe phase sign of each frequency. Forexample, the phase sign of fJ is negativeand Ni computes out as 1333.7. Ac-cording to the phase sign, Ni must bean odd number, therefore 1333 is enteredon the form.
(5) LI and L2 are determined from theformulas:
L = (Ni X U1) +K, and L = (N2 X U) +K 2
and then Li is subtracted from L2. IfL is larger than L2 the resulting value
will be negative.(6) U2 is subtracted from U1 and divided into
L2-L' to produce AN which is the
value to apply to N; to obtain the totalnumber of quarter-wavelengths.
(7) The resulting N 1 and N 2 should agree
with the phase sign for each frequency:
Phase sign -, N is an odd number
Phase sign +, N is an even number
A relationship exists between the N
values and is 100, 100.5 and 103. N3 is
determined by multiplying N 1 by 1.03.m. Determination of Slope Length. The unit
length is determined by multiplying the correctedquarter wavelength (U) for each frequency by
the N value for that frequency. The quarter
wavelength has been corrected for temperature,pressure, and humidity and will change for each
frequency. The slope length is determined by
adding the internal corrections (K) for each
frequency to the unit length (UXN) for each
frequency.
111
n. Electronic Distance Measurement Summary.
(1) The slope lengths for a series of measure-ments over a line are entered on DAForm 2854 (Electronic Distance Meas-urement Summary) (fig. 65). This formis provided for listing of the measure-ments, determination of statistics, and
reduction to a geodetic distance on thereference spheroid. After listing themeasurements, rejection of doubtful ob-servations should be made using rejectionlimits determined by Chauvenet's For-mula (table XI). The arithmetic meandistance is determined and residuals(v's) computed by subtracting the ob-served distance from the mean distance.The formulas for computing the probableerrors are given on the form. The ratiois the mean distance divided by theprobable error.
(2) The observed slope distance must bereduced to a geodetic distance on thereference spheroid. Care should be usedwhen abstracting the elevations of theinstruments as observations are madeover several days and the height of theinstrument will vary.
o. Horizontal Distance. The horizontal dis-tance (H) is computed using the Pythagoreantheorem:
H / (SLOPE DISTANCE)2H=
- (DIFFERENCE IN ELEVATION)2
(1) The difference in elevation (d) is obtainedfrom the following formula:
d= (ha+HIa)- (hb+HI,)
Where: ha= Elevation of occupied station
hb=Elevation of distant station
HIa,=Height of instrument at oc-cupied station
HIb--Height of instrument at distantstation
(2) The elevations of the stations are nor-mally determined by either differentialleveling or trigonometric leveling. Whencomputing the elevations by trigono-metric leveling, the reduction to lineformula. is:
Red.= (t-o)(2 06265) tan [H
when H is a horizontal distance and:
Red. (t- o) (206265) sinT
when T is a slope distance and
" is the observed zenith distance. Whenusing slope lengths, the formulas forh2-- hl are as follows:
h2-hl=T sin ( 2-'),
for reciprocal observations
h2 -h 1 =T sin (90°-- +k),
for nonreciprocal observations
(3) If nonreciprocal observations are used, avalue for (0.5-M) or 0.429 should beused in the computations.
(4) The, use of altimeter elevations todetermine differences of elevation is per-missible if more accurate methods ofdetermination are not required. The altim-eters must be carefully calibrated bothrelatively and absolutely, and fairlystable air conditions must exist betweenthe stations.
p. Chord-Arc Correction. The Chord-Arc Cor-rection (K) is applied to the horizontal distanceto change it from a chord distance to an arc dis-tance on the surface of the spheroid. This correc-tion is computed using the formula:
H 3
K- 24p2
or by the approximate formula:
K= 1.027H3 X 10- 15
Where: H=Horizontal distance
p=Mean radius of curvature from table
XX (app. III), using azimuth ofthe line and mean latitude as the
arguments.
49. Geodimeter Model 4
a. Method of Use. The Model 4 Geodimetermeasures distances indirectly by measuring thetime required for a light beam to pass from the
Geodimeter to a reflector and back to the Geo-dimeter. The approximate distance must be
known to within 1,000 meters. According to the
112
.DROJECTELECTRONIC DISTANCE MEASUREMENT SUMMARY
YUMAA - YUCCA TRAVER.5E (IN 5-237)L ORGANIZATION OATE
ARIZONA USAMS PE8 . 196,4TYPE OF LINK MEASURED BASELINE
LIBASE 0 TRAVERSE F- TRILATERATION OI TRIANGULATION
MASTER STATION INST. (7jp*dNo.). N. I. STATION KLKV.INTKLV SKRR
CANAL AMS 1961 /4/1 0.703 M. 56,0o30 5675 WTEREMOTE STATION INST. (7yp. and Me.). H.1. STATION ELKV. INST. ELEV. IOSERVERJUNiCrION LUSCf6S '934 /.9 8ANK PRIS4 .5.608 ". 42. 572 J48180 MP NY
Dos. (Slane) RESIDUALNO ITNE METERS (y) DISTANCE REDUCTION
/0533. 86604 + , 0111 DIFF. OF ELEVATION (d 8, 5552 . 8853 - . 0078 MEAN ELEVATION M) .52. 458 METERS
.__8751 -. 0024 Az. OFLINE (y 900 00' 00"
4 . 832 -. 00-57 MEAN LATITUDE (#a) 33 c 00' 0
S__ .. 8786 .0c// MEANRVADUSE OF 34 METERS____ 86 81 00 94 SLOPE DISTANCE (T) /05338775 METERS
7 .8751 + . 0024 HORIZONTAL DISTANCE (H) -" 0.00o35
___ 8786 -. 00/1 ECCENTRICITY CORRECTION--
___8772 *1. 0003 CHORD-ARC CORRECTION (E) 4. 0. 00/12
10 .8780 -. 000$ SEA LEVEL' REDUCTION (C) - , 00865
11 .8703 .,. 00 72 GEODETIC DISTANCE (1) /0533.7887 METERS
12 .849.5 +. ao080 REFERENCE SPHEROID CLAR KE /8(o6o
13 8.5 coSO-. 50 K H3 :/T
122 H=T- ZT _ dom.14.8823 -. 0048 2T ST
3_
.5 87,82 - . 0 07 Km02Hx05
C:-H h+ H2
S=H+K+C
168822 -. 0047 i'i.I cngicgtogpk. o ~C
17.88 /7 o. O4 2 REMARKS
___ 88 27 - . oo'S219~okse,-vdb ons / the 9 fakens
20 on 21 Sept. /96/.
22 Observatioas /0 Iffr4 /8takeni oej 22 Sept. 1961.
DISTANCEK /0533. 8776 .000550 32
+S3.84 +0t.9'/ I//600 000
PE(ODS) = t 0.674EV2
* - 1) n =NO. OF OBSERVATIONS
PE(Nn) = t 0.6715 ZY2/n(n- 1) ZVa = SUM OF RESIDUALS
SQUARED
1COMPUTED BY DATE CHCKD BYATE
(F Ji,,Qc a- A M 5 DEC . 3 J.L.an uw-- A MS 1JAN. '04DA FORM 2854, 1 OCT 64
Figure 65. Electronic Distance Measurement Summary (DA Form 2854).
757-381 0 - 65 - 8113
manufacturer, the Model 4 Geodimeter will
determine distances within 1 cm. t 5 ppm (parts
per million). The Model 4B is listed as having a
night range up to 3,000 meters, but it has been used
successfully on 8,000 meter lengths under ideal
conditions. The Model 4D employs a mercury
vapor lamp and the manufacturer claims a night
range of 25 miles and a daylight range of 3 miles.
(1) A quarter wavelength for any Geodime-
ter may be determined from the follow-
ing formula:
x C
4 4(F) (Na)
Where :=Quarter wavelength
C =Velocity of light
F = Modulating frequency
Na = Index of refraction
(2) The Model 4 Geodimeter has been de-
signed to have a quarter wavelength of
2.5 meters for Frequency 1, at a tempera-
ture of -6 degrees Celsius and a pres-
sure of 760 mm of mercury. If a color
sensitivity of 5,650 angstroms is adoptedfor the photomultiplier tube, the re-
fractive index for light waves at this
temperature and pressure is 1.0003104.
These values were used by the manufac-
turer to produce the tables furnished with
the instrument, but the manufacturer
used a value of 299,792,900 meters per
second for the speed of light. The
velocity of 299,792,500 meters per secondhas been adopted internationally.
(3) In order to use the velocity of light of299,792,500 meters per second with the
tables furnished with the Model 4
Geodimeter, it is necessary to change themodulating frequencies from F 1 =29,-
970,000 cycles per second; F 2=30,044,-
920 c/sec, F 3=30,468,500 c/sec as
furnished by the manufacturer to thefollowing: F 1=29,969,947 c/sec, F 2=
30,044,872 c/sec, F 3=30,468,445 c/sec.If the frequencies are not changed fromthe manufacturer's values, then the
distance measured can be multiplied by
0.9999986657 to correct the computed
distance for the velocity of light differ-ence.
(4) When the color sensitivity of the photo-multiplier tube is known, a new refrac-tive index should be computed and ap-plied as a refractive index deviation.
Computation of the refractive index andthe deviation is taken up later in thissection.
b. Geodimeter Reductions. The Model 4 Geo-dimeter measurements are recorded on DA Form
2855 (Geodimeter (Model 4) Observations andComputations) (fig. 66) which also serves as acomputation form for field and office use. Forpurposes of explaining the computation procedure
it is assumed that the headings have been com-pletely filled out including elevations, the approxi-
mate distance, and the calibration date. A sheet
containing the most recent calibration (delay line)
data must be furnished the computer. A sample
sheet containing the calibration data is shown in
figure 67.
c. Determination of Meters From Delay LineData. Phase 1, 2, 3, and 4 of Frequencies 1, 2
and 3 with the sign of each are recorded in the field.
The signs for F 1, F 2, and F 3 for the Reflex andGeodimeter are determined by the sign of the
initial setting for Phase 1 of F 1, F2, and F3 .These signs are determined by the direction the
null indicator moves in relation to the movementof the delay line control. The four phase readings
are meaned in each of the six columns. Usingthese mean values as the argument, the meters
are interpolated from the calibration sheet and
are entered on the form to the nearest millimeter.
Subtract the meters in the Geodimeter column
from the meters in the Reflex column for each
frequency. The resulting value must be positive
in each case. If a subtraction is impossible, add
U 1, U2, or U3 to the reflex meters under F1, F 2,and Fs as needed. The signs to be entered in the
Reflex-Geodimeter colurin are determined from
the signs at the top of the six columns of readings
and are paired for each frequency. For example,if the signs for Frequency 1 are both positive or
both negative then the Reflex-Geodimeter sign is
positive (even); if the signs are unlike, the Reflex-
Geodimeter sign will be negative (odd). However,if U 1, U2, or Us is added to the meters so a sub-traction can be performed, then the sign of the re-
flex minus Geodimeter will change. The L 1, L2,and L3 are the Reflex-Geodimeter values. How-
ever, L2 and L3 must be larger than L 1, and
therefore U2 and/or Us are added if necessary
and the signs changed again if they are used.
114
PROJECT
I/V/ADiA T GEODIMETER (Model 4) OBSERVATIONS AND COMPUTATIONS
DATE APPROX. DISTANCE METHOD OF DETERMINATION OBSERVER
28 FEB5./962 3.6 KM SCALED N.'G. K<INGINSTRUMENT NO. REFL ECTOR TYPE CALIBRATION DATE )/ngstroms) RECORDER
47 7 ,cRs~l 123 J&/,y_/96/ 550.0O J. M. WRITEGEODIMETER STATION H. 1. ELEVATION -INST. ELEV. AZIMUTH OF LINE
HOR0,SE, AM/S /962 /38 M, 379.78 m 311 2REFLECTOR STATION HI . ELEVATION INST. ELEV. MEAN LATITUDE
BASE 4~' A/kS /962 75S6 , 439.10 ,M 440.66~ M 33 °00F1 REFLEX GEODIMETER F2 REFLEX GEODIMETER F3 REFLEX GEODIMETER
PHASE SIGN + SIGN +f PHASE SIGN +j SIGN + PHASE__SIGN - SIGN +
1 ,'97-i+ 53+1 9'2+ 54 +1 235 - 55 +
2 /59S -1 52 - 2 88 - 53 - 2 237 + 53 -
3_ /9P2 + S/ + 3 64 + S2 + 237- 534-4_ 9 49- 4_ 8/ - S2 - 4 239 + so0-
MEAN /93. 75 5/ 25 MEAN 86 .25 .52. 75 -MEAN 237.00 52.75METERS 2.137 0.629 METERS /.04 o.647 METERS 2.623 0.439REFLEX - GEOD. $~ REFLEX - GEOD. $0 REFLEX- GEOD. 0(+ U
1 if required) 1. 508 M E (+ U2 if required) 0. 357 M E (+ U3 if required) 2.184 M -c-
_______ S508 M -E- L2
(+ U2if req.) 2.85/ M C L3 (+ U3ifreq.) 2. I84 M -E-
(L2 - L1) X(4) 5.72 HUNDREDS OF METERS (La - L1 ) (21) 6 X42 NI =LA -U 1 / 426(L 3 - L ) X (21) 64._/96 APPROXIMATE TENS AND UNITS OF METERS N2 =LA - U2 / 42.9
ADD 50 METERS IF SIGNS OF L3 AND L1 ARE NOT THE SAME N3 =LA -U 3 = /4.97
LA 3565. 000 M U2 X N2 3563.592 M U3 X1 N3 3564.28 5 M
1508 M L2
_ 2.85/ M L3
2/184 M
Dl :3566.508 M_ 3566.443 r4 ° 3566-46 9 M
9 9 9 9 9 8 6 6 (Dl+ D2 +D3)=3 66 6 METEOROLOGICAL READINGS
_3 6 .6 MTRSGEODIMETER CONSTANT } .23 ETR PRESSSURE TEMPERATURE. HMDT
ALTIMETERREFLECTOR CONSTANT - 0. 030 METERS DRY WET
TEMP.-PRESSURE CORRECTION METERS 28.78 0_5.0 02.2 60o
HUMIDITY CORRECTION + 0. O EES 2.8 0 , 22 6
REFRACTIVE INDEX CORR. (RC) . 08 METERS 28.68 076 03.2 4.5 7%SLOPE DISTANCE 3566. 762 METERS 28.68 076 03.2 45%/
MEAN MEAN MEAN MEANHORIZONTAL 0IST. OR CORR. - 0-.493 METERS 28.73 06.2 02.7 54 7ECCENTRICITY CORRECTION* __- FACTOR (From lNom agram) 25 2 x1 0-6
CHORD-ARC CORRECTION MEESASSUMED REFRACTIVE INDEX 1.0003104
SEA-LEVEL REDUCTION -0.229 MEESCOMPUTED REFRACTIVEINDEX (Ne')( 1.0002859
GEODETIC DISTANCE 3566. 039 METERS REFRACTIVE INDEX DEVIATION (RC) .0000245
N9=1 + L2876.4 + 3 (1J&.F + 5(00- 36)] 10-7 NOTE:
A A If the refractive index correction is used omit
the temp-pressure correction.-a' 1 +0397 N - )
t RC= DX RD
USE cC FOR TEMPERATURE AND MM/HG FOR PRESSURE. ______________________
COMPUTED BY DATE CHECKEDOBY JDATE 1PAGE
F J2S4 2 . 4 9 L 0 4 /SMS DEC. X63 J. Q P n~uo 4M EB. 6 4 ~ OF/
DA FORM 2855, 1 OCT 64
Figure 66. Geodimeter (Model 4) Observations and Computations (DA Form 2855).
115
CALIBRATION TABLES FOR NASM-4-47 JULY 1961
Nominal frequencies
F-1 29,970.000 kc/s
F-2 30,044.920 kc/s
F-3 31,468.500 kc/s
Inst. Constant
Mirror Constant
Unit lengths at -6 C/760 mm Hg
U-1 2.500.000 meters
U-2 2.493.766 meters
U-3 2.380 952 meters
0.235 m
3 & 7 Prism = -0.030 m.
Plastic Refl. = +0.004 m.
DELAY LINE DATA
F-1 F-2 F-3
diff. Div.
102
104
142
142
125
115
113
108
097
094
094
096
098
100
102
l05
111
116
124
130
128
128
128
132
128
123
123
120
106
106
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
270
290
300
0.103
0.208
0.350
0.488
0.616
0.730
0.841
0.943
1.041
1.134
1.230
1.332
1.432
1.532
1.635
1.745
1.860
1.977
2.108
2.234
2.360
2.486
2.611
2.734
2.854
2.976
3.102
3.218
3.318
3.418
diff. Div.
103
105
142
138
128
114
111
102
098
093
096
102
100
100
103
110
115
117
131
126
126
126
125
123
120
122
126
116
100
100
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
Figure 67. Calibration data, Model 4 Geodimeter.
116
diff.Div.
0
10
20
30
40
50
60
70
80
90
100
110
120
130,
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
0.102
0.206
0.348
0.490
0.615
0.730
0.843
0.951
1.048
1.142
1.236
1.332
1.430
1.530
1.632
1.737
1.848
1.964
2.088
2.218
2.346
2.474
2.602
2.734
2.862
2.985
3.108
3.228
3.334
3.440
0.082
0.164
0.250
0.340
0.418
0.494
0.570
0.654
0.742
0.840
0.958
1.094
1.248
1.408
1.570
1.730
1.888
2.035
2.163
2.279
2.388
2.480
2.568
2.647
2.717
2.788
2.864
2.942
3.030
3.134
082
082
086
090
078
076
076
084
088
098
118
136
154
160
162
160
158
147
128
116
109
092
088
079
070
071
076
078
088
104
d. Determination of a Length for Frequency 1.
Thousands of meters are determined from the
approximate distance. Multiply 4 times (L2 - L1)
to determine the hundreds of meters. (L3-L 1 ) (21)100
should approximately equal the decimal part of
(L2 - LI) X 4 and is used as a check only. In the
example (fig. 66), we have 3,500 meters at this
point (approx. distance 3.6 km and L 2-L 1 times
4= 5.372). To determine the approximate tens
and units of meters:
(L 3 - L 1) (21)
If the final signs of F and F3 are unlike, add 50
meters to (L3-L) (21). If the final signs of
F1 and F 3 are alike, the value of (L3-L 1 )(21) is
entered in the column. To determine the correct
tens and units of meters, the final sign of L1 is
used. If the final sign of L 1 is even, the tens and
units of meters are rounded to the nearest multiple
of 2.5 meters ending in 0 or 5. If the final sign
of LI is odd, the tens and units of meters are
rounded to the nearest multiple of 2.5 meters end-
ing in 2.5 or 7.5 meters. In the example, (L3-L 1)
(21) = 14.196; however, L 3 is odd and L1 is even
so 50 meters must be added making the result
64.196 meters. The sign of L1 is even and 64.196
is rounded to 65 meters. The resulting length
(LA) in the example is 3565.000 meters for
Frequency 1.
e. Determination of D.
(1) The number of unit lengths (N) for
each frequency may be determined by
the following formula:
N =LA+ Ul
N 2= LA U2
N3=LA+ U3
or
N1 =LA(0.4)
N 2=N 1 +-1 for each 1,000 meters
N1(0.1)N, =NI- 22
N1, N2, and N 3 must be rounded to agree
with the final sign condition of L1, L2,and L3.
(2) N2 and N 3 are multiplied by U2 and U3
respectively and L1, L2, and L3, which
were determined previously, are re-
entered on the form and added to producethree uncorrected distances. These three
lengths should agree very closely witheach other if all frequencies were operat-ing correctly and there is not an error inthe computations. For ordinary usewhen only one or two sets of measure-ments are taken (one set is one length for
each frequency), the three lengths canbe meaned. However, if a number ofsets of measurements are made, it maybe desirable to consider each lengthmeasurement separately.
(3) At this point in the computations, themean of the uncorrected distances is
multiplied by 0.9999986657 to correctfor the velocity of light. This wasdiscussed in a above.
f. Geodimeter Constant. The Geodimeter con-
stant is furnished by the manufacturer and will
be different for each instrument. This constant
is the distance between the centering point and
the electrical center of the Geodimeter. This
value would normally be checked or redetermined
during calibration of the instrument and shouldbe furnished with the calibration data.
g. Reflector Constant. The reflector constant
varies with the type of reflector used. The
constant for each reflector is stamped on the
reflector housing. The three reflectors most
commonly used are as follows:
3 Prism
7 Prism
Plastic Reflector
h. Temperature-Pressure Correction. The nomo-
gram for the Temperature-Pressure Correction
(chart 5, app. II) may be used for determining
this correction. The value obtained from the
nomogram is in units of the sixth decimal and
must be multiplied by the distance to obtain the
correction. The formula for temperature correc-tion is-
Ct= (Ng-1) 1-273.22732
The formula for pressure correction is-
1 (760-P) X (Ng- 1) (273.2)
273.2+t 760
Where: Ng = Index of refraction (group velocity)
t = temperature (Celsius)
P = pressure (mm of mercury)
Note. If the angstrom units of the photomultiplier
are known and the refractive index is computed, omit the
temperature-pressure correction.
117
i. Humidity Correction. The humidity correc-
tion taken from the correction for humidity graph
(appendix II) is negative when applied to the
refractive index. However, in the Model 4
Geodimeter computation, the value obtained fromthe nomogram is multiplied by the distance to
obtain the correction and is always positive when
applied in this manner. The nomogram tables
are derived from the following formula:
Hum. Corr.=- 5.5X10-et
1+273.2
Where: e = humidity in mm of mercury
t = temperature (Celsius)
j. Refractive Index Correction (RC). The as-
sumed refractive index is based on a temperature
of -6 ° Celsius and a pressure of 760 mm ofmercury. The manufacturer accepted 5,650 ang-
stroms as a mean value for the photomultiplier
tube. Then, from the formula:
N t (760)1+273.2
Simplifying :
N', - 0.359474(Ng- 1)P
273.2+t
therefore:
0.359474(0.00030361) 760273.2+(-6o)
N = 1.0003104
When the angstrom value for the photomultipliertube is known, a new refractive index must becomputed using the mean temperature and pres-
sure at the time of the observation. Ng valuesare tabulated for a series of, angstrom units (X)in table XVII (app. III). N' is the refractiveindex formula without the portion concerning thehumidity. It is more convenient to computeN' and apply the humidity correction separatelyas described in i above. In the example, theangstrom units are 5,500, mean temperature 6.20Celsius, and the pressure 28.73 inches of mercuryor 729.74 mm of mercury. Therefore:
N,-l 0.359474(Ng-1)P273.2+t
N'-l 0.359474 (0.00030453)729.74
273.2+6.2
NQ= 1.0002859
The difference between the computed refractive
index and the assumed refractive index is
0.0000245 which is called the refractive indexdeviation. Refractive Index Correction (RC)equals the refractive index deviation times the
length of the line being measured.
RD = Assumed Refractive Index - Na'
RC = RD X Length
RC = 0.0000245(3566.473)
RC = +0.087 m.
k. Slope Distance. The slope distance is thesum of Mean Geodimeter Distance, Geodimeter
Constant, Reflector Constant, Temperature Pres-sure Correction, or Refractive Index Correction,and the Humidity Correction.
1. Geodetic Distance. The corrections applied to
the slope distance to obtain geodetic distance
have already been discussed under Model 2A
Geodimeter (par. 48).
Section IV. MEASUREMENTS USING ELECTROMAGNETIC WAVES
50. Tellurometer--MRA1--MRA2a. Readings. Tellurometer measurements are
indicated in units of time called millimicroseconds
(10,740 mis, approximately, are required forradio waves to travel 1 mile, out and back).Slope distances up to 9% miles are read directlyfrom the instrument by use of four major patternfrequencies (A, B, C, D). For distances greaterthan 9Y miles, the measuring cycle is repeatedand the number of full 91%-mile intervals must bedetermined and added to the instrument measure-ment before reducing it to a linear value. A
118
negative "A" reading is provided to elimate zero
errors; readings are made with the "A" frequency
in a reverse direction to eliminate centering
errors. These readings, A+, A-, A+R, A-R,
B, C, and D are used to obtain the distance in
time units.
b. Interpretation of Coarse Readings.
Estimated A+ 13 A+ 13 A+ 13 A+ 13
Mileage B 14 C 14 D 40 A- 86
99 99 73 27
(1) Subtract the B, C, D, and A- readings
from the A+ reading to yield unadjusted
differences, adding 100 to the A+ readingwherever it is smaller than the subtrahend.
The coarse reading is composed of the
adjusted tens digit of estimated mileage,the B, C, and D differences, and all the
digits of one-half the "A" difference.
This coarse reading interpretation should
completed in the field by the recorder
before the frequency (cavity tune) setting
is changed. All coarse readings for a
measurement should agree with each
other within a few millimicroseconds
(units of the A scale). The procedure
for adjusting the difference is described
in (2) through (6) below; (7) below,discusses the adjustments for two lines
to clarify the procedures.
(2) Divide the A+A- difference by 2,adding 100 to this difference if necessary
to keep the quotient of the same order of
magnitude as the original A+ reading.
The quotient gives the tens and units
(and perhaps decimal) digits of the
coarse reading; these should then be
recorded.
(3) Adjust the A+D difference slightly if,necessary (adding or subtracting not more
than 3 units) to make its units digit
identical with the tens digit of the A
figures already set down. The tens
digit of the adjusted D difference is the
hundreds digit of the coarse reading.
This should then be recorded.
(4) Similarly adjust the A±C difference, if
necessary, to make its units digit identical
with the tens digit of the adjusted
D difference. The tens digit of the
adjusted C difference is the thousands
digit of the coarse reading and should be
so recorded.
(5) Likewise adjust the A+B difference, if
necessary, so that its units digit corre-
ne 1: Mileage
Difference 28
Adjusted 29
Interpreted Reading
sponds to the tens digit of the adjustedC difference. Use the tens digit of the
adjusted B difference for the tens-of-
thousands digit of the coarse reading.
(6) Scale from a map, or estimate the total
distance in miles. Adjust this estimateif necessary, by the method described
for the different modulation pattern
differences, so that the units digit of themileage corresponds to the tens digit of
the adjusted B difference. The tens
digit of the adjusted mileage becomes thehundreds-of-thousands digit, which must
be supplied to complete the coarsereading.
(7) Lines 1 and 2 below illustrate the methodof interpretation. The distance scaled
from the map is 28 miles.
(a) Referring to line one, the figures 13.5
or Y2 the A difference, are set down as
the final three digits of the interpreted
reading. The D difference is adjusted
to 71 and digit 7 is set down. The C
difference is adjusted to 97 and digit 9is set down. As the B difference does
not require adjustment, the 9 digit is
recorded. The mileage (28) must be
adjusted to 29 and the digit supplied
is 2. Note that for distances less than
10 miles, there will be no tens digit in
the mileage figure and the digit sup-
plied to the interpreted reading will be
zero.
(b) Referring to line two, one-half the A
difference, or 13.5, is set down. The
D difference is adjusted to 71 and the
7 set down. The C difference is ad-
justed to 07 and the 0 digit is recorded.
The B difference is adjusted to 100
(=00) and the 0 digit is recorded.
The mileage must be adjusted to 30
and the three digit supplied.
B C D A
99 99 73 27
99 97 71 13.5 (not recorded)
299 713.5
Line 2:
Difference
Adjusted
Interpreted Reading
Mileage B C
28 99 06
30 00 07T _.T __ .1--.T
300
D A
73 27
71 13. 5 (not recorded)
71- 3.713. 5
Li
119
c. Determination of Fine Readings.
(1) Subtract A- from A+, first adding 100to A+ if it is smaller than A-. Simi-larly, subtract A-R from A+R. These
two differences should be approximatelyequal and roughly twice the original A +reading. Add 100 if necessary; add dif-ferences and divide by 4. The quotientmust be of the same order of magnitude
as the A+ reading.(2) Do the same for each fine reading at the
different cavity tune settings, comparing
corresponding readings differences, andmeans for all readings. Any unusual
deviation from the trend should be exam-ined for rejection.
(3) Average all the accepted mean differences
and substitute the averaged mean differ-ence of the fine readings for the halvedA difference of the coarse readings to
obtain the uncorrected transit time
(UTT). This value is in terms of milli-microseconds (mps).
d. Determination of Auxiliary Readings.
(1) Compute the mean of the crystal tem-
perature readings (MRA-1) at the mas-ter station. The units of measurementare normally microamperes.
(2) Compute the mean of the altimeter orbarometer readings at both stations.Most altimeters are set to read 1,000feet at sea level. The field man nor-
mally makes this correction beforerecording the readings.
(3) Compute the means of the dry-bulb andwet-bulb temperature readings at bothstations.
e. Field Sheet Entries. The field readings onDA Form 5-137 are shown in figure 68. Theline is approximately 34 miles in length. Asshown on the form, the computations are com-pleted through the uncorrected transit time (UTT)by the field party. This is done using the methodsoutlined in the preceding paragraphs. Thesecomputations should be checked in the office beforeproceeding with the Tellurometer reduction.
f. Reduction. DA Form 5-138 (TellurometerReduction) (fig. 69) is used to reduce Tellurome-ter lengths. The following field data is enteredfrom the field sheet:
(1) Project, date of observation, location, andorganization.
120
(2) Names of stations, instrument numbers,heights of instruments (HI), eccentricityof instruments, and elevations of stations.
(3) Mean values for altimeter or barometerreading, dry-bulb and wet-bulb tem-peratures, and crystal temperature.
(4) Uncorrected transit time.
g. Corrected Transit Time.
(1) To determine the corrected transit time(CTT), scale the frequency deviation(FD) in parts per million cycles persecond from the frequency-temperaturecurve graph (fig. 70). (Each masterinstrument has its own graph.)
(2) Compute the crystal correction (CC) fromthe following formula:
CC=UTT X--FD X 10-6
(3) Apply the correction algebraically to theUTT to obtain the CTT.
h. Index of Refraction.
(1) The following explanations and formulasare based on the use of barometricpressures measured in inches of mercuryand temperatures measured in degreesFahrenheit. If other units are used inthe observations, the conversion to inchesof mercury and degrees Fahrenheit maybe made as follows:
(a) To convert degrees Celsius to degreesFahrenheit, use table XV (app. III).
(b) To convert pressure in millibars toinches of mercury, multiply the ob-
served value by 0.02953.
(c) To convert pressure in millimeters toinches of mercury, multiply the ob-served value by 0.03937.
(2) The formula for the index of refraction,n, is:
7=1+10- 6N
N= 4 7 3 0 p 8540e )
\459.688+t) ( P - 459.688+t/
Where:
7 = Index of refraction
P = Mean barometric pressure, in in. Hg.
e = Mean vapor pressure, in in. Hg.
t = Dry-bulb temperature, in ° F.
FIELD SHEET, TELLUROMETER DATA ENTRIES PAG NURNMBERO(TM 5-237)
OPERATION APPROXIMATE DISTANCE DATE METHOD OF DETERMINATION
TET34 MiLES 20 SEPT. 1957 TRIANGULATIONMASTER TATION N T. N . ENRICITY ELEVATION
THOMPSON , /846 A -A25 1524 M o 83 MREMOTE STATION INST. NO. H.1. ECCENTRICITY ELEVATION
5LL'E1-ILL, /845 RA - 26 /.524 mv +1 1.g00 m~ /94 MMASTER OPERATOR REMOTE OPERATOR RECORDER ICOMPUTER
Mc CALL WILSON WILL/AM145
COARSE READINGS FINE READINGS
IA+ A+ A+ A+N 70.5 70. 5 70.5 70.5 FRQ At+RMN
INO. FIEL A+ A-R MEAN.
B05.5 33.0 D 02.L ___ A081 DFF. - 0-DFF. ______
A 50 35 6. 2 /425 71.0 23.51o. 75 6. 12 7 27.0 82.5
INITIAL COARSE FIGURE 3(o 36,71.2 25 MUS ___'.0--- 71.250FA+ 710A+ 4 A+ I A+ 4 71.0O 22.5
B 065C32 4 8 8 _ _2_7.0 _ _8_2.0
ALB0. 4.o 04 284.0 o 4. /40.5 71.12S37L 21,4..0 71.0 23.0o
645 37767,5- 3 9 270 91.0FINAL COARSE FIGURE 36 36o 71.5 MUS -__ W- -a - -T4-2"o - 71.500
METEOROLOGICAL READINGS 70.5 22.5
XT AL. PRESSURE EMPERATURE 4 O 28.0 8/..oTEMP. ALT. R WET -r 3 -747:5. 71. 000
MASTER INITIAL ( //Fr75F65'c71.0 23. 5
REMOTE INITIAL 11sr75~.' 28.0 81.0455 8.5.5 6~9.0 __-m 71.375
MASTER FINAL 78 Ii 795 675 71.0 2.3.06 /2 3 275 _ _ 81.0
REMOTE FINAL 45.5 84.3 68.5 S__ 4: 14-2.0 7/. 375sum /43 /13/ 328.8 272.5 70.5 23.0
7 13 27.5 _81.0MEAN 71.5 MA 28 2 .8 FT 82.20 68.12 __ ___ /~ 0 2.0 71-250
COMPUTATION 70.5 .23.5
UNCORRECTED TRANSIT TIME MS s /4 - -728.0 12 8/ _10
_ _ 70..223.XTAL. FREQ. CORR. ___PP705 2.
CORRECTED TRANSIT TIME MUS 9 /5 275 81.5S-1_______ ______ _ 43.o /41 .5 X.125
DISTANCE IN FEET/METERS 70.65 23.010 /6 270 _ 8 _1.0 _
VAPOR PRESSURE IN. HG. (P) __________ .37_,t-3 g
EQUIVALENT PRESS. IN. HG. (E) 71.0 24.0TOTAL EFFECTIVE PRESS(P+E) 11 /7 /28..0 ,8. 0REFRACTIVE INDEX (N) 7/.0 24.0
REFRACTIVE INDEX CORR. 12 /9 2Z75 - $/.0 1.2
ZERO CORRECTION715 2.
SLOPE DISTANCE 74 7/2508.
COS. VERTICAL ANGLE
14
HORIZONTAL DISTANCE --- ------
ECCENTRICITY _______SUM 927 500
CORR. HORIZ. DIST. MEAN 71. 35
FORM -2D A I JUN ao5-3
Figure 68. Field sheet, Tellurometer MRA-1 or MRA-2 readings (DA Form 5-137).
PROJECT DATE
TES T 20 SEPT TELLUROMETER REDU5-232)LOCATION ORGANIZATION
MASS. 41SAMSMASTER STATION INST. NO. HI ECCENTRICITY ELEVATION
THOMPSON /846 MA -25 524 .METERS 83 M.REMOTE STATION INST. NO. HI ECCENTRICITY ELEVATION
SLE HILL /845 RA- 26 1524 M. 4 .900 METERS /94 M.MEAN ALTIMETER READING DRY BULB TEMP. (t) WET BULB TEMP. (t') DEPRESSION (t -t') CRYSTAL TEMP.
282.8 FEET 82.20 'F 68.12 F /4.0 .F 71. MA
UNCORRECTED TRANSIT TIME(UTT) 3636 7/ " 35 M$S FEQUNCY FD3.4 PPM
CRYSTAL CORRECTION (CC) / 24 MMS e 0. 6 93/ "HG
CORRECTED TRANSIT TIME (CTT) 363670 // M S A e - 0. /56 4 'HG
BAROMETRIC PRESSURE (P) 296 ~2 'HG e 0. 5367 "HG
B A 8.729 /376 N 332.4TELLUROMETER DISTANCE (T) 54 4 94. 672 METERS 1. 0003324
HORIZONTAL DISTANCE (H) .54 494. 559 METERS DIFF. OF ELEV (d) METERS
ECCENTRICITY CORRECTION /. 900 METERS MEAN ELEVATION (h) /40 MEERS
CHORD-ARC CORRECTION (K) .. 0. /6( METERS a MEAN i 36 1 42 0SEA LEVEL REDUCTION (C) _ , i MEAN RADIUS OF 63722051. 7 METERS CURVATURE(P) METERS
GEODETIC DISTANCE(S) 544 1. 628 ETERS
MASTER STATION INST. NO. HI ECCENTRICITY ELEVATION
METERSREMOTE STATION INST. NO. HI ECCENTRICITY ELEVATION
METERSMEAN ALTIMETER READING DRY BULB TEMP.() WET BULB TEMP. (t') DEPRESSION (t-t') CRYSTAL TEMP.
FEET 'F 'F 'F MA
UNCORRECTED TRANSIT TIME (UTT) MS FRDEVIATION CY(FD) PPM
CRYSTAL CORRECTION (CC) M S e' 'HG
CORRECTED TRANSIT TIME (CTT) MIS A e 'HG
BAROMETRIC PRESSURE (P) 'HG e "HG
B A N __ _ _ _
TELLUROMETER DISTANCE (T) M I
HORIZONTAL DISTANCE(H) DIFF. OF ELEV(d) FEETMETERS METERS
ECCENTRICITY CORRECTION MEAN ELEVATION(h) FEETMETERS METERS
CHORD-ARC CORRECTION (K) METERS MEAN i 0
MEAN RADIUS OFSEA LEVEL REDUCTION (C) METERS CURVATURE(P) METERS
GEODETIC DISTANCE (S)
CC=UTT x---FD x 10-6 T e' -e 6 =0.14989625 x- C=-H H-- K H3
S1J=1-{-1 N ...CTT=UTT+CC N=BP+Ae N =VTZ2d24or
H°1/T2 d2 or
B _4730A 8540 4730 or d2
d4
S=HKC K=1.027H3
X101 5
459.688 +t 459.688+ t 459.688+t H=T- 2 - 8T3COMPUTED BY DATE , CHECKED BY DATE
R. k L - As DEc. 63 J. Q A .5" zrs M JAN. '64DAI O o5-138
Figure 69. Tellurometer Reduction (DA Form 6-138).
122
I I I fill
II I I. I I I I 11111111 I I [1111111 1 I I I I I I
74 1 l 1 1 11 1 1 1 i ll :111 ii 1111111 11 I
.1 An r In irIr Ir I I I I I I
Figure 70. Frequency-temperature correction curve.
123
+4
+2
0
-2
-4
-6
-8
-10
- /2
wmz
-14)
f
JIXA IN 1,4.9- 1
100 Li is or
if[1_17 _T
i V.
i
LLT
lit
Substituting in the formula:
8540 4730459.688+t 459.688+t
4730B=-459.688+t
the formula becomes:
N=BP+Ae
(3) The mean barometric pressure (P) ininches of mercury is either observeddirectly or computed from formulas ortables.
(a) P is found by using table XVI, appen-dix III, with the mean reduced alti-metric elevation (ft) of the line as the
argument.
(b) To determine P from formulas use:
_ 288-0.00198h 5.256
Where:
h=Mean reduced altimetric elevation(ft) of line
Po=Atmospheric pressure at meansea level=29.9212 in. Hg.
(4) A and B are taken from table XVIII,appendix III, using the dry-bulb temper-ature t (oF.) as the argument, or com-puted from formulas.
(5) The mean vapor pressure (e) is deter-mined from tables or computed fromformulas:
(a) To determine e from tables, abstracte' in inches of mercury from tableXIX, appendix III, using the wet-bulbtemperature (t') (oF.) as the argument.At the same time, abstract Ae for 1inch of mercury and 1 OF. from tableXIX using t' as the argument. Multi-ply Ae by P(t-t') to obtain the totalDe and subtract the total be from e'to obtain e.
(b) To determine e from formulas, use:
e=e'-0.000367P(t-t')(1+t-32
Where:
t=Dry-bulb temperature, oF.t' =Wet-bulb temperature, oF.
P=Mean barometric pressure, in in. Hg.e'= Saturation vapor pressure of water
in in. Hg. at the temperature t'.
(6) The index of refraction (7) is now com-
puted using the formulas.
i. Tellurometer Distance.
(1) The Tellurometer slope distance (T) inmeters is determined by dividing thecorrected transit time by the index ofrefraction and multiplying the quotient
by the constant 0.14989625.
CTTT (meters) =0.14989625X
(2) The constant 0.14989625 was determined
using 299,792.5 kilometers per second asthe velocity of an electromagnetic wavein a vacuum.
j. Horizontal Distance.
(1) The horizontal distance (H) is computedusing the Pythagorean theorem:
H=AVT2-d2
or by the formula:
d2 d4 d2T 8T 3
16T 5 ....
where d is the difference in elevationbetween the master and remote stations.
(a) The difference in elevation (d) is ob-tained from the formula:
d=(h,+HIm)--(h,+HI,)
Where: hm = Elevation of master station
h,= Elevation of remote stationHIm= Height of Tellurometer at
master station
HI,= Height of Tellurometer at re-mote station
(b) The elevations of the stations arenormally determined by either differen-tial or trigonometric leveling. Thetrigonometric leveling method is ex-plained in chapter 11. Since only theTellurometer slope distance is normallyavailable, certain changes must bemade in the computation. The for-mula for the correction to the zenith
124
distances for the reduction to line
joining stations becomes:
Reduction (sec)= -(t-o) sinT sin 1"
-206265(t--o) sin
T
Where is the observed zenith dis-tance. The formula for the differencein elevation becomes:
h 2 -h 1=T sin (2-1).
If nonreciprocal observations are made,a value for (0.5-m) of 0.429 shouldbe used in the computation and
h2 -h 1=T sin (90°-~-+k)
(c) The use of altimeter elevations todetermine differences of elevation is
permissible if more accurate methodsof determination are not required.For accurate measurements, the altim-
eters must be carefully calibrated bothrelatively and absolutely; and fairlystable air conditions must exist be-tween the stations.
(2) The accuracy required in the difference ofelevation depends upon the horizontaldistance, accuracy required in the hori-zontal distance, and difference of eleva-tion itself. The allowable error may becomputed from the following formula:
H(aH)A= d
Where: Ad=Error allowed in difference ofelevation
H=Horizontal distance
AH=Error allowed in horizontal
distance
d=Difference of elevation
Example: H=5000 meters
AH= 1/50,000=5000/50,000=0.10meters
d=100. meters
Ad=H(H)-5000(0.10) ±5.0 metersd 100
(3) If zenith distances are measured to obtainthe difference of elevation, the error
allowed in the reduced angle can bedetermined from the formula:
A (in sec.) (AT) tan z ( 2- 1)
T sin 1"
206265(AT) tan 2 (P2-1)
T
Where: ('2-1)=Reduced zenith distance
from reciprocal observa-
tions
Alr=Allowable error in zenith
distance
T=Measured slope distanceT=Allowable error in slope
distance
Example: T =5000 meters
AT=1/50,000- 5000/50,000 =0.10 meters
( -- =) 88°51'00"tan ( 's- 1) =49.81573
206265(aT) tan ( 2- r1)
206265(0.10)49.815735000
A= ±206" = ± 00o03'26"
k. Mean Elevation of Line.
(1) In determination of the mean elevation(h) of the line, add the antenna heights(HI) to the elevations, if the elevationsof the stations are known, and then
compute the mean of the values.
(2) If the elevations are not known, scale theelevations from a map, if available, then
compute the mean.
(3) When the elevations and map are un-
available, use the mean corrected al-timeter readings.
1. Geodetic Distance.
(1) The horizontal distance is reduced to thegeodetic distance (S) by applying thesea level (C) and the chord-arc (K)corrections. If either instrument waseccentric during the measurement, an
additional correction for eccentricitymust be applied.
(2) The C correction reduces the horizontal
distance at the mean elevation of thestations to the horizontal distance at
mean sea level. This correction is nega-
125
tive if the stations are above sea level.
The formula for C is:
h h2
C=-H p ... .
Where: H= Horizontal distance
h = Mean elevation of the
stations
p= Mean radius of curvature
of the spheroid, and is taken from table
XX, using the mean latitude (,p) of the
stations and the azimuth (a) between the
stations as the arguments, or is computed
from the formula:
RNPR sin2 a- N cos 2 a
Where: R = Radius of curvature of the
spheroid in the meridian.
N= Radius of curvature of the
spheroid in the prime
vertical.
a = Azimuth of line
(3) The K correction is applied to the
horizontal distance to change it from
a chord distance to an are distance on
the surface of the spheroid. This is
computed using the formula:
K=
23p2
or by the approximate formula:
K=-1.027H3 X 10-' 5
For distances less than 5 miles, K is
negligible.
(4) The application of these corrections-
eccentricity, sea level (normally nega-
tive), and chord-arc (always positive)-
results in a geodetic distance at sealevel on the spheroid of reference.
m. Zero Correction. The zero correction is
determined during calibration of the instrument
by making repeated measurements over a known
distance using paired instruments. This cor-
rection may be as large as several centimeters.
If the form has no provision for a zero correction,it may be combined with the eccentricity and
entered on the form under eccentricity correction.
51. Tellurometer MRA-3a. Capabilities. The Tellurometer MRA-3 con-
sists of two identical units which measure the
phase delay of a radio wave transmitted between
the two units. According to the manufacturer,the Tellurometer MRA-3 will determine distances
ranging from 100 meters to over 60 kilometers
with an overall measuring error of 2 centimeters + 3ppm. As with all electronic distance measuring
equipment, the atmospheric unknowns introduce
more error into the measured lines than the
inherent accuracy of the instrument. To reduce
these unknown conditions to a minimum, only
calibrated thermometers, psychrometers and altim-
eters should be used.
b. Measurement Reduction.
(1) Determination of uncorrected distance.
(a) The field observations, consisting of
meteorological readings, initial and
final coarse readings, and a series of fine
readings are entered on DA Form 2856
(Field Sheet, Tellurometer Data En-
tries (MRA-3)) (fig. 71). The head-
ings should be completely filled out
including an approximate distance.
(b) The initial and final coarse readings
are resolved from the readings A, E,D, C, and B in that order. The second
digit of each value should agree within
three units with the third digit of
the succeeding value (para. 50b).
Should these differ by more than three,the coarse readings should be repeated.
The resolved values for the initial
coarse and final coarse readings should
agree within approximately one-tenth
of a meter under normal conditions.
(c) A sufficient number of fine readings
should be taken over the entire range
of carrier frequencies to provide a
smooth sine curve when graphically
representing the readings against the
frequencies. The "A Forward" and
"A Reverse" readings are meaned and
replace the units, decimeters, centi-
meters and millimeters in the resolved
coarse reading to obtain the uncor-
rected distance. Comparison of thefinal mean of the fine readings with
the individual fine readings will yieldan indication of the relative accuracyof the observations. An excessivedeviation of any one of these readings
126
FIELD SHEET, TELLUROMETER DATA ENTRIES (MRA3 MNKII) PAGE NO. NO. OF P AGES
(TN 5.237)
PERATION APPROXIMA D1
3 i. 27 Feb. 1964 MAP SCALE7MASTER STATION INST NO. H.I1. ECCENTRICITY ELEVATION
ROOF MRA. 188 / 54 M. -- 99-3 m',.REMOTE STATION INST NO. N. I. ECCENTRICITY ELEVATION
G. w. PARKWAY MRA. /80 1.46 Al. - 33.5 Al.
MASTER OPERATOR REMOTE OPERATOR RECORDER ORGANIZATION
R.E. RU'SSMAN W.A. HOFFDAHL W D. moTr USAMISCOARSE READINGS FINE READINGS
TIME CAVITY TIME 1CAVITY RDG. A A RDG. A A
/ 4: 15 20 /S5 00 200 NO. CAVITY FORWAR REVERS NO. CAVITY FORWAR RVERS
IIIL A 81291IA A8Q20O 1 20 829 835 13 /40 820 826
E 4 80 E 4 62 2 30 824 832 14 /50 836 832
D 0 6 9 D 0 6 3 3 40 8/9 827 15 /60 8/0 8/7
c 5 16 4 8 8j 4 50 827 824 16 /70 817 824
B 0 4 9 B1014 1 5 60 816 820 17 /80 820 818
_______6 70 829 .832 is / 9o 816 8/7
101-510141812191 10 5 0 4 8 2 01 7 80 820 826, 19 200 820 8 23
METEOROLOGICAL READINGS a 90 818 830 2DPRA RE DtTEMPERATURE 9 Q 82 p2O 1
PES DY() WET t ) - /O aI80 2
MASTER INITIAL 29.80 3 99 OF 33.2 'F 10 //o 8)5 " 815 22
REMOTE INITIAL 29 9/ 38.2 292 11 120 825 828 23
MASERFIAL 29 80 399 33.2 12 /30 8/5 822 24
REMOTE FINAL 29. 91 35.7 32.0 SUM SUN 155.97 156.68
sum /19.42 153.7 /27.6t MEAN MEAN 8.20/ 8246MEAN 29.86 38.42 31.90 FINAL MEAN _ ___ 8.228BAROMETRIC PRESSURE (P) 2 9 86 "HG DEPRESSION 6 2O
B A 9.496 /(62.807 0.1797 "HG
ZRO CORRECTION (Z) -METERS a 0.1082 "HG
RERCIEINDEX CORR. (RC) + 0. /20 METERS N 30/. 2
[R F A TVECR E TE DS A CE(!).
0 4 8 .22 METER
E VITO 0.07)
"
SOPE DISTANCE (T) J 4 4 METERS REFRATIVE INDDEX 2 Q. PPM
HRIZONTAL DISTANCE (N) 50 47. 9 /9 METERS DIFF OFELEV () 65.8 -ECCENTRICITY CORRECTION - METERS MEAN ELEVATION () //.4 A f
CHORD- ARC CORRECTION (K) -.. METERS 8 MEAN40 335 0 39
SEA LEVEL REDUCTION (C) - 0.0.53 MET ERS CURVAT ADUS OF ~ 2. MTR
GEODETIC DISTANCE (S) 5047. 866 MTR EAK U.rvre
* 473) A '8540 x4730 HO C :N L~I 498 t A 4 5968 j§ jt x 459.6888+t H=VT - d K "; ... C:- p+'NP .
e= e De R= UDx RD x-.f H= d2 d4 =107'X01.-A.RCU X ~ NT....... - K 1.27I~ XO" S=H+K+C
N=UP+Ae RD35NCT=UD+ Z +RC 2T 8T5f
COMPUTED BY DATE CHECKED BY DATE
R. F. IQGu~aPrU V - A MS MAR 44 J1. Qc41~u, - A M5 IMAR.'& 4DA FORM 2856, 1 OCT 64
Figure 71. Field Sheet, Tellurometer Data Entries (MRA-3) (DA Form 2826).
127
from the final mean indicates that
possibly one or more readings should
be rejected.(2) Determination of refractive index of radio
waves. Refer to paragraph 50i.
(3) Zero correction. Refer to paragraph 50m.
(4) Refractive index correction.
(a) The preset refractive index for the
Tellurometer MRA-3 is 1.000325.
(b) The refractive index correction is de-
termined by subtracting the computed
refractive index from the preset re-
fractive and multiplying this deviation
by the uncorrected distance. The
formula is as follows:
RC=(Preset Refractive Index--v) x UD
(5) Horizontal and geodetic distance. Refer
to paragraph 50j, k, 1, and m.
52. Micro-Chain (MC-8)
a. Capabilities. The Micro-Chain MC-8 dis-
tance measuring equipment consists of two
identical units which measure the phase delay
of a radio wave transmitted between any two
units. According to the manufacturer, the Micro-
Chain will determine distances ranging from 200
meters to 50,000 meters at an accuracy better
than 1.5 centimeters ± 4 ppm. As with all of
the electronic distance measuring equipment,the atmospheric unknowns will introduce more
error into the lines measured than the inherent
accuracy of the instrument. To reduce the
unknown conditions to a minimum, only cali-
brated, thermometers, psychrometers, and altim-
eters should be used.
b. Determination of Uncorrected Distance.
(1) For the purpose of explaining the com-
putations necessary to determine thelength of the line measured, it is assumed
that the headings on DA Form 2857(Field Sheet, Micro-Chain Data Entries)
(fig. 72) have been completed, including
the date of calibration (zero correction)and the approximate length of the line.
Also on the field sheet are the field
observations consisting of the coarsechannel readings for frequencies 1 and
9 (channels 3 through 6), the fine channelreadings for frequencies 1 through 9
(channels 1 and 2), and the meterologicaldata.
(2) The difference between each channel
reading and the channel 1 reading pro-
vides the distance data. Subtract chan-nel 1 data from each of the other channels.
In each case, use the channel 1 data
that was observed at each particular
frequency setting.
(3) Total the 2 minus 1 column and divide by
10 to obtain the mean M2. The two sets
of coarse readings are meaned and theresults are transferred to the offset blocks
M6, M5, M4, M3, and M2.
(4) If the channel 1 reading is greater than
any one or all of the other channel read-
ings so that a direct subtraction is im-
possible, add 1000 to the smaller channel
readings so that a subtraction can be
performed.
(5) The mean M2 value is considered correct
and is used to resolve the M3 data which
is correct only to within ± 50 parts. In
the sample computation (fig. 72), theM3 data is 368 ±50. Therefore, the true
number is somewhere between 318 and418. Since the first figures of the "cor-
rect" M2 value are 43, the correct
number for M3 must be 343. The re-
mainder of the distance is resolved as
above and produces an uncorrected dis-tance in millimeters.
(6) Comparison of the mean 2-1 value withindividual 2-1 readings will yield anindication of the relative accuracy of the
observations. The digits in the 2-1
channel are meters, decimeters, centi-
meters, and millimeters. An excessivedeviation of any one of these readingsfrom the mean indicates that possibly
one or more readings should be rejected.
c. Determination of Refractive Index of Radio
Waves. Refer to paragraph 50i.
d. Zero Correction. Refer to paragraph 50m.
e. Refractive Index Correction. Refer to para-
graph 51b. The preset refractive index is the same
as the MRA-3.
f. Horizontal and Geodetic Distance. Refer to
paragraph 50 j, k, 1, and m.
53. Electro-Tape DM-20
a. The Electro-Tape DM-20 equipment con-
sists of two identical units which measure the
phase delay of a radio wave transmitted between
128
PROJECT
TES T FIELD SHEET, MIR-HI DATA ENTRIES
ORAIAINDATE APPROX. DISTANCE
C. of E. /0 JUINE 163 40 KM.AZ. OF LINE MEAN LATITUDE CALIBRATION DATE OBSERVER RECORDER
17.9 IN 38030, JUNE '63 J. MAaxovICH J. SUCZAKMASTER STATION H. 1. ELEV. ELEV. INST. ECCENTRICITY INST. NO.
TA PP FCC. /. 5 4 M. /P797 . 199.6/ - /REMOTE STATION H. 1. ELEV. ELEV. INST. ECCENTRICITY INST. NO.
CLARKE /.54 m. 324.19 M. 325.3J 2
METEOROLOGICAL READINGS CHANNEL
PRESS. TEMPERATURETIME ALT. 6 5 4 3 2 1 2- 1 FNKQp
A DRY (t) WET(t#) ___ ______ ________
MASTER INITIAL 0925 736 88 OF74 'p 508 246 536 460 522 096 426 1 HI
REMOTEINITIA 0935, 726 84 74.5 0960 096 096 096 .527 097 430 2 HI
MASTER FINAL 0945 36k. 89 76' 412 /SO 4.40 364 528 097 4,31 3 HI
REMOTE FINAL 0945 727 85 75 530 097 433 4 HI
SUM 2925,( 34(o 298.5 1. INDICATE *COR* O 6 35 0 98 437 1s HI
MEAN 731.2 186.574.(o TEMPERATURE.
BARONERCPESR P 28,79 'HG 2. RECORD CORRECTED
B A 8.660 /354 ALTIMETER READINGS IF USED.___
a____083 H . ALWAYS ZERO THE NULL 2____e_ 0. 12 92 "HG INDICA TOR WITH A CLOCKWISE 5"28 088 440 5L
e 0. 7344 "HG ROTATION OF THE DIAL. .529 089 440 L
N 348.8 11 1.ooo34.88 52______ 0 95 4 33 LPRESET IDX 1.000325 .6 5__ 4 3 526 091 435 LREFRACTIVE INDEX
/ Qj, ~ CO A~ q 9 3DEVIATI N (RD) - 0.003 .5/ 24 ___ 529 46 .5-03 3
DIF FEE.(d) 126.22 t~S093 4093 093 .093 SUM 4343MEAN ELEV. (b) 262. 22 FEE 4)7 152436 37I MEA
MEAN RADUS OF 6(008.3 METERS 829 302 8 76 735 SUM
438m 368 MA
M44 3 8 B= 4730 A- 8540 X 4730459.688+It 459.68a+t X459.688+t
M3 3 6 8 e-el-ne N=BP+A1e 7)=1+10- N
M2 A 3 - 3O .035? R DxR
UNCORRECTED DISTANCE (UD) £f. j 434 343 METERS
ZERO CORRECTION* (Z) - 0 ./ 6 5 METERS T= UD ± Z ± RC H=-,/T2
)-(d 2,
REFRACTIVE INDEX CORR. (RC) - 0. 98(6 METERS K- 3 C=~ Hh
SLOPE DIST ANCE (T) 414.33 1 92 METERS K242-Hp }
HORIZONTAL DISTANCE (H) A'A /133 o0 METERS. or
ECCENTRIC CORRECTION -i METERS K1O73XO1 SH}K
CHORD- ARC CORRECTION (It) + 0. 0 73 METERS + Obtained fgom metr nt Caibreato.
SEA-LEVEL REDUCTION (C) - 1. 7 // METERS NOTE: Apply .cowntui coireclic to H befog.
GEODETIC DISTANCE (S) 4'4.3 I 3 e62 METERS caautinEK ad C.
REMARKS
COMPUTED BY DATE ) CHECKED BY DATE PAGE
DA FORM 2857, 1 OCT 64
Figure 72. Field Sheet, Micro-Chain Data Entries (DA Form 2857).
757-381 0 - 65, - 9 129
any two units. According to the manufacturer,the DM-20 will determine distances ranging from
50 meters to 50,000 meters with an accuracy of
1 centimeter ±3 ppm.
b. The computations for the DM-20 are identi-
cal to those for the Micro-Chain MC-8 with the
one exception that the preset refractive index for
the DM-20 is 1.000320, while for the Micro-Chainis 1.000325.
c. Field observations and office computations
may be completed on the same form as the Micro-
Chain.
130
CHAPTER 5
TRIANGULATION
Section I. PREPARATION OF DATA FOR ADJUSTMENT
54. Introductiona. Adjusted survey data is the result of a
continuous process of planning, selection, andadjustment. The field survey parties, since theyare most familiar with the field methods andequipment used, should be the first to examinethe field data. The final selection and adjustmentof the survey data is performed by experiencedcomputers.
b. Definite rules in regard to acceptance orrejection of field data cannot always be stated.However, a great many years of experience andtesting have produced methods and rules that arealmost as reliable as basic specifications. Thesemethods and rules may not be the complete answerin all cases but the exceptions are rare. Manyof these methods and rules have been included inthis chapter.
55. Abstract of Horizontal Directionsa. An Abstract of Directions is prepared for
every station at which horizontal angles or direc-tions have been observed. DA Form 1916,Abstract of Horizontal Directions (fig. 73), is usedfor abstracting directions from field record books.This form provides spaces for recording readingsof 16 separate positions of the instrument circle.After the field books are checked, the readingsshould be entered opposite the proper circle posi-tion as noted in the field book. For example,8 readings may be taken at positions 1, 3, 5, 7,9, 11, 13, and 15 of the instrument circle. Thedegrees and minutes for each direction are enteredonly once, at the top of each column, and onlythe seconds are entered for each circle position.All readings are recorded on DA Form 1916opposite the appropriate position number with theexception of those specifically rejected in thefield book.
Note. All field rejected values must be accompanied
by a "statement of reason" in the field record book, i.e.,bumped tripod, wrong light.
b. After all observations not rejected in thefield book have been abstracted, any observationdiffering from the others so greatly that it is anobvious blunder in observing or recording shouldbe rejected immediately. The remaining obser-vations are now meaned. Any readings varyingfrom this mean by more than ±4" for first-order,±5" for second-order, or ±5" for third-order arerejected. (These rejection limits apply basicallyto observations made with direction theodoliteson which the micrometers can be read to ± 1".)The rejection limit for third-order triangulationcan be extended to ± 15" when a transit typeinstrument is used for the observations. For anyposition still having more than one reading, a meanvalue is determined for that position. At this
point, there should be only one value for each
position, and no values for those positions whosereadings have been rejected. A new mean value
(of the remaining readings) is computed and therejection limit is applied again. This procedureis repeated until all the values that remain arewithin the rejection limit. Once a value has beenrejected, it cannot be used again, even though itmay fall within the rejection limit of a subse-
quently determined mean. The mean value ofthe remaining position readings is the final valueof the observed direction.
c. In the example shown (fig. 73), station
Burdell is the occupied station. Directions are
turned to stations Red and Hicks with an initial
pointing on station Lincoln. The first eightsettings of the instrument circle were used.
The first observation of Red for position 2 was
06'7 which the observer considered too high,and the position was reobserved after completion
131
ABSTRACT OF HORIZONTAL DIRECTIONS(TM 5-237)
LOCATION ORGANIZATION STATION
_______________ .2t nI Bar/ell(2v~f vS)
OBSERVER DATE INST. (TYPE) (NO.)
4.1. R. ES vutA 2 2 Av3 . 1 936 w, /d T-2 #146875
POSITION STATIONS OBSERVEDNO.
LINCOLN Red HICKS(c/sc +G S) (29069f) (c'sc "Gs)
(INITIAL) 0 0 .O .3 I, 0 0 0
" o0 00' 294 46 337 10
1 0.00 01.6 3.
2 0.00 (09.4 __ _ _ _ __ _ _ _ __ _ _ _
3 0.00 3 8 1.8
4 0. 00 002.9_ __
5 0.00 (07.6) R 28.2e...4s 32. 3
6 0.00 S /3 .
10 0.00
11. 0.0
12 0.00
13. 0.0
14 0.00
15 0.00
16 0.00
Sum, 6.
l Mean, 44 32.9________COMPUTED BY DATE CHECKED BY DATE
W "C.A. 27yJ
DA1'B 57D 1916Figure 73. Abstract of horizontal directions (DA Form 1916).
132
of the set and a second value of 00'4 was read.The first reading for position 5 was 07'"6 which the
field man also considered too high, and the positionwas reobserved at the same time as position 2.
This time the reading was 04'.'8. The two values
for positions 2 and 5 are included on the abstract,and a mean of all ten values is found to be 02'.4.
The 07''6 reading for position 5 falls outside the
5" limit and is rejected. Now the values for
position 2 are meaned. A new mean of theremaining values (1'"6) is computed. The 06'"7
reading for position 2 is rejected at this time.The mean of the eight values remaining (1 for
each position) is found to be 01'2, and all readings
are within the limits. The final value of the
observed direction to Red is 294°48'01':2. Noticethe values for positions 3 and 6 which are recorded
as 57.8 and 59.1 respectively. The bar over the
numbers is used to indicate that this value of
seconds goes with an angle 1 minute less than thatrecorded at the top of the column. The angle
observed for position 3 was actually 294°47'57':8
and position 6 was 294°47°591 1. When summing
up the column, these values should be treated as
minus 02':2 and minus 00.9 respectively. The
direction to Hicks shows a set of nine observations
(for the eight positions), all of which are within
5" of the mean, including the mean at position 5.
None are rejected. The final value of the direction
is 337°10'32.9.
56. List of Directions
a. A list of directions must be prepared for
each occupied station. The directions to all
objects observed appear on the list (DA Form
1917, List of Directions). This list is used to
obtain the angles for all subsequent computations
and adjustments. It must be complete and
accurate.
b. In some cases, more than one initial direction
is used for the observations at a station. When
this occurs, a preliminary list of directions (fig.
74) is prepared and refers to the respective ini-
tial directions. From this list, the various direc-
tions are consolidated into a final List of Direc-
tions (fig. 75). All of the directions are referredto a single initial, and one direction is shown to
each observed object. Normally, these- directions
are listed with the initial direction being the one
most counterclockwise in the triangulation scheme
or network, with the others following in order of
their increasing directions. This procedure allowsthe internal angles of the scheme to be readily
obtained as the differences between desired
directions.
c. Whenever observations of directions have
been made using different initials, or where the
values vary too greatly to be meaned by normal
procedures, it may be necessary to compute a
station adjustment before consolidating the di-
rections. Generally this is done only for first-
and second-order observations. A station ad-
justment is a least squares solution of all of the
directions observed from a station, with proper
weights applied to each. Statistically, it provides
the most probable direction to each observed ob-
ject. A sample station adjustment is presented
in paragraph 61.
d. Any eccentricity of instrument or object
should be clearly explained by sketch and descrip-
tive information on the list of directions. The
computation of the eccentric reduction provides
the correction which, when properly applied to
the observed direction, reduces it to the value it
would have been if the instrument or target or
both had not been eccentric. A sample compu-
tation of an eccentric reduction is presented in
paragraph 58.
e. The sea-level correction or reduction com-
putation is determined by the formula in para-
graph 14b(12), but the numerical value of the
correction may be taken from the "Sea Level
Reduction Chart" (chart 3, app. II). The arguments
used in the chart are: latitude of the occupied
station, elevation of the observed station, and the
azimuth of the observed line.
f. The eccentric reduction and the sea level
reduction are listed in their appropriate spaces on
DA Form 1917, and the application of these two
corrections provides the "Corrected Direction
with Zero Initial". The values in this column
are used in all subsequent computations and ad-
justments.g. The final column on DA Form 1917, "Ad-
justed Direction", is completed only after a final
adjustment of the network has been accomplished.
57. Triangle Computation
a. Basic Factors.
(1) The basic computation in triangulation
is the solution of a triangle in which at
least two angles and one side are known.
(Ordinarily all three angles in the tri-angle are observed). The solution of
this problem is made by the law of sines
for a plane triangle. Since the observed
133
PROJECT ORGANIZATION LIST OF DIRECTIONS1-243 z 9 Engs. (TM 5-237)
LOCATIONa/fra STATION
OBSERVER INST. (TYPE) (NO.) DATE
C21 RE smith k/i/d T-2 10144687S _______2 Auq -ifOBEREDSTTONOBERE IRETIN ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADWvsTLjD
O ERE STTOOSEVDDRCIN REDUCTION REDUCTION' WITH ZERO INITIAL DIRECTION'
0 I if / if if 0 N f0 i
L//vcoLN (ac 0 00 00.00 _________0 00 00.00 ______
14/cA's (SC#6s) 337 /0 32.9 ___ ____________
£etNitd=7342m (a 27 37 ______ ____
OW o. Z da A. 91 84 1 2
=0 03am 7 42~ _____ _____
" These eolumns are for office use and should be left blank In the field.
COMPUTED BY DATE1 CHECKED BY DATE
DA IFE 7 1917Figure 74. List of directions (preliminary) (DA Form 1917).
134
PROJECT ORGANIZATION LS OFDRCIN1-439-' Engrs (TM 5.237),,,
I T O I E T OS23LO
C AT IO N STAT I O N
Ca//vrnia U eOBSERVER IS.(TYPE) (NO.) DATE
Cpt PE. stn;m Wild T-2 #14685S _ _ _ _ 22 Aut sOBSERVED STATION OBSERVED DlIRECTION ECCENTRIC SEA LEVZ L CORRECTED DIRIECTION ADJUSTED
REDUCTION RDUCTION' WITH ZERO INITIAL DIRECTION'
_____________ 0 00 00.00 0 00 600.00
8W, No. 1,sZ 342m 6 27 .37_____
"These columns are for office use aid should be left blank in the field.
COMPUTED BY DATE CHECKED BY C .DATEw
/A. FORM 111 EB 571917
Figure 76. List of Directions (Final) (DA Form 1917).
135
angles are not plane angles, but sphericalangles, the sum of the three angles ofthese triangles is slightly larger than the180 ° associated with plane triangles.The amount that this sum is larger thanthe plane triangle sum is known as thespherical excess (e), the computation ofwhich is explained in f below. It hasbeen rrwven that the spherical trianglecan be solved by using plane angle for-mulas if one-third of the spherical excessis subtracted from each of the sphericalangles. Therefore, one use of thismethod of triangulation computation isas a preliminary step to finding thespherical excess in the triangle.
(2) Since the true total of the angles in a
triangle is known to be 180°±e, itfollows that the observed angles must
add up to this total. Seldom, if ever,
does the occasion arise when the observed
angles do total exactly 1800°+e in anytriangle. Therefore, some correctionmust be made to each observed angle to
perfect the total. Various methods areused to find the values of the corrections,such as a most probable value by least-
squares, or approximate corrections asthree equal values. Whatever method is
used, the algebraic sum of the corrections
must equal numerically, but with op-
posite algebraic sign, the sum of the
observed angles minus (1800+e). This
difference between the sum of the
observed angles and 1800+ E is known asthe triangle closure. In the approximate
adjustment of the triangle, one-third of
the triangle closure is applied, with
opposite sign, to each of the observed
angles. If the triangle closure is not
exactly divisible by 3, the odd value
correction is applied to the angle nearest
900. Applying corrections to the ob-
served angles produces the spherical
angles which total exactly 180°+ e. One-
third of the total spherical excess of the
triangle is subtracted from each of the
spherical angles to obtain the plane
angles. If the spherical excess is not
exactly divisible by 3, the odd value
correction is applied to the angle nearest
90 °.
(3) Figure 76 illustrates the triangle num-bering system. For computation, thevertices of the triangle are numbered inclockwise order, starting with number1 at the vertex to which the lengths areto be computed. This means the ends ofthe known side are numbered 2 and 3.The station names and the anglesobserved at the station are entered besidethe appropriate number. By applyingthe corrections and spherical excess tothe observed angles, the plane angles areobtained.
b. Computation By Logarithms. For computa-tion by logarithms, DA Form 1918 (Computationof Triangles) is used. Enter the log of the knowndistance on the line 2-3 and the 1 )g sines of theplane angles in the column headed "Logarithms".The computation process for the unknown sidesis then:
(1) Log (2-3) minus log sin (1) plus log sin
(2) equals log (1-3).(2) Log (2-3) minus log sin (1) plus log sin
(3) equals log (1-2).(3) A numerical example is shown in figure
77. In the example, Burdell is station 1,Hicks station 2, and Lincoln station 3.The observed angles are 22°49'27':1 atBurdell, 4900'55'.2 at Hicks and 108009 '
31'.'4 at Lincoln. The distance Hicks-Lincoln is 4,870.241 meters (log dist.
3.687 55045) and the spherical excess is
0' 1 for the triangle. The names of thestations are listed in the station column
on the form-(1) Burdell, (2) Hicks, and(3) Lincoln. Then the observed anglesare listed in the observed angle columbeside the station where they wereobserved. Total the angles (179°59'53''7)
i
Unknown lengths
to be computed
3 Known Length 2
Figure 76. Triangle diagram.
136
PROJECT DATE
1-24 12 4u 55 COMPUTATION OF TRIANGLES(TM 5-~n7
LOCATION ORGANIZATION
Ca khrnja ______2En,__ ___SPHEICALSPHEICAL PLANuLOcRIow:
STATION OBSERVED ANGLE CORRECTION. ANHRCL SERCAL PANE LOAITHMANGLE Exc~a ANLE G NATURAL
___ 2-3 - -__ _ _ _ _ __ _3£7A A
_-1Byl 22 4!?2ZL # 2.1 29.2 0.0 2f,2 IF, s58 7q-9
2_ t LIcu 8 g14#2.1 5. 3 0.L0 53. 9Zg7 8 I2
__ _ _ _ _ _ 79 9 53.7 *6-+ 0./ -0.1 000 _ _ _ _ _
____ 2-3-__________
1 ______________
____ 2 __________ ____________
____ 3 _______________ __________
____ 1-3 ____ __ __________ ____
-~ Luios 4870241
1_ ___ 22 44 271 2. / 29.2 0 f~ 0. Od21827 914212 k~ks49 00 55: *2.1 57.~2 523i 0. 75489/80
__-3 L/,VCOLA/ /0 O9 31.4 ±42 .3 b0~ -. 0 90 9
1-3 -~Z64
__1-2$ t ,.I.N~' 11929.627__________ 79 5? 53.7 # 6.4 Oa / -0.1/ 00.0 ______
2-3
1
2
3
____ 1-3 _____________ _________
____ 1-2
COMPUTED BY npDATE CHECKED BY DATE
DAFORM 3E4DAIFEB5713 !8
Figure 77. Computation of a triangle using three angles and a known side (DA Form 1918).
137
and subtract (1800+e) from the sum to
find the triangle closure.
(179059,53"7) - (180 +0'.1) = -6'.'4
The correction (+6.4) is divided into
thirds (+2.1, +2.1, and +2.2) andapplied to the angles; the odd part to
the angle nearest 90° (large angle),in the correction column, to give the
spherical angles in the spherical anglecolumn. Then the spherical excess is
divided into thirds (0':0, 0'0 and 0'1)and placed in the spherical excess column.These values are subtracted from the
spherical angles to give the plane angles
in the plane angle column. Finally, thelogarithms of the plane angle and the
logarithm of the distance (2-3) are
listed in the logarithms columns, and
the distances (1-2) and (1-3) are foundusing the law of sines. The logarithmof (1-2) is 4.07662688, and the logarithm
of (1-3) is 3.97669947.
c. Computation By Natural Functions. For
computation by natural functions, the same DAForm 1918 is used. Enter the length of the knowndistance (2-3) and the natural sines of the planeangles in the column labeled Logarithms (strikeout Logarithms when using natural functions).The computation process is then-
(1) Length (2-3) divided by sin (1), thenmultiplied by sin (2) equals length (1-3).
(2) Length (2-3) divided by sin (1), thenmultiplied by sin (3) equals length (1-2).
d. Computation Using Two Sides and IncludedAngle.
(1) On DA Form 1919, Triangle ComputationUsing Two Sides and Included Angle(fig. 78), sides a and b and angle C areknown. The problem is to find side c,and angles A and B.
(2) Call the longer known side a and enterlog a, log b, and the measured angle CSon the form. If the triangle is large anda precise solution is required, the sphericalexcess in the triangle should be computedby adding log m (from table of log m inTM 5-236), log sin CS, log a, and log b.This sum will be log spherical excess.This computation is illustrated in theupper right-hand corner of the form.Subtract % of the spherical excess from C,to obtain CP the plane angle. Subtract3C, from 900 to find %(AP+Bp). Sub-
138
tract log b from log a, the result being logtan (45°0+). From a table of logarithms
of trigonometric functions, extract theangle (45°0+). From this angle, sub-
tract 450 to obtain angle 4. From the
table, find log tan 0 and log tan 32(Ap+Bp). Add these logs to find log
tan %(AP-BP). From the table, findangle Y(AP-B,). Add angle %(A,+B,p)and angle %(A,-B,) to obtain angle A,.Subtract angle 2(A,-B,) from angleY(A,+B,) to obtain angle B,. The sumof angles A,, B,, and C should be 180 ° .
Side c is now computed by the sine law.Log a plus log sin (Cp, minus log sin Ap,equals log c. Since the solution forangles A, and B, can be in error and thesum of the angles A,, B,, C, still equal1800, a further check must be made onthe solution. This check can be madeby solving the triangle twice by the sinelaw, using first the fixed length a andthen the fixed length b as the startingline. Agreement of the computed lengthsin these two computations will prove thevalue of A, and B,.
e. Machine Computation of Triangles.
(1) DA Form 1920, Triangle Computation(For Calculating Machine) was designedespecially for use with the calculatingmachine (fig. 79). All four commoncases for triangle solution can be solved;and a solution of the threepoint prob-lem can be made on this form.
(2) The four cases to be solved:(a) Given one side and either two or three
angles.(b) Given two sides and an angle opposite
one of them.
(c) Given two sides and the included angle.
(d) Given three sides.(3) Since there is space for four triangle com-
putations on the form, one case will becomputed in each space. The three-point problem will be solved separately.All of the problems are taken from thetriangulation scheme shown herein.
(4) A general explanation of the form isgiven to explain a few of the headingsthat might not be too clear. The Sym-bol column is for use when the triangu-lation is to be adjusted and the anglesare designated by some kind of symbol.
PROJCT 3 TRIANGLE COMPUTATION USING TWO SIDES AND INCLUDED ANGLE!-2T'3(TM 5-237)
LOCATION ORGANIZATION DATE
Ca-aa2 9 V' E'ng 922 Aug38
[b-tan (45°-+40) (Call longer side a). tan 1 (Ap-Bp)=tan 4'tan 1. (Ap+B~ c a sin 0*
z z p)sin A 1C. !Z90. Log a 3.90704529 Logi m .4047
Spb exes Log b 3. 687?S04S Log sin C. I4qqg-C 91. 2q e'q., Log tan (45*+04) 0.2qq8 Log a
O5_-3 o C 45 44 34.25 (4510+4') 08 5 o 85
90 0 1- Q=(A.±Bv) 44 1S 25.75 0' / Log sph. ex. p qqgj16j (AD--Bp) /3 33 2.3 Log tan, 4' 95 0083 Sph. excess 00MSumc=A 0 574 49.31 Log tan 3 (Av+ B0) 3 7 ___ ___
Dif=0 30 42 g2, g Log tan 3 (Ap-Bp) 9. 382 236820________
CO g ~o. LINCOQLN (Sketch)
Log a 3. 30704529 b
Log sin0 Qgaag
Colog sin AD rc14~L~ CIKLog c 3c73d49 _________________
CHECK COMPUTATION_______
No. STATION SPHERICAL ANGLE SPHERICAL P'LANE EXES AGE LOGARITHM
2-3 Rd-ICS3. 90704529
___ LINCOLA' 57 48 49.31 4 1.92 ~g753488
1-3 IIAcoLN - 1ICK 3.Z5 68____1-2 L INCoL N - Red 3. q793643g
____2-3 HI/K - LbI 3.L85S4
____1 Red 30 0221 o~.9___ a g 7080o0
___2 Ilcks g1 2g 86 , 08. 1. 9* 5qp
____3 L INCOLIV 57 48 49.31 ____49.31 9. q Z7S3488.__ _1-3 Re-LINCOLN _____ 3____ i93~6441____1-2 -HIK 1 ____ ___ 3. q0704531
"The subscripts s and p on this form refer to spherical and plane angles respectively.
C OMPUTED BY DAECHECKED BY DATE
FORMD AI FEB 6711
Figure 78. Computation of a triangle using two sides and the included angle (DA Form 1919).
139
PROJECT 1-243 TRIANGLE COMPUTATION (FOR CALCULATING MACHINE)(TM 5-717)
LOCATION ORGANIZATION
Ca/,(ornia .2____._
SPIIER'L SPIIERIS PLANE SDSYMBOL STATION OBSERVED ANGLE CORR'N ANGLE EXCESS ANGLE SINE DISTANCE
7oH 2 Y/3 a_1 0296 0.0 .29 902-32 #0. 1-3
3 1.4.9 34. / 34.. M U72 1-2 -
Cas
2 (/2F .$ - l16 R-13 -
D=Ratio, side/sine 2539.36041 -
0'
1 - o. . 2-3 g
2 -- 49.G 496 1-3/? - 45.9 6 45.8 /6 2. 1-2
fen = 0805/532C -+D=Ratio, side/sine 1OZS0.08g9
2 9 2. 991 -- O.0. 389yl- 48o212-3 IK-
141 IS . 1-3 ujl
3 Liol / f 3. 3-ta 42rO.532 1-2
/. I9L/69437 25 - 2. 6277277 D=Ratio, side/sine /25CosA a 0. ?2437 ae . /30622/I (s-a) 0.82 683985
Case I a/sin A~b/sin'Bc/sin C Case III tan A-a sin B/c-a cos B COMPUTED BY OATE
Given: 3 angles, I side Given: 2 sides and included angle le& 23 ASS -Case II sin B=b sin A/a CaseIV c AI2s(sa)/bcll s=1/2(a+b+c) CHECKED BY DATE
Given: 2 sidekand an angle opposite Given: 3 sides . C.3FORM109
DA, FEB 57 92
Figure 79. Computation of triangles (DA Form 1920).
The Corr'n column contains the correc-
tion to be applied to each angle as de-termined by an adjustment. The Sidecolumn provides space to label thelengths found by computation by nameas well as by number. The ratio, D, ofside to sine is the length of a side dividedby the sine of the angle opposite that
side. The ratios of the sides to sinesshould be constant in a triangle. For
instance, in case (1) side 2-3 (4,870.241
meters) divided by the sine of angle 1(sine 220 49' 29'.'8 = 0.38791690) is a ratio,D, of 12,554.8565. Now side 1-3 di-
vided by the sine of angle 2 should bealmost exactly the same as the ratio2-3/sin 1, and it is 12,554.8560. Thesame hold true for the ratio of 1-2/sin
3=12,554.8569. These ratios can beused as a check on the computations.
(5) For case (1), one side and either two orthree angles in the triangle are given.If only two angles are given the third isconcluded, and if three are given theymust be corrected to close the triangle.For the illustration of case (1), the tri-angle Burdell-Hicks-Lincoln was usedwith the three angles and side Hicks-
Lincoln (4,870.241 meters) given. Thecorrections to the given angles are from
an adjustment; the spherical excess is
computed as explained in paragraph
57a and is applied to the spherical anglesto obtain the plane angles. The sines
of the plane angles are entered in the
column so headed. The given distance
(2-3) of 4,870.241 meters is entered in the
distance column. The triangles arewritten as previously explained for the
triangulation net. The given distance
140
is divided by the sine of angle 1 (the angle
opposite), 4,870.241-0.38791690 to findthe ratio, D. The ratio is multiplied bysin (2) to obtain the length 1-3, and again
by sin (3) to obtain the length 1-2.
The computation of the triangle for case
(1) is now complete.
(6) In case (2), two sides and an angle op-
posite one of those sides are given. For
this illustration, the triangle Red-Hicks-Lincoln will be solved with the known
components of the triangle being the
angle at Red (30°41'59"7) and the sides
Hicks-Lincoln (4,870.241) and Red-Hicks
(8,073067). The general formula forsolving this case is-
b sin Asin B= a
in which angle A and sides a and b are
known. Angle B is found from its sine
and the third angle in the triangle is
concluded. In this example the ratio,D, is found by dividing side 2-3 by sin
(1) to get 9,539.3604. The sine of angle
(3) is now obtained by dividing side 1-2
by D. Angle (3) is found from its sine.Angle (2) is concluded, its sine obtained,and side 1-3 found by multiplying D
times sine (2). The triangle is now com-
pletely solved. Notice the first two cases
use only three lines of the five provided
for each triangle. The extra two linesare needed only for cases (3) and (4).
(7) Case (3) is a triangle is which two sidesand the included angle are given. The
general formula for this solution is:
a sin Btan A sinB
c-a cos B'
a and c being the given sides and angle
B the angle between them. The trianglechosen to illustrate the case is Red-
Black-Hicks, and the known values arethose obtained from the angle method
of adjusting the triangulation. Thissame triangle was solved on DA Form
1919 as part of the angle method of
adjustment. The known data are sidesBlack-Hicks, Red-Hicks, and the angle
at Hicks. Numerically, these values
are 6,428.344 meters, 8,073.192 meters,
and 89°11'45'"9, respectively. The firststep in the solution is to obtain the sineand cosine of the known angle fromtables. The correct algebraic sign of thecosine is essential to the correct solution.The cosine for angles between 90° and180 ° is negative. The cosine is enteredon the form on the fourth line of thesine column. To avoid confusion, theword "cos" may be written beside the
value if desired. In this case, cos (3)=0.01403102. For this solution, it isusually better to call the shorter knownside a. In this example, then, a= (2-3)
= 6,428.344, c = 8,073.192, B = 89'11 '45 '.9.
The denominator of the fraction in theformula, c- a cos B, is solved and written
below cosine B and can be labeled"Denom." The numerator of the frac-
tion, a sin b, is multiplied on the machine
and divided by the denominator pre-
viously written down to obtain tan A,which is the tangent of the angle oppositethe side a. Notice that the product,a sin b, need not be written in theObserved Angle- column on line 4
(0.80517532). The angle A [in thisexample, angle (1) at Red] is taken outfor tan A and entered in the "Plane
Angle" column. Remember that allthe angles used in these triangle compu-tations are plane angles. Only the
seconds of the plane angle are written
in that column; the degrees and minutes
are written in the Observed Angle
column. Angle (2), at Black, is now
concluded. The sines of angles (1) and
(2) are found and the ratio, D, obtained,and the missing length found by multi-plying D times the sine of (3). The ratio
of the side to sines should now be usedto check the computed and concluded
angles. The solution of the triangle is
now complete. A comparison of thissolution with the solution made on DA
Form 1919 in the angle method of
adjustment of the same triangle, shows
that exactly the same results are ob-
tained.
(8) Case (4) is the solution of a triangle whenthree sides are given. For this example,the triangle used for case (1), Burdell-
Hicks-Lincoln, was used with the lengths
141
found by solving case (1) becoming the
given data for case (4). The generalformula for solving case (4) is-
cos A= 2s(s-a)be
in which s = % (a+ b+c) ; a, b, and c being
the given lengths. The angle A is oppo-
site side a. In this example, side a is
Hicks-Lincoln (4,870.241), b is Burdell-
Lincoln (9,477.506) and c is Burdell-
Hicks (11,929.532). Side a should not
be a very short line as compared to the
other two sides. For the sake of con-
venience throughout the computation,the decimal point in the given lengths
may be moved the same number of
places to the left in each to keep the
numbers near unity. A four-place shift
was sufficient in this example. No
change in the shape of the triangle or
size of the angles will be incurred by this
shift of the decimals. The lengths arenow used as 0.4870241, 0.9477506, and1.1929532, although they are not written
down as such. The solution of theformula is now begun by finding 2s, be,and (s - a). 2s is found simply by adding
a+b+c. This result is 2.6277279 and is
written on line 4 of the form. It can belabeled "2s" if desired. One-half of 2s
can be found mentally and from this
quantity subtract a. The result is (s-a)
and is labeled and entered on the form in
any convenient space on lines 4 or
5. The product (be) is found andentered on the form. Numerical valuesfor these three items in this exampleare: 2s = 2.6277279, (s- a) = 0.82683985,bc=1.13062211. Cos A is now obtained
by dividing the product of 2s X (s - a)by be and then subtracting 1 from theresult. Angle A is now found using cosA as the argument. If cos A is positive,angle A is less than 90°; if cos A is nega-
tive, angle A is between 90° and 180 ° . In
this example, cos A = +0.92169437 (posi-
tive, so less than 90 ° ) and angle A =
22049'29'.'8. Next find sin A, and from
this the ratio D. Using D and the given
sides, the sines of the other two angles in
the triangle are found by the relation,
side bsine = from which sin B = and
sin C = D" From the sines the angles
may be found. If angle A is less than90 ° , it is possible that either angleB or C may be more than 90 ° . Theangle over 90° can never be oppositethe short side. In the illustration, Dis found to be 12554.8565; sin B (sin 2)
equals 0.75488764, sin C (sin 3) equals0.95019262. From sin B, angle B couldbe 49° or 1300. Using the statement
concerning the short side, the correctangle for B in this triangle is 49°00'56'.0,as side b is shorter than side c (9,477 to11,929). Now from sin C, angle C
could be 71° or 108 ° . Since the threeangles in the triangle must total 1800,it is easy to see that the correct value
for angle C in this triangle is 108°09'342.
The solution is now complete.
(9) Comparing the angles computed for thetriangle by solving case (4) with the
angles used in case (1), it is evident thatthe case (4) solution is correct, since thecomputed angles are exactly the same as
the original angles of case (1).
f. Spherical Excess Computation.
(1) It is known from spherical trigonometrythat the three angles of a spherical tri-
angle add up to slightly more than 1800.The difference between 1800 and the
total of the angles in the triangle is called
the spherical excess (E). As this spherical
excess amounts to approximately 1
second for every 75 square miles of area,it is evident that for lower-order tri-
angulation or for triangles covering asmall area the amount of spherical excess
is negligible and need not be computed.
(2) The formula for computing spherical ex-
cess is:
albl sin Cl(1-e 2 sin2 0)2e- 2a
2(1-e
2) sin 1"
In the formulas for the spherical excess,al, bl, and C1 are two sides and the in-
cluded angle of the triangle; e2 is the
eccentricity squared; a the semimajor
axis of the spheroid of reference; and0 the mean latitude of the vertices of the
triangle. Since e2 and a are constant
142
values for each spheroid it has been
possible to tabulate a factor m for
(1-e 2 Sin 2 n)22a 2(1 e2 ) sin 1" using the mean latitude
as an argument. Thus the formula forspherical excess becomes e=albl sin Clm.
A table for log m for the Clarke
18C6 spheroid expressed in meters is given
in TM 5-236 and USC&GS Sp. Pub.138 and 247. Sp. Pub. 247 and TM
5-236 have tables of the natural value
of m for -the Clarke 1866 spheroid.
USC&GS Sp. Pub. 200 has a table of logm for the International Ellipsoid. For
an equilateral triangle with 200-kilo-meter sides, the above formula will give
a value of the spherical excess too smallby 1/100 of a second. Therefore, thisformula is entirely adequate for allnormal triangulation.
(3) Since spherical excess is a function of the
area on the sphere, the total sphericalexcess in a triangulation figure is con-stant. For example, the sum of thespherical excess in each pair of trianglesmaking up a quadrilateral should beequal.
(4) Figure 80 illustrates the computations andapplication of spherical excess. Giventhe quadrilateral Lincoln-Burdell-Red-Hicks:
Log
a1 = Lincoln-Burdell = 3.97670b = Lin coin-Hicks =3.68755
C 1 =Hicks-Lincoln-Burdell= 1080 09' 31"log sin C 1 =9.97781
Burdell Red
Lincoln Hicks
Figure 80. Quadrilateral diagram for spherical excess.
Mean latitude of Lincoln,Hicks, and Burdell is38008'
Log m (Clarke 1866) =1.40471-10
The spherical excess of only one triangle(Burdell-Hicks-Lincoln) is illustrated,since the computation for the other threetriangles in the quadrilateral is merely
a repetition of the process. Using theformulas: E=a lb1 sin Clm, and arrangingthe given data in columnar form:
log al
log b1log sin
log m
=3.97670
=3.68755
C1= 9.97781
=1.40471
log e = 9.04677e =0.111 sec.
This computation can be made on aregular triangle computation sheet as
a preliminary step. Logarithms need becarried only to five places. Splitting thequadrilateral into two pairs of triangles
to facilitate recording the spherical excessfor each triangle, shows the sums foreach pair of triangles to be equal (fig. 81).
Burdell Red
0 Sum-O3
® Sum -O3
Lincoln Hicks
® First pair of triangleso Second pair of triangles
Figure 81. Equality of spherical excess.
143
58. Reduction to Center
a. Need for Reduction. If, when observing
horizontal directions, the instrument or the signal
is not set exactly over the triangulation station,the instrument or the signal is said to be eccentric
to the true station. Directions measured from an
eccentric instrument, or to an eccentric signal,must be corrected to the directions that would
have been observed if the instrument or object
had been centered over the station. The compu-
tation necessary to correct these directions is
called reductio to center or eccentric reduction.
b. Computation.(1) The formula for computing the reduction
to center (symbolized c) is as follows:
d sin ac sin 1"
and is derived from the law of sines.
Figure 82 illustrates the general form of
the problem in which-d=measured distance from eccentric
to true station.a-clockwise angle from the true
station to the distant station asmeasured at the eccentric station.
s =distance from the true station tothe distant station from prelimi-
nary triangle computation. At
times, it may be possible toobtain only an approximation of
s which means that successivecomputations must be made until
no change occurs in the computedvalue of c.
From figure 82 and the law of sines, therelation can be written as follows:
d s
sin csin a
from which:
d sin asin c=d
s
Since the basic nature of an eccentric
is such that the eccentric distance, d,is always very small as compared to thedistance, s, it is evident that angle c,
True
Sd ~ C Distantd
Station
Eccentric
Figure 82. Reductzon to center diagram.
the eccentric reduction, will always be
very small. Under these conditions, itis possible to use the approximation thatthe sine of a small angle is very nearly
directly proportional to the angle.Therefore, the value of a small angle inseconds can be found by dividing the
sine of the angle by the sine of 1 second,thus:
sin csin 1'
Using this approximation, the equation,
sin d sinasin c-s
is converted to the formula for eccentric
reduction by dividing both sides by
sin 1,',
sin c d sin a
sin 1" s sin 1"sin c
and then substituting c for sn c whichsin 1'
gives:d sin a
s sin 1"
(2) In the development of the formula, a
was called a clockwise angle from the
true station to the distant station as
measured at the eccentric station, but
it would be just as correct to call a
direction from the eccentric station to
the distant station by observing the
rule that the direction from the eccentric
to the true station is always zero.Following this rule, it can be stated that
for all directions (a) less than 1800, c
is positive; and for directions more than
180 ° , c is negative.
(3) When an eccentric instrument setup is
used to measure horizontal directions,the initial direction of the list of di-
rections receives an eccentric correction
as well as do the rest of the directions
on the list. The correction to the initial
direction naturally affects all the di-
rections referred to the initial direction.
In order to retain the initial direction as
zero, the correction to the initial di-
rection is applied, with opposite sign,to all the other directions on the list.
(4) To illustrate the reduction for an eccentricinstrument, the sketches in figure 83
144
were drawn showing the two cases of the
direction (a) greater and less than 1800.In each case, a dashed line has been
drawn through the eccentric, parallel
to the line from the true station to the
distant station. It is the direction of
this dashed line that is computed, as
its direction is the same as that from thetrue station to the distant station. In
figure 83 O in which a is more than180 ° , it can be seen that direction a
is larger than it should be by the angle
c' (which equals c). Therefore, thereduction, c', is minus when applied
to the observed direction. In figure 83Q, a is less than 180 ° , and is smaller
than it should be by the angle c' (equals
c); therefore, the reduction, c', is plus
when applied to the observed direction.
(5) Figure 83 also illustrates the condition
when an eccentric signal is observed from
the distant station. The only differences
to be pointed out are that the eccentricreduction in this case is angle c and thereduction is applied to the directionobserved at the distant station eventhough it is computed at the eccentric.
Distant
Distant
C
-f
o<> 180°
C is negative
o < 180°
C is positive
Ecc.
® When c is negative
O When c is positive
q) When c is negativeo When c is positive
Figure 83. Eccentric reduction, sign of c.
757-381 0 - 65 - 10
The directions used in this computation
are the one observed at the true station
with the initial direction changed tothe eccentric. Following the rule stated
previously, the direction from the trueto the eccentric must be 180 ° . There-
fore, to prepare all directions on thelist of directions for use in the compu-
tation of the reduction, add 180 ° to
the direction to the eccentric as givenon the list, and then subtract this sumfrom each of the other directions on the
list, for lines on which an eccentric
reduction is required.
(6) The situation sometimes arises where
an eccentric instrument observes an
eccentric object. The solution of this
problem is merely a combination of the
two previously discussed reductions. A
reduction is computed for each eccentric
separately, and then the two reductions
are combined algebraically on the list
of directions.
(7) The eccentric reduction may be computed
on DA Form 1921. (Reduction to
Center) (fig. 84). Notice that d and sare labeled "meters" on the form. It is
not absolutely necessary that d and s be
in meters, but they must both be in the
same unit of measure. The form is set
up for use with logarithms, and the colog
of sin 1" is printed on the form for
convenience. Use only 5 decimals in the
logarithms in the computations, and the
directions (a) to the nearest minute.
(a) The procedure in the use of the form is
first to enter all the given data:
distance d, station names, direction
to stations (a), and log s from the
preliminary triangle computations.
From logarithmic tables, obtain log d
and the log sines of the directions, and
enter in the appropriate spaces. Add
log d and colog sin 1", and record the
sum. Subtract log s from log sin a,and enter the results in the column
headed logSi as . Add the sum of
log d plus colog sin 1" to the values
in the log (Sin a) column. The results
are the logarithms of the reductions
(in seconds) for each direction. Antilog
145
True
m
PROJECT REDUCTION TO CENTER37- 553 (TM 5-237)
LOCATION fTYPE OF STATION: ®ECCENTRIC STATION K'enDIStract of Co,0umnio. Q ECCENTRIC OBJECT AT STATION
ORGAN IZATION [Log ds 0.540O20o Distance (d) (meters)
DATE CIgsnV
21 J1Qn. 56 ___Sum3.6
Lo.a(BIN Q LOGARITHM Or
(aAIO In LOeters)Lc. LOG J REDUCTION REDUCTION
(TAIO in LOterI) IN SECONDS -C
Center 01
___ __ ___ __ 1la A &.. 8.7601.5..~5 78-9
Ta7 5.00t 0f 0. -u7..32k.1Z~.i3A4~8~iL i2ofO
qgFor.st 6Ia ~iape -13(p .9M~ 3420hAi ~.~ I65~q7or 43__ ab stI
, .( 4 5 ,u of St8AE . W IvolessftL J17 S s 8 8 4 3. q 4 4 .,gpI 7 0 - 5 3 + .. 71 "
COMPUTED BY A.DATE iCHECKED BY DATE
DAFR 71921
Figure 84. Computations of a Reduction to Center (Eccentric Station) (DA Form 1921).
146
these results to obtain the reduction inseconds. Attach the algebraic signto the reductions according to therules previously stated.
(b) Apply the reductions to the list of direc-tions, and recompute the preliminarytriangles using the reduced directionsto obtain the angles. From this set ofpreliminary triangles, take the values
of log s required on the reduction tocenter. Using these new values oflog s, recompute the eccentric reduc-tions and repeat this process until nochange occurs in the reductions. Ifthe recomputations of the preliminarytriangles produce no change or only aninsignificant change in log s, it is notnecessary to recompute ' the eccentricreductions.
(8) An illustration of the reduction for aneccentric instrument is described below.The example shows the original list ofdirections observed at Ken Ecc. (fig.85), the preliminary triangle computa-tions (fig. 86), and the eccentric reduc-tion computation (fig. 84). The figuresfor the second computation of thepreliminary triangles and the secondcomputation of the eccentric reductionare written above the original figures.
(a) For the first computation of thepreliminary triangles, Ken-Home-Renoand Ken-Home-Chevy, the angles atKen are concluded. This will give abetter approximation of the distance sthan would be obtained by using theobserved angle at Ken Ecc. For theother triangles, the directions at Ken
Figure 85. List or directions at eccentric station (DA Form 1917).
147
PROJECT ORGANIZATION LIST OF DIRECTIONS(TM 5-237)
LOCATION STATION
District at Colum8cia Ken Ecc.
OBSERVER INST. (TYPE) (NO.) DATE
G.L. Berkin WjjI T-2 * 145974 I Dec. 55
ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDOBSERVED STATION OBSERVED) IECTION REDUCTION REDUCTIONe WITH ZERO INITIAL DIRECTION
O I if I i // 0
0 00 00.00 -- L.L 0 00 00.00
Tdrdc w~est of a [)uic 2q 03 37.0 -1L.8~± 2g 02- 34.4_____
K (center) 3.449 M. 116 4225
F 3i- 24 53.0+3 01.3 31? 28 o _
Ncarve 32.6 31 +02 ±L 19 3l2s 32 05-45 _____
&u,. Q; S*. Wpnl.M Qa 35a 171 20.8 + 5.7 351 11 33.8
eo 357 28 184 - I 16 357 28 54.
Ken Ecc.
NCO
PROJEC3 SATE COMPUTATION OF TRIANGLES7-.'553 1 9 .ae1 L re~ianoary (TM s-237>
LOCATION Mbaryindc ORGA__ ENIZAT
ION A 45, Inc._____
TATION OusavmD ANGLE C'OaauCION. ANGLE SPRI xLECMnm1A N ANGLE LOGGA ID
2_ _ _ 47 0, .7.7 3.4 .
__3 2*no 101 S3 06.7 3.4 oJ1 9.isA
_ _ 1-24q7
__1KT. (3A 2LA 52 - g44g78
__2 4ome 2S .20 .54.6 +2. '56, g.8u~
- 2 BaD II 488 2.2 to.3 ___ g.g32217
23 Kan. Chv-
2 ~ .38 34 47.2.__
2-3Ta* q
12 07k-p 3..0 110
2-36 3. 16&39.1~ e .3 3~s.tar4 ..
1- F Ge. - _ _ __ __ 3. 4 1201-2 Tan - ___ _ 3 3o&
2-3 Ren *- Ken ____2&32
-- ~~~ 13 pol 303 8 6
__2 Rom 6 o J 4.2__-_ 9.750
3__ k e n _ °O 5 .. a__ ___ 7_
K enrd~e e ___ ___ __ _ _ 3. 24
2492-3 R p&no ____ 3. I6432(
Fig ur8. Preliminaryd co puaton of trage o edcint etr D om11)
148
Ecc. must be used because in eachcase, one of the stations was notoccupied. For the second computationof the preliminary triangle, the reduceddirections at Ken should be used toobtain all angles at Ken, and thetriangles should be corrected wherenecessary to close to 1800.
(b) Attention is now directed to thechange in the list of directions. Al-though the list was originally labeledKen Ecc., the application of theeccentric reductions has changed thelist so that the directions in theCorrected Direction with Zero Initialcolumn are at station Ken.
(9) A second illustration shows the reductionto center for an eccentric object observed.In this illustration, the lists of directionsfor all the stations involved and thereduction to center computation areshown in figures 87 and 88, but thepreliminary triangles are not shown as nonew principle is involved in computingthe triangles. The two important pointsto be brought out in this illustration arethe methods of obtaining the directionsfor the eccentric reduction computationand the application of the reductions tothe appropriate lists of directions.
(10) To obtain the directions (a's) for thereduction to center form, subtract 180 °
from the direction from Home to HomeEcc. as referred to Park on the list ofdirections for Home (222°55'20"--180 °
= 42°55'20"). Since this direction(Home Ecc. to Home) must be 00
according to the rules stated previously,the procedure is to subtract the directionHome Ecc. to Home (42°55'20") fromeach direction on the list of directionsfor Home. The results of this operationare the directions from Home Ecc. toeach station on the list and referred toHome as 0° . These are the directionsentered on the reduction to center form.For example, on the list of directionsfor Home, the direction to Park is00000'00"00 as it is the initial; subtracting42°55'20" from it gives the direction317°04'40", from Home Ecc. to Park as
referred to Home. This value is roundedoff to 317°05 ' for use on the reduction to,
center form. The direction from HomeEcc. to Cedar is 47007'49'6-4255'20" '
=4°12 ' . The rest of the directions arefound in the same manner.
(11) The eccentric reductions are now com-puted as explained previously, and thealgebraic signs attached (fig. 88). Thereductions are applied to the individuallist of directions. The reduction forCedar of +3'18 is applied to the direc-tion to Home Ecc. on the list of directionsfor Cedar; the reduction for Gerst of
+44':84 is applied to the direction toHome Ecc. on the list for Gerst, andso on. The application of the eccentricreductions automatically changes the ob-served station from Home Ecc. to Homeon the various lists.
(12), In all cases, a sketch of the eccentricconditions should be shown on the list ofdirections and on the reduction to center
computation form.
59. Strength of Figure
a. Introduction.
(1) For computation purposes, it is oftennecessary to know the relative strengthof the figures involved in the triangula-tion net. The strength of a figure de-
pends on the size of the length angles inthe triangles through which the length is
computed, on the number of directionsobserved in the figure, and on the number
of conditions to be satisfied in the figure.(2) The strength of a figure is designated by
the letter R and is computed from the
equation:
R=(D-C 1\ [S-SDAB
in which D= the number of directions
observed in the figure; C= the number of
conditions to be satisfied in the figure;
SA and 8B are the logarithmic differences
in the sines for 1" change in the distance
angles A and B of a triangle.(3) In table XLI, TM 5-236, the values
tabulated are [bf+SA2B+ 8 in units of
the sixth place of logarithms. The two
arguments of the table are the distance
angles in degrees with the smaller dis-
tance angle being given at the top of the
table.
149
PROJECT 3753ORGANIZATION ¢ ICLIST OF DIRECTIONS37-53 A S IAB.(TM 5.237)
LOCATI ON MaySTATION ryanCe r
OBSERVER INST. (TYPE) (NO.) DATE
j OBSERVED STATION OBSERVED DJRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDREDUCTION REDUCTION WITH ZERO INITIAL DIRECTION'
O i n n O i al e 1 0 00 00.00 -0 00 . 00.00
o2 crn 4 a o7.4 + '..2 ___43 02 10.6
Taknma 5&5TA2 -7_7._ 58 59 11.6
Phr-k5q 21 -. ____ 5 q 29 285
PROJECT 37-553 ]ORGANIZATION A , :r. T LIST OF DIRECTIONS
LOCATION Macry lad ~STATION
OBSERVER INST. (TYPE) (NO.) DATE____________ W,14 T-2 # 137A46 17____ lJsn u
OBSERVED STATION OBSERVED DIRECTION ECCENTRIC SEA LEVEL CORIIECTED DIRECTION ADJUSTEDREDUCTION REDUCTION' WITH ZERO INITIAL DIRECTION
o / II / f if 0 ,/ i / if
I nsane 0 00 00.00 0 00 00.00 ______
Womne Ecc 62 57 2S.g 9 - 51.3 62 58 17.2.
_________88 .o oo 88 '0 O'.0____
PROJECT 3- 53ORGANIZATION 4./fw LSOFDRCIN
LOCATI ON Mryn fSTAT IONGet
OBSERVER INST. (TYPE) (NO.) DATEF. T. A. Wi T2 Nci. 1319I46 ______ ainlf6
OBERE SAIO BSRVDDIETIN ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDOBEVDSTTO BSRE JRCIN REDUCTION: REDUCTION' WITH ZERO INITIAL DIRECTION'
O / f / if n 0 / if I i
0 00 00.00 -0 00 00.00
____________ 12. 13 22.0 - 1 _ 12 13 22.0
_______ _46. .2a 53.2 + 44. ___46 21 38.o
OSREJETORGANIZATION ILIST OF DIRECTIONS
LOCATION Iand STAT ION
TK tan Pairk __OBEVRINST. (TYPE) (NO.) °DATE
__________ Wild T-Z No. 137946 _______18 Ja~nOBSERVED STATION OBSERVED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTED
REDUCTION REDUCTION* WITH ZERO INITIAL DIRECTION'
O / it / if if 0 . f I if
(Acr0 00 00.00 0 00 00.00 _____
Wal la oc e 44 4554.8_____ 44Ai454.8H o4rn e E..c. 116 2 6 12.1 L i , L L6a17 3 8. - 1 - - 7 324 8 . __
Figure 87., Corrected lists of directions (DA Form 1917).
150
Figure 87=Continued.
b. Use of Table XLI in TM 5-236. To com-
pare two alternative figures, either quadrilaterals
or central-point figures, so far as the strength with
which the length is carried is concerned, proceed
as follows:
(1) For each figure, take out the distance
angles (to the nearest degree if possible)
for the best and second-best chains of tri-
angles through the figure. These chains
are to be selected at first by estimation,and the estimate is to be checked later
by the results of comparison.
(2) For each triangle in each chain, enter the
table with the distance angles as the two
arguments and take out the tabular
value.
(3) For each chain, the best and second-best,through each figure, take the sum of the
tabular values.
D-C(4) Multiply each sum by the factor D
for that figure, where D is the number of
directions observed and C is the number
of conditions to be satisfied in the figure.
151
PROJECT3 53 ORGANIZATION LIST OF DIRECTIONSPROJECT A 4S, Ic (TM 5-237)
LOCATION STATION
OBSERVER INST. (TYPE) (NO.) DATE
TKE \4W d T-2 No. 137446 IsJan 6S
OBSERVED STATION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDREDUCTION REDUCTION WITH ZERO INITIAL DIRECTION
0 I ID I t ft 0 I ft . f
ar 0 00 00.00 0 00 00.00
Cedcir 41 07A q.
Gerst 1742J15A. ____ _
Geiv~iald 25 427 ___ _
I.1ma -Ec. 2.7gmnr. 222 55 20 ___ __
To r_ _ _
3KiI
Ecc. o tob To s
,Par
To
PROJECT REDUCTION TO CENTER37.353 (TM 5-237)
LOCATION TYEO STATION: 0 I TTO
a~ w CCW4TRIC ouxcT AT STATION Worn*
ORGANIZATION AtS. Inc. Log d= 0.44 S560 Distance (d) (meters)
DATE Cologasin 1"= 5. 31 4 43 22 1 J n . 56 Sum = .7 6 03 2 .19Lo RI Mor R OO M
STATION a Loa a (rl m) Loa C ), RamucrouI R3fON
Center
C e 4 , '4 A A L 8 , 6 4 7 4 A J .Z J8 1 t A 1 4 2 h . 1 £ h J5 Q .2 B { 1 8L
.
Gefdl.I 1QS. S1& dM a 4.10&l JSW15 1. 6IQIL f
0-5
-omyfts
DA I F 71921
Figure 88. Computation of a reduction to center (eccentric object) (DA Form 192?1).
152
The quantities so obtained, namely:
(D-C)D / 5j[SA+A -I'SB
will for convenience be called R 1 and R 2
for the best and second-best chains,respectively.
(5) The strength of the figure is dependentmainly upon the strength of the best
chain through it, hence the smaller the
R1 the greater the strength of the figure.The second-best chain contributes some-
what to the total strength, and the other
weaker and progressively less independ-
ent chains contribute still smaller
amounts. In deciding between figures,they should be classed according to their
best chains, unless the best chains are
very nearly of equal strength and their
second best chains differ greatly.D-C
c. Some Values of the Quantity D
(1) The starting line is supposed to be com-pletely fixed.
4-1(a) For a single triangle, 4 = 0.75.
10-4(b) For a completed quadrilateral, 10
10
= 0.60.
(c) For a quadrilateral with one station on8-2
the fixed line unoccupied, 8 2 0.75.
(d) For a quadrilateral with one stationnot on the fixed line unoccupied,72 0.71.
7
(e) For a three-sided, central-point figure,10-410 = 0.60.
10
(f) For a three-sided, central-point figurewith one station on the fixed line un-
occupied, 8 2 = 0.75.
(g) For a three-sided, central-point figurewith one station not on the fixed line
7-2unoccupied, = 0.71.
(h) For a four-sided, central point figure,14- 5 = 0.64.
14
(i) For a four-sided, central-point figurewith one corner station on the fixed
12-3line unoccupied, 12 = 0.75.
(j) For a four-sided, central-point figurewith one corner station not on the fixed
line unoccupied, 11-= 0.73.11
(kI) For a four-sided, central-point figurewith the central station not on the
10-2fixed line unoccupied, 10-20.80.
10
(1) For a four-sided, central-point figurewith one diagonal also observed,16- 0.56.
16
(m) For a four-sided, central-point figurewith the central station not on thefixed line unoccupied and one diagonal
observed, 12 4 = 0.67.12
(n) For a five-sided, central-point figure,18 = 0.67.
18
(o) For a five-sided, central-point figurewith a station on a fixed outside line
16-4unoccupied, 16 4 0.75.
(p) For a five-sided, central-point figurewith an outside station not on the
15-4fixed line unoccupied, 15 = 0.73.
(q) For a five-sided, central-point figurewith the central station not on the
13-2fixed line unoccupied, 13 -0.85.
(r) For a six-sided, central-point figure,22--722- 0.68.22
(s) For a six-sided, central-point figurewith one outside station on the fixed
20-5line unoccupied, =20 0.75.
20(t) For a six-sided, central-point figure
with one outside station not on the
19-5fixed line unoccupied, 19 =0.74.
(u) For a six-sided, central-point figurewith the central station not on the
fixed line unoccupied, 16 0.88.
(2) To illustrate the application of thestrength table, the R, and R 2 for figure102 will be considered. Let it be as-
sumed that the direction of progress is
153
from the bottom line toward the top
line. It will be found that the smallest
R, called R 1 for this figure, will be ob-
tained by computing through the three
best-shaped triangles around the central
point. The next best R, called R 2, will
be obtained by computing through thetwo triangles formed by the diagonal.
The R 2 is easily computed as follows:
From the known side to the diagonal,
the distance angles are 890 and 27° .
Using these angles as arguments in the
strength table, the factor 17.5 is obtained.
Similarly, from the diagonal to the top
line, the distance angles are 91 ° and 26°,and the corresponding factor is 18.8.The sum of the two factors is 36.3. If
the central point of the figure is an oc-
cupied station,D -=0.56 (see above),
and R 2 = 36.3 X 0.56 = 20. If the control
point is unoccupied, as shown in figure
D-C102, ---0.67, and R2 = 36.3 X 0.67 = 24,
as given opposite the figure.
(3) The R 1 may be computed in a similar
manner by using the distance angles in
the three best-shaped triangles around
the central point.
d. Examples of Various Triangulation Figures.
(1) Figures 89 through 102 show some of the
principles involved in the selection ofstrong figures and illustrate the use of
the strength table.
(2) In every figure the line which is supposedto be fixed in length and the line of
which the length is required are repre-
sented by heavy lines. Either of thesetwo heavy lines may be considered to be
the fixed line and the other the requiredline. Opposite each figure, R, and R 2,as given by the table, are shown. Thesmaller the value of R 1, the greater thestrength of the figure. R 2 need not beconsidered in comparing two figures un-less the two values of R1 are equal, ornearly so.
(3) Compare figures 89, 90, and 91. Figure89 is a square quadrilateral, figure 90is a rectangular quadilateral which is 12
as long in the direction of progress asit is wide, and figure 91 is a rectangularquadrilateral twice as long in the di-
154
® All stations occupied.
® Any one station not
occupied.
5
45
Figure 89. Strength of figure diagram.
) All stations occupied.
O Any one station not
occupied.
27
63 63
Figure 90. Strength of figure diagram.
( All stations occupied.
® Any one station onixed line not occupied.
Figure 91. Strength of figure diagram.
A stetions occupied.
Figure 92. Strength of figure diagram.
All stations occupied.
Figure 93. Strength of figure diagram.
R2
m6
RI 1R2=1
R1 =2R2 =2
R, -22R2.22
Rl m2782 .27
RI 21
R212
I
rection of progress as it is wide. The
comparison of the values of R 1 in figures
89 and 90 shows that shortening a
rectangular quadrilateral in the direction
of progress increases its strength. A
All stations occupied. RI -164(approx.)R2
= 176 (opprox.)
Figure 94. Strength of figure diagram.
® All stations occupied.
( One outside station on
fixed line not occupied.
comparison of figures 89 and 91 showsthat extending a rectangular quadri-lateral in the direction of progress
weakens it. Figure 92, like figure 90,is short in the direction of progress.
/ Unoccupied station
on fixed line.
not RI = 36R2
.102
Figure 99. Strength of figure diagram.
RI -2R2 .12
RI "3R2=15
Unoccupied station atsection of fixed lineline to be determined.
inte
and
Figure 95. Strength of figure diagram.
Figure 100. Strength of figure diagram.
® All stations occupied.
® One corner station notoccupied.
) Central station not
occupied.
RI -13
R2 -15
RI -16R2 =16
RI -17
R2 -17
Figure 96. Strength of figure diagram.
() All stations occupied. R = 10R
2 15
) Any one outside sto- RI
IIItion not occupied. R
2= 16
® Central station not
occupied.
All stations occupied.
(A strong and quick
expansion figure.)
Figure 101. Strength of figure diagram.RI
= 3R
2= 19
Figure 97. Strength of figure diagram.
All stations occupied. R1 = 5R2 • 5
D-C 28-16S 28 0.43
Figure 98. ° Strength of figure diagram.
Central station. not
occupied.
r- RI v4
d R2 -20
RI =9R2 =9
R I =18
R 2 =24
Figure 102. Strength of figure diagram.
155
Such short quadrilaterals are in generalvery strong, even though badly dis-torted from the rectangular shape, butthey are not economical as progresswith them is slow. Figure 93 is badlydistorted from a rectangular shape, butis still a moderately strong figure.The best pair of triangles for carryingthe length through this figure are DSRand RSP. As a rule, one diagonal ofthe quadrilateral is common to thetwo triangles forming the best pair, andthe other diagonal is common to thesecond-best pair. In the unusual caseillustrated in figure 93, a side line of thequadrilateral is common to the second-best pair of triangles. Figure 94 is anexample of a quadrilateral so muchelongated, and therefore so weak, thatit is not allowable in any class of tri-angulation. Figure 95 is the regularthree-sided, central-point figure. It isextremely strong. Figure 96 is the reg-ular four-sided, central-point figure. Itis much weaker than figure 89, the cor-responding quadrilateral. Figure 97 isthe regular five-sided, central-point figure.Note that it is much weaker than anyof the quadrilaterals shown in figures89, 90, or 91. Figure 98 is a goodexample of a strong, quick expansionfrom a base. The expansion is in theratio of 1 to 2. Figures 99 and 100are given as a suggestion of the mannerin which, in second- and third-ordertriangulation, a point A, difficult or
TONY
impossible to occupy, may be used as aconcluded point common to severalfigures.
(4) Many of the figures given are too weakto be used on first-order triangulation,but for convenience or reference and toillustrate the principles involved, theyare included with the figures which canbe used.
60. Side Equation Test
a. Experience has shown that the requirementfor triangle closures is not always sufficient, andthe agreement in length of the various lines inthe figure as computed through the two bestchains of triangles must be checked before leavingthe station. This is done by a logarithmiccomputation of the triangle sides, and a compar-ison of the log-lengths of those for which there isa double determination. In a quadrilateral, thesewill be the three exterior sides other than theknown side. For first-order, the log-lengthsshould agree within about one and one-half totwo times the logarithmic difference for onesecond in the sine of the smallest angle involvedin the computation of the length; for second order,2 to 4 times the difference; and 10 to 12 times thedifference for third order.
b. When a quadrilateral (fig. 103) has unsatis-factory triangle closures, it will be necessary tomake an inspection of the closures to determinewhat stations should be reobserved.
(1) Usually, it will be found that the triangleclosures (fig. 104) will show two triangleswith large triangle closures which havea common line. This indicates that poor
BILL
WALLY
MILLER
Figure 103. Quadrilateral for side-equation test.
156
PROJECT DATE
P'ETCAI Y JA.4 COMPUTATION OF TRIANLELOCATION DE v]ORGANIZATION
SIPuERRCAL SPRICAL PLN 0LUOGAR :STATION 03S33v30 ANOLKR CORRucttON. ANnLR Excxaa ANOLR AN. ipii
*COL G
2-3 ______4.306 6864
1+2 1 MILLER 56 14 39.34 0.0o80 1827w-10+12 2 TO0NY 76 /3 21.69 .9,987 3215
-8+9 3 BILL 47 32 02.17 9. 867 86641-3 4,374 1906
__1-2 4,.254 7355
03.20 0.90______
2-3 4.3741906
-4+6 1 WALL Y 86 46 2775 0,00/ 1822-2+3 2 MILLER 38 22 35.98 9,792.97/7
-748 3 SILL. 5550S 53.32 ___9.9177956
1-3 __4.16 83445
1-2 4.2 931684
__ _ __ _ 705 0.73 _ _ _ _ _
__2-3 ___4.306 6864
-.5+6 i WALLY 45 25 13.12 ___0. 1473524*-/0+1I 2 TONY 31 1I 32.28 ___9. 7143256-7+9 3 BILL 103 22 55.49 __ __5. 9880452
__1-3 ___4. /68 3644
___1-2 __ __4,442 0840
______00.82 0.74
2-3 ___4.44 2 0840
-1+.3 1 MILLER 94 37 /5S32 __0.00/1414 0*-11+t2 2 TONY' 45 0/ 29.41 ___9. 8496732
-4+5 3 WALLY 40 2) /4.63 9._______ 81124591-3 ___ __4.293/7/2
1-2 4__ ______ _4254 743-9
______59.36 10.829COMPUTED myDAE CHECKEDDY DT
R. 9.Akox- A MS IF8. (04 T8nCL - A MS FEFF&64
U. &. Gwanmmuw "UIffe" OPiI IS17 0-4"m66
Figure 104. Triangles for test quadrilateral.
157
FORMDA S FEB 571,918
observations were made at one or bothof the stations at the ends of the line.
(2) Applying a trial correction to the ob-servations at one of the stations to im-prove the closure and by recomputing,the triangle may give better side checks,which definitely establishes the fact thatthis station must be reoccupied. If theside check is not improved but madeworse, then the same procedure should befollowed at the other station. In themajority of cases, the above procedurewill show which station should beoccupied.
(3) Now and then the closures will be sodistributed with poor side checks thatthe above procedure will not be con-clusive. In this case, it will be necessaryto make a side equation test to de-termine what station or stations mustbe reobserved. This example of makinga side equation test is for first-ordertriangulation.
(4) The numbers on the sketch are measureddirections. For example, 1 at MILLERmeans the direction from MILLER toTONY; 2 is the direction from MILLERto BILL; 3 is the direction fromMILLER to WALLY, etc. Referringto the angle at MILLER from TONYto BILL, we use the expression -1+2,the left hand direction is always givena negative sign. At WALLY, the anglefrom TONY to BILL will be expressedas -5+6. If we consider the trianglesin which directions were measured atstation MILLER, then the followingrelationship can be expressed:
MILLER-TONY MILLER-BILLMILLER-BILL XM ILLE R -WALLY
MILLER-WALLYX MILLER-TONY
-
(5) Since the equation was written aroundMILLER, it is therefore called the pole.Equations similar to the abovecan bewritten for the other stations when theyare used as poles. The sides in anytriangle are proportional to the sines ofthe opposite angles and, therefore, wemay write the following equation forthe above:
Sine (-8 +9) Sine (-4 +6)Sine (-10 +12) Sine (-7 +8)
Sine (-11 +12) 1XSine (-4 +5) -
or, using logarithms, it will become-
Log Sine (-8+9)+Log Sine (-4+6)
+Log Sine (-11+ 12) - Log Sine
(-10 + 12) - Log Sine (- 7 + 8) - Log
Sine (-4+5)=0
(6) The above relationship would be exact ifthe measurements were perfect, butsince there is always a small error in thedirection measurements, there will be aresidual. This residual is the indicationof the errors so far as the equation isconcerned, and is referred to as the con-stant term of the equation.
(7) This constant term, divided by the sumof the tabular log difference for onesecond of the sines of the angles involved,will give a quotient which is the averagecorrection to be applied to the angles inthe equation so as to eliminate theresidual. By experience of many years,it has been found that for triangulation,this quotient must be less than 0.7" forfirst-order, and 2" to 4" for second-order.
(8) If there are large closures and it appearsthat the error is at one end or both endsof a common line, the test should beapplied by using both stations as poles,and if the quotient is greater than 0.7"for one of the equations, the error inangular measurements will be found atthe opposite end of the line from theselected pole. This is true because theangles at the station selected for the poledo not enter into the equation, and thesource of trouble must be at the stationat the opposite end of the line from thepole.
(9) A side equation test is given for thequadrilateral MILLER-TON Y-BILL-
WALLY. A study of the triangle clo-sures shows that errors are present attwo or more stations. Therefore, equa-tions are solved with each station as a
pole, below.
158
Select Pole at MILLER
Value
47-32-02.17
85-46-27.75
45-01-29.41
76-13-21.69
55-50-53.32
40-21-14.63
Angle
-8 +9
-4 -6-11+-12
- 10+ 12
-7 +8
-4 +5
Log Sine
9.867 8664
9.998 8178
9.849 6732
+9.716 3574
9.987 3215
9.917 7956
9.811 2459
-9.716 3630
+9.716 3574
-56
Angle
-- 11+12
-7 +9
-2 +3
-1 +3
- 10+11
-7 +8
Value
45-01-29.41
103-22-55.49
38-22-35.98
94-37-15.32
31-11-52.28
55-50-53.32
Log Diff. 1"
+ 19. 3
+ 1.5
+21. 1
+ 5.2+ 14. 3
+24. 8
86.2
-56Average correction -56 - 0.65"
86.2
-171Average correction 1 -1. 65"
103.5
Select Pole at TONY
Angle
-5+6
-1+3
-8+9
-7+9
-4+5
-1+2
Value
45-25-13.12
94-37-15.32
47-32-02.17
103-22-55.49
40-21-14.'63
56-14-39.34
Log Sine
9.852 6476
9.998 5860
9.867 8664
+-9.719 1000
9.988 0452
9.811 2459
9.919 8173
-9.719 1084
+9.719 1000
- 84
Log Diff. 1'
+20.7- 1.7
+19. 3
- 5.0
+24.8
+14. 1
85.6
-84Average correction -84 =- 0. 98"
85.elect Pole at BILL6
Select Pole at BILL
Angle
--2 +3
- 10+12
-5 +6
-4 +6-1 +2
- 10+11
Value
38-22-35.98
76-13-21.69
45-25-13.12
85-46-27.75
56-14-39.34
31-11-52.28
Log Sine
9.792 9717
9.987 3215
9.852 6476
+9.632 9408
9.998 8178
9.919 8173
9.714 3256
-9.632 9607
+9.632 9408
-199
Log Diff. 1'
+26. 6
+ 5.2
+20. 7
+ 1.5
+14. 1
+34. 8
102.9
- 199Average correction = -- 1. 93"
102. 9
A summary, therefore, shows the following results:
Pole at MILLER-average correction is - 0.65"
Pole at TONY -average correction is -0.98"
Pole at BILL -average correction is - 1.93"
Pole at WALLY -average correction is -1.65"
(10) The equations show that the angles at
MILLER enter into the three which give
an average correction greater than the speci-
fied value of 0.70" while the angles at
MILLER do not enter into the test
where the average correction is less than
that amount. It is definite that the
directions measured at MILLER are
probably in error. A test should be
made before actually reoccupying the
station by assuming a change in angles.
This can be done very easily by multi-
plying the tabular log difference for 1
second by the number of seconds the
angle is changed and adding this value to
the sine or subtracting according to sign.
Suppose the direction at MILLER to
BILL to be in error. The angle at
MILLER from TONY to BILL may be
decreased with a corresponding increase
in the angle BILL to WALLY. If we
assume a decrease of 3" from TONY to
BILL and the corresponding increase
from BILL to WALLY, then the side
equation test will result as follows:
159
Log Sine
9.849 6732
9.988 0452
9.792 9717
+9.630 6901
9.998 5860
9.714 3256
9.917 7956
-9.630 7072
+9.630 6901
-171
Log Diff. 1"
+21.1
- 5.0
+26.6
- 1.7
+34.8
+ 14.3
103.5
Select Pole at WALLY
Pole at TONY-average correction-- 4242 - 0.49"85.6
Pole at BILL-average correction-
-78 .- -0.76"
102.9
Pole at WALLY-average correction-
-92 - 0.89"103.5
(11) The correction for pole at MILLERwill remain -0.65 since the angles atMILLER do not enter the test. Anexamination shows that the change atMILLER has improved the corrections
and station MILLER should be RE-OBSERVED.
(12) The average corrections with poles atBILL and WALLY are still too large,while with the pole at TONY, a satisfactoryvalue was obtained. The angles meas-ured at TONY do not enter into theequation with the pole at TONY, but doenter into the tests for the poles at BILLand WALLY. It is probable that theangles at TONY are in error. If weassume a change in direction to WALLYfrom TONY of 2" in order to balance theclosures of either side of the diagonal,i.e., decrease the angle from BILL toWALLY at TONY by 2" and increasethe angle from WALLY to MILLER bythe same amount, the test will give thefollowing results:
Pole at BILL-correction will be-
-08 - 0.08"102.9
Pole at WALLY-correction will be-
+10+10 = +0.10"103.5
Pole at MILLER-correction will be-
-14 - 0.16"86.2
(13) The correction at TONY will remain-0.49" since the angles do' not enterinto the test when it is selected as thepole. Tests show that changes at TONY
will result in improved corrections andTONY should be reobserved. In this
example, both MILLER and TONYmust be reobserved to obtain satisfactoryresults.
(14) The side equation test will be effectiveprovided that either the angles are ap-proximately equal or that the error in-volves fairly small angles. If the badangles are close to 900, the test will not beconclusive because the tabular log differencefor 1 second of the sine is very small, andan error of several seconds in such an
angle might still give a quotient for theequation which will be less than 0 70".
61. Station Adjustment
a. Many times, during high-order triangulationobservations, duplicate directions are observedto a station from two or more initials. Theseduplicated directions cause condition equationswhich can be properly satisfied only by a leastsquare solution in which each observed directionis weighted according to the number of setsinvolved in its determination. This least squaressolution is referred to as a station adjustment.In the case of a station adjustment containingonly one condition, the solution is referred to asa weighted mean. The station adjustment pro-vides statistically the most probable value forthe direction to each observed object.
b. An example of a station adjustment ispresented with the intention of clarifying thecomputation, and to emphasize the value of itsapplication. The mathematics involved in thecomputation are presented in USC&GS SpecialPublication 138 (pages 8-16) and are not repeatedhere. The station chosen for this example iswithin a complex first-order triangulation networkand has a total of 27 directions referred to fivedifferent initials (fig. 105). These directionswere each determined by meaning a set of notless than 12 circle position readings observedusing a direction theodolite on which the hori-zontal circle micrometer may be read to within± 0.2". The readings were abstracted from thefield books according to the methods outlined inparagraph 55.
c. Following is the procedure for preparing thecondition equations:
(1) A preliminary list of directions (fig. 106)is made from the abstracts of directions.All directions are evaluated and those
160
BOB
EARL
HOMER
JOHN
FRED
Figure 105. Diagram of observed directions.
757-381 0 - 65 - 11 161
PRELIMINARYPROJECT TA- ORGANIZATION U SA M S LIST OF DIRECTIONS
'rl (TM 5-237)
LOCATION KE UKSTATION T
OBSERVER INST. (TYPE) (NO.) DATE
F WIL-SON WILD T-3 NO._/2345 MAR 160OBSERVED ST'ATIO V ORSERVEn ISIRECTION ECCENTRIC SEA LEVEL CORRETFED DIRECTION ADJUSTED
I;EDUCTION REDUCTION' WITH ZERO INITIAL DIRECTION'
0 r n J it .T 0 r IF I" i
AL 0 00 00.00 _____0 00 00.00
BOB 46 54 45.51 ___
45.55DON 10/ /4 32.49____
31.09
EARL /08 40 58.86
58.71FRA NK /37 42 2756
GEORGE 145 53 40.88 ___
40.98 ___
HOMER 180 02 S56,63
5749JOHN 195 26 49.88
50.60
____ ___ ____ ___49.85
EARL [ 00 00 00.00 NEWl INJTIA L
JOHN 86 45 50.88 __
51.78______
ROY /66 1 2/ 29.39 _ ___
GEORGE 00 00 00.00 NEW INITIAL
JOHN 49 33 o8.57 __
ROY -00° 00' 00.00 NEW INITIA
WILBUR 35 56 24.20 __
JOIN 0. 00,00,00 NEW iN/TI IL
LLOYD 41 01 45.21___
45. /8 __
PETE 62'19 51.91 __
RO-Y 79 1 536,69______FRED 9S 56 55/7______
S ~54.87 __
FRANK 30o2 /5 3754______H1O M ER f344 36 06.92________________
" These columns are for office use and should he left blank in the field. CH KE y JX,".
COMPUTED -BYS DAY I 60 CHECKE BYM/ 4'AM DATE
- AMSJ MA 0IJIc~de M MAY 60II. S. GOVERNMENT PRINTING OFFICE :l195 0-420665
Figure 106. List of directions (preliminary) for station adjustment.
162
DAI FEB 5T 1917
directions which cause excessively largecondition equations are rejected andnot shown on this list.
(2) An adjustment list of observed angles(fig. 107) is made repeating all the angles
from the preliminary list. Means are
determined for all of the angles having
more than one value and a column
headed WT. (weight) lists the number
of values used in determining each mean
angle. Another column headed v showsthe identifying number at each correc-tion (v) which will be applied to the
observed angles upon completion at thestation adjustment. The final columnheaded adjusted final seconds is leftblank at this stage and will be com-pleted when the computed correctionsare applied.
(3) The list of directions for adjustment(fig. 108) shows a direction to every ob-served station, all referenced to a singleinitial with the v's associated with the
angles used to determine those direc-tions. This list is completed utilizingthe minimum number of v's required torefer all directions to the single initial.Again, the final seconds column (ad-justed directions) is left blank and will
be completed when the computed v's areapplied.
(4) Every v which was not used in the "list
of directions for adjustment" creates a
condition. Each unused observed angleand its associated v is listed in turn and
compared to the corresponding anglefrom the "list of directions for adjust-ment." In this example, the list angleshave been subtracted in each case from
the observed angles in order that thealgebraic sign in the condition equations
will be correct. This algebraic difference
is set equal to zero in forming the condi-tion equation. Since there were five
angles not used in the sample adjustment
list, there are five condition equations
(fig. 109).d. The correlate equations are prepared from
the condition equations. In figure 110, column 2,headed alp, a is a selected constant, and p is the
weight of the particular v. Normally, a is chosen
as the least common multiple of all the weights,p, in order that the values of a/p be integers. In
this example, a equals 6.
e. The normal equations (fig. 111) are obtainedby taking the algebraic sums of a/p times the
products of the various columns in the correlate
equations.
f. The Doolittle method is used in the solution
of the normal equations (fig. 112). This method
of solving normal equations is covered in para-graph 65.
g. After the C's are determined, the v's are
computed by substituting the values of the C'sinto the correlate equations taking into account
the weights in the a/p column.
163
PROJET TMTABULATION OF GEODETIC DATA(TM 5-237)
LOCATION, KEN TUCK Y ORGANIZATION
LOOKOUT
OBSERVED STATIONS OSREANLS W.()VADJUSTED
FROM - o .OOSAE NLS W.~FINAL SEOD
AL - 808, 4,O 54 45.51 ___
mml. 45.53' 2 I 45.53
AL -DON /01 14 32.49
31.09
MN 31.79 2 2 .31.79
AL - EARL 108 40 58.86 ______
58.71
MN. 58.78 2 .958.62
AL - FRANK 137 42 2756 / 4 2760
AL - GEORGE 145 53 40.88
40.98 ______
MN. 40.93 2 5 41.13
AL - H~OMER /80 02 56.63
57.49
MN. 5706 2 6 57.04
AL - JOHN /95 26 49.8850.60
49.85
MN. 50./I 3 7 50.09-
EARL -JOHN 86 45 50.88_____
5/. 78 _____
MN. 51.33 2 8 -51.47
EARL - ROY /66 2/ 29.39 ______ 9 28.78
GEORGE- JOHN 49 33 08.57 / /0 08.96
JOHN - LLOYD 41 0/ 45.2/
45.18
MN. 45.20 2 ii45.20
JOHN- PETE 62 /9 .791 I /2 51. 91
JOHN -- ROY 79 35 36.69 I/3 3731
JoNN- FRED 95 S6 S55.17__________ 54.87
_____ MN. 5.02
2 14 55.02
JOHN -FRANK 302 /5 3754 I 15 375/
JOHN 11-HOMER 344 36 06.92 / /6 06.95
ROY -W IL BUR .35 S6 24.20 / 17 24.20
TABULATED BY DATE 1EKED BY DATE
-AMS APR.0 I J.12a . 4o - AMS APR.6~0DA 1ORM .1962 GO921961 U. S. OVRMN 7891789 OFFICE: MI0- 19Z
Figure 107. List of angles for adjustment.
164
PROJECT wAORGANIZATION V AIA LIST OF DIRECTIONSTM USAt 1SM (TM 5-237)
LOCATI ON SATION
KENTUCKY LOOKOUTOBSERVER INST. (TYPE) (NO.) DATE
F WILSON WILD T-3 NO.- /2345 ______MAR. 60OBERE SATO. BSREDDIETIN ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTED)
OBEVDSATO.OSREDDRCIN REDUCTION REDUCTION' WITH ZERO INITIAL DIRECTION'
AL 0 00 00.00 ____0 00 00.00 00 00.00
808 46 54 45.53 + Vi _______ 54 45. 53
DON /1 14 31.79- + V2 _______ 4 .31.79
EARL /08.40 58.78 + V3 _ _______40 58, 62
FRANK______ /37 42 27.56 + V4 42 2760
GEOR_____E /45 S3 40.93 +' Vs __ 53 41.13.
HMR180 02 5706 + V6 02 5704JON 95 26 50.11 + V____ 26 50.09
LOD236 28 3.531 + V7 + y_ 28 35.29
PEE257 46 42.02 + A, + )2 46 42.00
R~OY 2 75 02 26.80 + V1 f'- g. 02 2740
FRED .291 23 45.13 V7 + 14_ 23 45.11
WI LB UR 3 /0 58 51.00 + V7 + *3 + 1 58 61. 60
*These columns arc for ofliee use and should be left blank In the field.
COMPUTED BY DATE I CHECKED BY DATE
G. te n'p - A44IS APR. 6~0 J.4 ka.w-- AMS APR. 60
U. S. GOVERNMENT PRINTING OFFICE :1957 0-420665DAFR M71917
Figure 108. List of directions for adjustment.
165
PROJECT TM TABULATION OF GEODETIC DATA.,IVI I(TM 5-237)
LOCATION, KETUK ORGANIZATION SM
STATION
LOOKOUT______
EARL -JOHN O83. ANGLE .8 5 l 9+V
L IS r ANGLE 86 45 51. 33 - V3 * V7
CoN, I TlO N / 0 0.00 + V3 -V 7 . V8
EARL -ROY 08S. ANGLE /66~ 21 29 39 + VyLIST ANGLE /66 21 -28.02 - V3 + V7 + V,3
CONDITION 2 0=+/.37 + Vg-V7 + Vv - a
GEORGE- JOHN 08S. ANGLE 49 33 08.57 + Vo_______
/-/Sr ANGLE 49 33 091/8 - vs + V7 ______
CONDITION 0 0-0.61 + VS -V 7 + V 0 ____
JOHN - FRANK 08. AI GLE 302 /5 ,37.54 + Vis
LIST. ANGLE 302 /5 37 45' + V4 - V7
CONDITION 4 Of +0.0 9 - V4 +V7 +V1
JOI4N- HOMER oBS. ANGLE 344 36 06.92 + V,6_________LIST ANGLE 344 3( 06.95 + V6 - V7
COND1 TION 5 0:-0.0-3 - V6 + V7 + V6
ETAUULATED BY A DATE R.(0CHECKlnED BY D ATE.- AMaP.60z-A4S AR6
DA , FR x1962 GPO 92,961 U. S. GOVERNMENT PRINTIN~G OFFICE: 1957 0 - 21182
Figure 10.9. Condition equations-station adjustment.
166
Sta. LOOKOUT___
Correlate Equations ___
___AdOPrED
Vs al. 2 3 45 1c V VT 3 0.000 0.002 3 __0.000 0.003 3 +1 +/ +2 -0.164 -0.164 6 -/ _ -/ *0.034 *0.04
IS 3 t0___ +1 +0./96 +0.20
6 3 -I -/ -0.0/7 -0.027 2 -/- I# 1-/ -0.021 -0.02
8 3 i-' __ +#0'./43 +o.149 6 +0 +1 --0.614 -o.61
/0 6 +1 _ +/ +o.392 +0.39
II3 _ _ 0.000 0.00._2 6_ 0 .o0o 0.00
13 6 _ - - I +o. 614 +0.62/4 3' 0.000 .00/5 6 .. . -0.034 -0.03/6 6 + IJ +1 +0.034 +o.o3/7 6 __0.000 0.00
Figure 110. Correlate equations-station adjustment.
Normal Equations -
/ 2i 3 n c1j +8+ + 2-2 0,00 +11.00 40+&04757
2 +1 +2-2- +1.37 +21.37 -0.102273 11k JM 0.0 -0.61 + 10.39 +0.06540
4 ~ 4~ + 0.09 + /0.09 -0.0o572___ __ Jj 00 + 6.97 +0.00571
Figure 111. Normal equations-station adjustment.
Sta. LOOKOUTI________
Solution of Normals - _ _ ___
/ 2 3 4 S n1 ____ nc=
+ 8. +5 +-2. 1-2. - -2. 0.00 +/100C, = -0.62500 -. 25000 +0.25000 +0.25000 0.00006 -1.37500 +0.o4737-
____+17 +2. -2. -2. #-1.37 +21.37
______ 13.87500 +0.75000 -0. 75000 -0.75000 # 1.37000 +14.49500____
C2 =-0.05405 +0.05405 +0.05405 -0.09874 -1.044(68 -0.10227-____ ___ 11. -2. -2. -0.61 +-10.39 ___
______~~ ___ 10 /c45946 -1.45946 -1.45946 -0.68406 +6.85649____
____ ____C 3 = # 35 0.13953 + oo6540 -o.45553 +0.06540
________+14. +2. +0.09 +10.09
_____ _____+13.25582 #1.25582 +0.06861 +145802-T
______ ______ C4 = -0.09474 -0,00518 - 1.0999/ -0.00572-____ __ _ ____ ___ __ 41. -0.03 46.97 _ _ _ _
________~~~~~~~~ __________ ____ 10.13684 - 0.05790 +10.07894 ____
_____ ______ Cs = 4-0.0057/ -0.99429 +0.0057/1
Figure 112. Solution of normal equations-station adjustment.
Section II. QUADRILATERAL ADJ USTMENT (LEAST-SQUARES METHOD)
62. Introductiona. The most common figure . occurring in
triangulation is the quadrilateral with bothdiagonals observed, because it embodies both
strength and simplicity. A thorough understand-
ing of the computation and adjustment of such a
quadrilateral is basic to the understanding of a
net adjustment. For this reason, a quadrilateral
167
ISta LOOKOUT
adjustment will be explained and an example
computation shown. Because this is to be
background material, the quadrilateral in the
example will be adjusted by the least squares
method. DA Form 1925, Quadrilateral Adjust-
ment.(Least Squares Method), has been set up
for this solution.b. Field observations always contain small
errors which cannot be distributed in any way
except by the laws of probability. Because of
these errors, certain geometric properties of a
quadrilateral are not satisfied by the field observa-
tions. It is the purpose of the least-squares
adjustment to find the most probable values of
the field observations that will satisfy the geo-
metric conditions. If the following conditions
are satisfied, all the geometric properties of the
quadrilateral will also be satisfied. The conditions
are as follows:
(1) In the four triangles in the quadrilateral,the sum of the three angles in eachtriangle must total 1800 plus the spheri-
cal excess.
(2) The length of the common sides ofadjacent triangles must be the same no
matter which of the two adjacent
triangles is used to compute the length.c. By solving certain equations known as con-
ditions equations, small corrections are determined
which are applied to the observed angles. These
corrected angles will then satisfy the aboveconditions.
d. The two types of equations occurring in a
quadrilateral are angle equations and side equa-
tions. In a completed quadrilateral (fig. 113),three angle equations and one side equation areneeded. The number of angle and side equationsin any figure can be determined by the formulain the section on triangulation adjustment.
63. Direction Method
a. This illustration will be made using thedirection method in which an angle is consideredto be made up of two directions. The corrections,therefore, will be found for each of the twodirections making up the angle.b. The adjustment is begun by numbering the
observed directions on the sketch (fig. 113),starting at the left end of the fixed line lookinginto the quadrilateral. Any directions not ob-served are not numbered, and such directions areshown by a dashed line on the sketch. Thenumbers on the lines are used as the subscripts
Red
Lincoln Hicks
Figure 113. Quadrilateral sketch.
on the symbol v used to designate the corrections
to the observed directions. The designation of
the corrections is then vl, v2, v3, and so on. For
convenience, the symbol v is usually not written,thus the subscript is written as (1) and the symbol
is understood.c. Each angle is designated by the two direc-
tions forming the angle. Always considering theangle as being measured clockwise, the first direc-
tion is negative and the second direction is posi-tive. For example, the angle at Lincoln from
Burdell to Red is designated -1+2, while theangle at Hicks from Lincoln to Burdell is -4+5.
In this way, not only the angle, but also the cor-rection to the angle is designated. For instance,the correction to the angle in the first example
above is - 1 +v 2 and when numerical values are
found for vl and v2, they are algebraically added as
(-v 1 +v2).d. DA Form 1918 (fig. 114) is used for the com-
putation of the triangles in this example.e. In order to be certain that all the triangles
in a net are written on the triangle computationsheets, some sort of order should be establishedfor writing the triangles. The system outlinedhere will establish a pattern which will continuethrough all subsequent computations. In the
quadrilateral Lincoln-Burdell-Red-Hicks, startwith the fixed line Hicks-Lincoln and proceedclockwise around the figure. The first station en-countered is Burdell which will be number 1 in thefirst section of the triangle computation sheet.At Burdell, find the first clockwise line to a fixedstation which will be the line to Hicks. StationHicks then is number 2 in the first triangle. Still
168
PROJECT DATE
1-2 43 1.2 Aug if COMPUTATION OF TRIANGLES
LOCATION aSrnaORGANIZATION2
SPHEICA SPERICL PANRFUNCTION:STATION OBSERVED ANGLE CORRECTION. , E~LSRRA I, LOGARITH
ANGLE EXCESS ANGLE
2-3 NICKs " LINCOLN _ __ 6754
11512 NI &,r.e/ S2ZlL L 51, 0!0. 28 A.8136
-. 5 u a / 2 /ic1s 49. 000 55.
5N. 5 Z 0 Q l 6 Z & 3 f
-143 3 LIMccUN 108 Of31. 43.2 34.6 0,LI.L-59.9778511441__ -38~,~lLACL _ _ 3. 9 767 03/a
__1 - 2 fil
3. +6.4 00.1 Q2/ 00.0 _____
___2
-3
,HICKS -Ij.lvcouI ______ 3.68754-__ 1 Rea 304 5. 2.5 QMA0 97840
-4d+6 2 9/ 29 09.2 43.. 10.1 O.L 10-o 9.9f98539-24o3 3 LINCoL A 57 48 45.3 ~ Z~ ao Z .9738
___1-3 &ed-INCOLN ______ .79,364291- __________ 3.90704327
______51.2 +8.9? oo.1 0.1 00.0
___2-3~, 8Udl ______ _ 4.07663020
-7*9 1 Red q5Q?110+3 14.2~ .0 J4. 9. 998240546
-So-6 2JCKS _ 42 28 11.0 *2- L3, 0 4k1 J2A9 82 /
/O/ 3 urdell 42 22 31.7 ±L/.3 331 4 32 A265,111-_ 3 _. udl ______3 078171-2 ed-11S _ ____3f014L
__________ 53.7 +6.5 00.2 0.2 00.0_____
-8f9 1 Re___ 64 27 IL 3 ±J 1 o. / LE355L9-/1+2 2 LNOA - 02 4. 0.7 .46.8 0., A6. 19,88644309
-10*12 3 5urdeIJ 6 A5LL 6L ±2A 6 O.4 .24 ~Z1-3 ed-ukl________ __39724
____ 1- - -_____ 3.97936462
________ Z:: 1 14.0 00.21 0.2 00.0 - vCOMPUTED BY 0"DATE CHECKED BY DATE
DA 7oM1 918
Figure 114. Computation of triangles for quadrilateral.
169i
at Burdell, find the next clockwise line whichwill be the line to Lincoln. Lincoln is number 3.The triangle thus defined is Burdell-Hicks-Lincoln. Now move to the next clockwise sta-tion which is Red. This is number 1 in the secondtriangle. The first clockwise line is to Hicks.Hicks is number 2. The next clockwise line is toLincoln which is again number 3. The triangleRed-Hicks-Lincoln is now defined. Still at Red,the first clockwise line is to Hicks; the next clock-
wise line not previously used is to Burdell. Thetriangle now completed is Red-Hicks-Burdell.
All the triangles with Red-Hicks as the first clock-
wise line are now written, so shift to the nextclockwise line which is Red-Lincoln. Lincoln isnow number 2 in the fourth triangle. The nextclockwise line is Red-Burdell. The fourth tri-
angle is Red-Lincoln-Burdell. Notice that sta-tion Burdell is number 1 in the first triangle, andstation Red is number 1 in the second, third, andfourth triangles.
f. In the left-hand margin of the form, the desig-nations of the angles are shown. The observedangles are from the list of directions. (In thisexample, the angles are from the list shown in thesection on triangulation adjustment.) The sumof the observed angles is recorded for each tri-angle. Only the seconds of the sum need bewritten on the form. A bar over the seconds in-dicates that the sum is less than 180 ° . Forexample, in the first triangle in figure 114 thethree angles total 179°59'53'.7 and the seconds arerecorded as 53.7. The three angles in a triangleshould total 1800 plus the spherical excess (E).(Spherical excess computation is discussed inparagraph 57.) 1800+e minus the sum of thethree angles is the error of closure of the triangle.The algebraic sum of the corrections to the threeangles (each angle correction made up of directionscorrections) must equal the error of closure of thetriangle. This statement leads to the conditionequation known as the angle equation, which canbe stated as: The algebraic sum of the v's in atriangle must equal the triangle closure. As anexample, the angle equation for the triangleBurdell-Hicks-Lincoln is-
-(1) (3)-(4) + (5)-(11) + (12)= +6.4
remembering that (1), (3), (4), and so on are thesubscripts of v's.
g. An angle equation can be written for eachof the four triangles in a quadrilateral, butvalues for the v's which will satisfy three of the
170
equations will also satisfy the fourth, becausethe fourth equation is a combination of the otherthree. When choosing the three-angle equationsto be solved, triangles with small angles shouldbe avoided if possible. In the example, there-fore, the angle equations for the second, third,and fourth triangles are selected. The threeangle equations are as follows:
0 = - 8.9 - (2) + (3) - (4) + (6) - (7) + (8)
0 = - 6.5 - (5) + (6) - (7) + (9) - (10) + (11)
0 = - 4.0 - (1) + (2) - (8) + (9) - (10) + (12)
The eq'uations are numbered in ascending orderof v's for convenience in the solution. Theequations are shown in correct order on the ex-
ample form.h. The condition of side agreement is satisfied
by including a side equation in the solution.One way to set up this equation is to select astation as a pole and write the product of theratios of the lines running to that pole as equal to 1.By the laws of sines, the sines of the anglesopposite the sides can replace the sides. By re-placing natural sines by logarithms, the expressioncan be reduced to one which can be solved byaddition and subtraction.
i. In this example, the side equation is writtenusing the pole at Lincoln in order to include thesmall angles at Burdell and Red. The ratio ofthe lines intersecting at Lincoln is-
Lincoln-Burdell Lincoln-RedLincoln-Red XLincoln-Hick
Lincoln-Hicks -
Lincoln-Burdell 1
Substituting the sines of the angles opposite thesesides and using the symbol designation of theangles, this ratio becomes-
sin (-8+9) Xsin (- 4 + 6 ) sin (-11+12)1sin (-10+12) sin (-7+8) sin (-4+5)
Replacing natural sines by log sines:
log sin (-8+9)+log sin (-4+6)
+log sin (-11+12)-log sin (-10+12)
-log sin (-7+8) -log sin (-4+5) = 0
When written on a side equation form, the logsines of the angles in the numerators are enteredin the left-hand column and the log sines of theangles in the denominators are entered in theright-hand column.
j. For each angle, the tabular difference is en-tered in the column headed "Tab. Diff." (fig. 115)beside the angle. The tabular difference is the
PRO-243 QUADRILATERAL ADJUSTMENT (Least Squares Method)LOCATIONORAIAINDT
Ca'lifAorni iq nv 0SKETCHANLEQAIS
sup-del 10 9 Re12 7 (a)1 0= -.- 2+3-4+i-7+8
L~nc~In..2(1) 0= -6.5-(5+()-(7).(q)-(0)+('ii)
SIDE EQUATION
SYMBOL ANGLE LOG. SINE TAB. Div SYMBOL ANGLE LaO. SINE TAB. I)1IFF
0= +76¢+1 99(4)-I. 83(5 -a o6(,+355(7)-4.S(ce)+ .v40(9) fo.97Io)-.00eI)+4.o34i
__________ CORRELATE EQUATIONS _____
1 2 3 4 Ecv ADOPT. V V
1 - ____ -°1.0 -i±2M3 -1.3 1.6E!2 +j 1 _____ -,335 -06 o-4 2
4 + 1 +1.89 +.8 178 4.L9 3AL 61
5 -1 -1.03 -2 2hQ3 ____ 5
64+1 +1 -00 +-9q 4~~4 g 6
7- I -1 .3.55 + .5 -1.L12 -1L9 3,(L 78 -, +l. -/ a -4.5 +Q~9 o. 5q26 _06 03
10, -g -1 Q7 1i3 L3IL 10 ~f~
12 +1 +-40Q1 i - .32061 +1.3 1 ,11NORMAL EQUATIONS
1 2 '3 4 E
2 +6- 2 *.6/ -10.6 1
2 1-6 +2 -100 -8. - 2
3............................... ------- - 3
4. . ....................................................... 1- 3 1. . 1____ __________SOLUTION OF NORMALS
1 2 3 4 71 E C
f6. -2.0 +.2. + -4.0 1!10.6101 + -. 4350 66L - 1. 7683 + 1. 2843 CI
#S 333 2t 4 . - 10. 23-33 t.AAS02 zQ±b.~- +I. +46 . 9IL& 1.2610 +.±LiLZL....02
PROBABLE ERROR OF -f~QQ -. 00 2 QA_____OBSERVED DIRECTION - '~ - 029 =.75. -2C . 0 . Q 0 21.1 C
x - f 23.11 2A. _____
d =± .6~ +5. °g A L ... Z~.____C=NUMBER OF CONDITIONS C0 " -o g g + C. .6 04CMUTDCOPUE B Y 4 DATE /jCHECKED BY ,fDATEr
fAFORM 12DAI FEB 5712
Figure 115. Computation of Quadrilateral adjustment.
171
change in the logarithm for 1 second change inthe angle. Express the tabilar difference in units
of the sixth decimal place of the logarithm.
Notice that the tabular difference for an angleover 90 ° is minus. When this tabular arrange-
ment of the side equation is used, the constantterm of the equation is found by subtracting the
sum of the log sines in the right-hand column
from the sum of the log sines in the left-hand
column and pointing the difference off in units ofthe sixth decimal. If the right-hand sum is larger
than the left-hand sum, the constant term has aminus sign. The constant term is written with
unchanged algebraic sign on the same side of the
equation as the v terms. The tabular differences
are used as the coefficient of the v's as shown in
the Symbol column. If a v appears on both sides
of the form, the coefficients for that v must becombined in the equation. The tabular differ-
ences at the right-hand side reverse the algebraic
sign for this operation. The side equation in the
example, before coefficients of v are combined, is
as follows:
0= +7.64+(-0.06) (-4)+(-1.83) (-4)
+(-1.83) (5)+(-0.06) (6)
+(-3.55) (-7)+(1.00) (-8)
+(-3.55) (8)+(1.00) (9)
+(-0.97) (-10)-+(5.00) (-11)
+(5.00) (12)+(-0.97) (12)
When the coefficients of the v's are combined, theequation is as follows:
0= +7.64+ 1.89 (4)-1.83 (5) -0.06 (6)
+3.55 (7)
-4.55 (8)+1.00 (9)+0.97 (10)--5.00 (11)
+4.03 (12)
k. At this point, a check should be made toinsure that the sum of the coefficients of correc-tions to directions (v's) radiating from any onestation equals zero.
1. After the condition equations are formed,they are tabulated in correlates. For each equa-tion, there is a numbered column in the arrange-
ment of the correlates. On the horizontal lines
at the sides of the correlates are the numbers of
the v's. The coefficients of the v's in each equa-tion are written in the correct column on theappropriate numbered line. (Coefficients in equa-tion 1 are written in column 1, and so on.) Thecolumn headed "e," contains the quantities ob-tained by adding algebraically the coefficients on
172
the same horizontal line in the four columns. Thecolumns headed "v", "Adopt v", and "v2", arefilled in after the solution of the normal equa-tions. The correlates are used to form the normal
equations.m. The normal equations are arranged using a
column for each condition equation (correspondingto the numbered columns for the correlates) plusa column headed "7" and a column headed "2]".The q column contains the constant terms of thecondition equations, while the values in the 2,,column are used as a check corresponding to thevalues in the 2, column of the correlates. Thenormal equations themselves are numbered on thehorizontal lines.
(1) The normal equations are formed byfinding the algebraic sums of the prod-ucts of the values in one column of thecorrelates multiplied by the values inthe various other columns in the cor-relates. The products are found onlyfor values on the same horizontal line.In other words, a value on line 1 ismultiplied only by other values on line 1and not by values on any other line.
(2) The first normal is found as the summa-tion of the products of the values incolumn 1 of the correlates, multipliedby-
(a) The values in column 1 of the correlates.
(b) The values in column 2 of the correlates.(c) The values in column 3 of the correlates.(d) The values in column 4 of the correlates.(e) Plus qj, the constant term of the first
condition equation.(J) The values in column 2, of the correlates
(after the summation of this multi-plication, add 11).
(3) The values obtained in (2) above, aretabulated on the form for the normalsas follows: (a) in column 1, (b) in col-umn 2, (c) in column 3, (d) in column 4,(e) in column q, and (f) in column 2,,;all on the first line. The value in the 2column is [,q+column 2 times 2]j. Asa check, the sum of the values in columns1, 2, 3, 4, and 77 should equal the value inthe 2, column.
(4) The second normal is found as thesummation of the products of the valuesin column 2 of the correlates multipliedby:
(a) The values in column 1 of the correlates.
(b) The values in column 2 of the correlates.
(c) The values in column 3 of the correlates.
(d) The values in column 4 of the correlates.
(e) Plus 2, the constant term of the
second condition equation.
(f) The values in column Z of the correlates
(after the summation of this multi-
plication, add 2).
(5) The values obtained are tabulated in the
.same order as explained for normal
equation 1.
(6) Notice that item (a) in (4) above, for
normal 2 is exactly the same as item (b)
for normal 1. If both these items were
written in the normal equation, there
would be a repetition of the value in
line 1 column 2, and line 2 column 1.
A repetition of this type occurs in each
normal for all values to the left of the
diagonal term. The diagonal term is the
value obtained when a column in the
correlates is multiplied by itself in the
formation of the normals. Each normal,therefore, can be formed by finding the
diagonal term and the terms to the right.
The terms to the left of the diagonal
term appear in the column above the
diagonal term, in order, reading down
from the top. The complete normal
reads down the column to the diagonal
and then across the line to the right.
When checking the equation as explained
for normal 1, remember to add down and
across. Each term that falls in a num-
bered column in the tabulation of the
normal equation is the coefficient of a
constant (or C) corresponding to the
column. The C's also correspond to the
same numbered column in the correlates.
(That is, C, is the constant for column 1
of the correlates and the normals.)n. The normal equations in the example could
be written as follows:
S 6C01-2C2+2C38.61C4-4.0=0
® -2C1+6C2+2C-- 10.05C4-8.9=0
0i ±2C+2C2+6C0s-6.75C4-6.5=0
o +8.61C0- 10.05C2-6.75C3±
83.4114C4+7.64=0
These are 4 simultaneous equations. (Notice that
the terms to the left of the diagonal term are
included.)
64. Solution of Normal Equations by SuccessiveSubstitution
a. The solution of the example normal equa-
tions (par. 63) by successive substitution is as
follows: Equation Q is solved for C1 in terms of
C2, C3, C4, and a constant.
(1) C,= +0.3333C2-0.3333C3
-1.4350C4+ 0.6667
This value of C, is substituted in equa-
tion @:
-2(0.3333C2-0.3333C3--1.4350C4
+0.6667)+6C2+2C3-10.05C4-8.9=0which is:
- 0.6667C2+0.6667C+2.8700C4
-1.3334+6C2+2C-- 10.05C4-8.9=0
Collecting terms, equation @ becomes:
+5.3333C2+2.6667C3- 7.1800C4
- 10.2333=0
Solve this equation for C2 in terms of C03C4, and a constant:
- 2.6667C3+ 7.1800C4+ 10.2333
C2= 5.3333
(2) C2= -0.5000C3+ 1.3463C4+1.9187
Substituting the value of C from ( and
the value of C2 from @ in equation Q;
reduced equation @ is found to be:
4.0000C03- 6.0298C4- 0.0500 =0
Solving this equation for Ca in terms of
C4 and a constant:
(3) C3=+1.5075C4+0.0125
Substituting the reduced values for C1,
C2, and C3, from above formulas in
normal equation ) and solving the
resulting equation for C4 as a constant,the forward solution of the normals is
complete. The reduced equation for C4
is:
+52.2997C4-0.4714=0
Solving for C :
+0.4714(4) - 0.4714+0.0090S+52.2997
The numnerical value of Ca is obtained by
substituting the value of C4 into equation
(3). The numerical value of C2 is ob-
tained by substituting C03 and C4 into
equation (2), and C1 is found from equa-
tion (1). This process is known as the
back solution.
173
b. The solution of the normals can be checked
by substituting the numerical values of the C's
into the original normal equations. The v's are
now found by substituting the C's into the cor-
relate equations. For example, in this problem,v= (-1)XC1
or
v=- (-1)(+ 1.2843)= -1.2843.
The correlate equation for v2 is-
v2- (+1)(C)+ ( 21)(2),
or
v2= (+ 1) (1.2843)+ (-1) (+1.9178)
+ 1.2843 -1.9178- -0.6335.
The value for each v is found in the same manner.
Carry these values of the v's to at least 1 more
decimal than is required in the final result. Sub-
stitute the values of the v's into the original condi-
tion equation as a check on the whole solution.
After this check has been made, the v's can be
rounded off to the desired number of decimals.
The process of rounding off may disrupt the closure
of the angle equations by 1 or 2 in the last decimal
place. If this occurs, 1 or 2 of the v's should be
arbitrarily raised or lowered 1 in the last decimal
to insure the closure of the angle equations.
65. Solution of Normal Equations by the
Doolittle Methoda. Although the discussion in paragraph 64
covers what actually is taking place in the solution
of normal equations, the solution should be made
by the Doolittle method. The Doolittle method,a form of successive substitution, is preferred asbeing the easiest for longer solutions due to the
deletion of the diagonal terms in the equations.
The form which the solution takes is shown at the
bottom of the example form in figure 115.b. For this example, where there are four nor-
mals to be solved, the form has four horizontal
spaces, columns corresponding to the normal equa-
tion arrangement, plus an additional column in
which the numerical value of the C's can be
written. The top horizontal space contains two
lines, and the other three spaces have three lines
each. On the top line of each space the normal
equations are entered directly as formed from the
correlates. On the bottom line of each spaceare the divided equations corresponding to
equations T(i, (, 0, and ® as illustrated in para-
graph 64a. The middle lines in the second, third,
and fourth spaces are for the reduced equations
found by the successive substitution of the C's.
c. The Doolittle method simplifies the reduction
of the normal equations greatly, as successive
substitution rapidly becomes too laborious to be
practical. In the Doolittle method, use is made
of the coefficients from the reduced equations as
well as from the divided equations.
d. In the following explanation of the Doolittle
method, the solution of the normals as shown in the
example (fig. 115) will be used as reference, and
equations (0, Q, 0, and ® are used here with
the terms to the left of the diagonal terms re-
moved. The first normal is written on the form
on the first line. This equation is then divided
by minus the diagonal term which is - (+)6.
The result is a divided equation giving C 1 in
terms of C2, C3, C4, and a constant. Normal
equation 0 is written on the first line of the
second space. This equation is reduced by the
product of the coefficient of C2 in equation ( i(which is -2), times each of the divided coeffi-
cients of equation (i), algebraically added to each
of the coefficients of equation ). Products are
added only to coefficients in the same column as
the divided coefficient making the product.
For example, the diagonal term in equation
( (+6) is reduced by the product of (-2)
(+0.3333) which gives a reduced diagonal of
+5.3333. The second term of normal 2 (+2)is reduced by the product (-2) (-0.3333)
which gives a reduced value of +2.6667. The
third term (-10.05) is reduced by (-2) (-1.4350)
to give -7.1800, and so on for n and 2,. As a
check on the solution, the algebraic sum of the
coefficients and n in the reduced equation should
equal the value in the 2,, column with the possible
exception of 1 or 2 units in the last decimal.
Always change the value in the 2,, column to agree
with the addition of the coefficients if the differ-
ence is in only 1 or 2 units. To check the division,the algebraic sum of the coefficients and q in the
divided equations, should equal the value in the
2, column. Remember to include a -1 as
the coefficient of the C found by dividing the
equation. The reduced normal 2 is now divided
by minus the reduced diagonal, [- (+5.3333)],which gives C2 in terms of C 3, C4, and a constant.
Normal equation ( is written on the first line of
the third space. This equation is reduced by
the product of the coefficient of C 3 in normal 1
(which is +2), times each of the divided coeffi-
cients of normal 1, plus the product of the coeffi-
cient of C 3 in reduced normal 2 (which is +2.6667),times each of the divided coefficients of normal 2.
174
Both products are added algebraically to thecoefficients of equation ( in the same columnas the divided coefficients making the products.Thus the diagonal term of normal 3 (+6) is re-duced by the product of (+2) (-0.3333) plus theproduct of (+2.6667) (-0.5000) which gives areduced diagonal of +4.0000. The second term
in normal 3 (- 6.75) is reduced by (+2) (- 1.4350)
plus (+2.6667) (+ 1.3463) which gives a reduced
term of -6.0298. The third term (-6.5) is
reduced by (+2) (+0.6667) plus (+2.6667)
(+1.9187) which gives -0.0500 for the reduced
third term. The same procedure is used to
obtain the reduced 2, term. The reduced equa-
tion is checked by adding across and then dividing
by minus the reduced diagonal term which is
[- (+4.0000)]. Repeating the process for nor-
mal 4 produces a numerical value for C4 of
+0.0090. The solution of the C's and v's ismade as previously explained.
e. After the v's have been computed and checked
through the equations, the adopted v's are applied
to the angles on the computation of triangles sheet
in the column headed "Corr'n". Applying the
correction to the observed angle produces the
spherical angle from which the spherical excess is
subtracted to obtain the plane angle. The tri-
angles are then computed as explained in para-
graphs 57 through 61, and the geographic positions
computed as explained in paragraphs 67 or 68.
f. As part of the adjustment of a quadrilateral,the probable error of an observed direction is com-
puted as shown on the form in figure 115.
Section III. GEOGRAPHIC POSITION
66. Introduction
a. When the geographic position of a station is
unknown, but the azimuth and distance to the
station from a station of known position are
available, the unknown position can be deter-
mined. There are many acceptable formulas,
varying in accuracy, for this computation. In
this manual, the USC&GS formulas were selected
for all computations. These formulas are sufficient
for all triangulation lengths in normal latitudes.
b. Normally, two known stations are used to
give a check on the position of the unknown
station. The procedure generally followed is to
solve a triangle, with the unknown station as 1
and the known stations 2 and 3, after correcting
the observed spherical angles by some type of
adjustment. Then using the azimuths between
the known stations, and the spherical angles of
the triangle, determine the azimuths to the un-
known station from the known stations. Finally,the formulas are solved for the position of the
unknown station and azimuths from the unknown
station to the known stations. Due to the con-
vergence of the meridians on the earth's surface,the back (or reverse) azimuth of a geodetic lineis not exactly 180 ° different from the forwardazimuth. The difference between these azimuths
is known as convergence and is dependent on the
difference in longitude of the ends of the line.
c. It sometimes becomes necessary to compute
the geographic azimuths and length of a line
joining two stations which are fixed in position,but have not been directly connected by the obser-
vations. In order to compute this line, an inverse
or back computation must be made. The mathe-
matical basis of the inverse position computation
is exactly the same as that of the position compu-
tation. The computation is based upon the
solution of the right spherical traingle formed by
the line connecting the two known stations, and
the line representing A4 and AX.
67. Direct Position Computation, Logarithmic
Solution
a. The formulas used in the solution of this
problem are as follows:
-- 0A=s cos a. B+s 2 sin2 a. C
+(54) 2D-hs2 sin2 a .E
-s 2kE+(3/2)s2 cos2a . kE
+_J2 cos 2 a sec2 2 A' 2k sin2 1";h=s cos a . B;
-S¢=s cos a . B+ 2 sin2 a. C
-hs 2 sin2 a . E;
k=s2 sin2 a C;
sin AX= sin sec 4' sin a;
log AX=log s+Co A-Cog 8,+log sin a
+log A'+log sec 4';
-tan (Aa)= -- tan (AX) sin ('+ )
cos (€'-€)-Aa=AX sin 1(4'+4) sec 1(A4) +(AX) 3F.
Where: C (log AX) and C (log s) are the arc-sine
corrections, the arguments log AX and log s
being indicated in parentheses.
A logarithmic solution of a first-order position
computation is shown in figure 116. The dis-
175
PROJECT 33- 147 LOCATION U7TAH- ORGANIZATION A #' S ,/ DATE MYS
TVSHAN 20 40536.08 _ a Tv-SHAR_ t2 MT AEB0-- 19 4/085
2dZ 48 04 05.50 3__ __ _ -88~ /6 30.92
Oa 2 MT NEBO to I WHEELER PEAK 68 109 41.58 a 3 TUSHAR? _ 0 1 WHEELER- PEAK 11,'4 3766
'a_ -- _ -- - -- - - / 37 0/.4___ _ _ _ _ _ -/ i 20.22
180 00 00.00 180 1 00 00.00
«a I WHEELERPEAK to 2 MIT. P4E80 246 32 401 a-' 9 WHEELER PEAK to 3 TUSHAR~ 290 / 13 /744First Angle of Triangle -- 43 40 37. 34- -
0 i 0 r it 0 ,, ".0
0 39 48 38,3/6 2 M-T. NEB80 1/ 45 56.235 0~ 38 25 0955/ 3 TUSH'AR. a 112 24742.1~68°- 49 29.300fr --_- 2 -32 50.783 °0 + 33 59.467 +~ 15f490481
0r 38 5S9 109A016 WEELER PEAK 1~ 114,18 47.0/8 (P' 38 59 . 08 1'WHEELER PEAK N' /14184 9
Logarithmls (1) Los~ Lo~~~ 39 r'it5.474 Log 5 IS5376/505 (1) 2867800 9 9 m+)3 23 5367 S 5.2478407l (1) -2094,8860 9699 (±)38 42 0928
COS a 9.5705323 (2) ±+1030505 / 07152 Logarithms cos ar 9.5623485 (2) + 54,6500 S2 10.496 Logarithnis
B 8.5108661 (4) - 1/7599, K 12.0/3 S_ 5.376/505 B 8.5/0971,x' (4) + o,67o8~ 1.738 S 522478407(1)=h3 4575489 a + 2 990 906I E 6/I00 sinla 9.9676586 (1)=h 3 3 2 11 604 i_0-2039,5652 E6.072 sama 929689446
.S3 10,.75230 (3)1__ 0.2140 ' (5) 8.564 A' 8. 509144 0 s2 /0. 49568 (3) + 0,1000 (5) 8,0051 A' 85091440
sinea 9.93532 (5) - 0,0366 3 0. 477 sec& 0'0,194/05 sine a 9. 9378 9 (5) -00/02 .3 0. 47.7~ sec0'. o,o094105
C 1.32543 (6)+ 0.0152 e°S2 a 19/!41) S11111 3.9623636 C 1.3040 2 cfi) + 0.004 0 cos a. 2 25J s1111] 3 8353398(20/305 7 0/(7 (6 18 corr. i - 428 ()K173759 (7) + 0,0046 (6) 7407 corr. +24
(,0)2 6.9452 -1012992999 (colog) F 13.900 1" 43. 9624064 (bm2I a. 6/91 -' -20394668 ,(colog) E 13.928 AX 3.8353640I) Aarc ~ ineA~nr""- sine
D2.3852 2 1484.65 :~o~*592('& 8272 D 2.3807 S 10/97 _-Ka 5.9j2 ( ±&)9.7960730
(3) 93304 Sec20 0.229 Sec 2//3 (3) 8.9998 scc~o 0.212 e25
-1_ 3.40 7 !8.223______ APa' 37649909 -13. 21 (7) 7659 ____-Aa 363/4423
s2hin2a 10. 6876 Are-sinl corr. 110 + f5820,"910 S2Sin~a /0. 4336~ Arc-sin corr. do 4279. "985,
E 6.1004 fors -1003 (°0X)3 /1.887 (8) +0.571 E 6,0718 fors -S55 (°x)
3 11.506 (8) +0,'?40
(4) 02455 for AX±/43/ F 7870 z A. '5821. 481 (4) 98266 for AX + 797 "'7874 . 428022'5
'Total 1+ 428 (8) 9757 9+ 9/70. 7828 lotal + 242 (8) 9.380 6L(.844:8511
COMPUTED BY amDATE ICHECKED BY DAE NOTE: Far log s Io a.9, omit terms below heavy black line NOT inl
/9 qSZ - AMS M4AY 56 ~ WlC. OQ.wup.nr.-AMS MAY 56 Theavy boll tIpe or underlined.
DA IOCT 4 1922 REPLACES DA FORM 1922, 1 FEB 57,WHICH IS OBSOLETE.
POSITION COMPUTATION, -. FIRS r ORDER TRIANGULATION (Logarithmic)(TM 5-237)
Figure 116. Position computation, first-order triangulation (logarithmic) ( DAForm 1.922).
tances used are the sides of a triangle, and theazimuths are the angles the vertical section makeswith the meridian, measured clockwise from thesouth. The solution is made using the Clarke1866 Spheroid, and all factors are taken fromUSC&GS SP 8. For the International Spheroid,SP 200 would be used to obtain the necessary
factors. Wheeler Peak, the unknown station, isnumber 1. Mt. Nebo and Tushar, the knownstations, are numbered 2 and 3, respectively.The azimuths (a) from 2 to 3 and from 3 to 2 are
entered on the form, and the second and third
angles from the triangle computation are appliedto these azimuths, giving the azimuths from the
known stations to the unknown station. The
log distances (s) are taken from the triangle
computation, log sin a 2-1 (3-1), and log cos
a 2-1 (3-1) are taken from the tables to seven
decimal places, and the factors B, C, D, E, and F
are taken from USC&GS Sp. Pub. 8 using the
latitude (4) of the known station as the argument.The sum of log s, log cos a, and B is the log of
(1), the sign being determined by the sign of
cos a. In the Northern Hemisphere, the cosine
is plus in the first (0° 90°) and fourth (270 ° -
3600) quadrants, and minus in the second (900-
1800) and third (1800--270 ° ) quadrants; in the
Southern Hemisphere the reverse is true. The
sum of log s2, log sin2 a, and C is the log of (2),and the sum of log (-h), log s2 sin2 a, and E is
the log of (4). Find the antilogs of (1) (2) and
(4) and add algebraically to get (S¢). The sum
of log (84)2 and D is the log of (3). Log (5) is
the sum of log 2, log s2, log K, and log E. Log
(6) is the sum of log (5), log 3, and log cos2 a.
Log (7) is the sum of log (6), colog E, log
A2 arc2 1"'and log sec2 4.
3
Note. If the distance (s) is small, (4), (5), (6), and (7)
will be small and the characteristics must be carefully
watched to avoid error.
Find the antilogs of (3), (5), (6), and (7) and
add these algebraically to the sum of (1), (2),and (4) to get -A0 in seconds. Apply -A0 to
the latitude (4) of the known station to obtain the
latitude (4') of the unknown station. 4' is used
to interpolate A', from Sp. Pub. 8, and log sec 4'from the tables. Add log s, log sin a, A', and
log sec 4'. This sum is used as an approximate
AA to find the arc-sin correction, and takes its
sign from sin a. In the Western Hemisphere, the
sine of a is plus in the first (00-90 °) and second
(90°-180° ) quadrants, and minus in the third
757-381 0 - 65 - 12
(180°-270 ° ) and fourth (2700-360 ° ) quadrants; inthe Eastern Hemisphere, the reverse is true. Thearc-sin correction is equal to the correction forAX minus the correction for s. Both of thesecorrections can be fouhd in Sp. Pub. 8, page 17,or computed from the formulas in Sp. Pub. 8,page 18. After the arc-sin correction is computed,it is added numerically to the previous sum to
obtain the log of AX. Applying AA algebraically
to the longitude (X) of the known station gives
the longitude (X') of the unknown station. The
sum of log (AX)3 and F is the log of (8), the sign
being the same as the sign of AX. The sum of log
A4AX, log sin (4+40'), and log sec 0- is the log of
- Aa (approx.), the sign being the same as the
sign of AX in the Northern Hemisphere, west of
Greenwich, and in the Southern Hemisphere, east
of Greenwich, and opposite the sign of AA in the
Northern Hemisphere, east of Greenwich, and
the Southern Hemisphere, west of Greenwich.
Add algebraically the antilogs of - a (approx)
and (8) for -Aa in seconds. Apply -Aa to the
azimuth (a) from 2 to 1 (3 to 1) to get the azimuth
(a') from 1 to 2 (1 to 3). These steps are used on
both sides of the form. The positions computed
for station 1 should check within 1- or 2-thou-
sandths of a second for both latitude and longitude,and the azimuth should check within 1-hundredth
when the first angle of the triangle is applied to
either side of the computation.
b. Figure 117 is a logarithmic solution of a
third-order position computation. The unknown
station is Parson, numbered 1, and the known
stations are Outer and Hard, numbered 2 and 3,respectively. The known azimuths are entered
on the form, and the second and third angles of
the triangle are applied to give the azimuths from
the known stations to the unknown station. The
distances (s) are taken from the triangle computa-
tion; log sin a, and log cos a to six decimal places,are taken from the tables; and factors B, C, and
D are taken from USC&GS Sp. Pub. 8, using the
latitude (.4) as the argument, and all are entered
on the form. Log s, log cos a, and B are added
to find h, which is the first term in -A4. The
sign of h is determined by the sign of cos a. The
sum of log s2; log sin2 a, and C is the second term,
and the sum of log h2 (enter in (60)2 space) and D
is the log of the third term of - A4. Add the
three terms algebraically to find - A4. Apply
- A0 to the latitude (4) of the known station to
177
get the latitude (0') of the unknown station. Use
0' as the argument for A' in Sp. Pub. 8, and find
the log sec gyp' from the tables to six decimal places.Adding log s, log sin a, A', and log sec 0m' gives
the log of A. The sign of A is determined by
the sign of sin a. Apply AX to the longitude (X)
of the known station to get the longitude (X') of
the unknown station. Add log AX, and log sin
~(0)+0") for the log of -Da. Apply -Da to theazimuth (a) from the known station to the un-known station to get the azimuth (a') from the
unknown station to the known station. The
positions computed for the unknown station shouldnot differ by more than 1-thousandth of a second
D ,OCT 8412REPLACES DA FORM 1922, 1 FEB 57. POSITION COMPUTATION,- Third ORDER TRIANGULATION (Logarithmic)
(TM 5-237)
Figure 117. Position computation, third-order triangulation (logarithmic) (DA Form 19232).
178
PROJECT 5 688 LOCATION NEW
." HARD12 _oUrE.R-2«
2 OUTER __'PARSON
'PAPARSON t° 2
.OUTER_Fi rst Anigle of Tlriansgle
2' 40 .35 18.742 2 OUTER X 73 .36 33.9614 0' 40 37 20.5/4 3 HARD X 7.3 38 27008
°0+ / 5983- ---. A + 3/.763 m - 01-919 ~ - / 21.281
40 37 8.595 it PARSON iX' 73 37 05. 727 4"40 37 18595' PARSON 7Z 3 37 05.727
Logarithms " Los '4) Logaritbms () " L~ogs 4'4" /96S3.57652b ()19854/ .s x" 40 36 18.7 S 3.281346 ( _) _+1911 9699 "01)40 37 /.
cos 9. 991 320 (2) +_0.0012 'Logarithnms cos 8.489222 (2) +, o,0079 s' Logarithmus
B 8.510 807 (4 K S .576 526 B 8.5/0804 (4) s 3.281346
(')=h 2 0 78 4o53 ] _E_ ina «9 296 554 (')=.h 0:281 372 LO- E si 385 ,999 793S__ 7153 05 -(3) ,F 0.0004 (5) A' 8.509 /03 Sa 6.56269 (3) + 0 (5) A' 8.509 103
a1°' « 8.5931 (5) - 3 0.477~ sec 4" 0.1 /9 745 81n°a « 999959 (5) - 3 0.47.7 sec 4 0. 119 745
C 1.33 7 31 6 + coca «I D C111 .337 83 (6i) + cost a Munm
(2)=K 70 3 7 (7) ~+(6) I Arc-sin - 2)-K() Arc- sins7_ 08 47__ __ corr. _____-- 7.200/11 ()+() _ corr.
(a _ 41573 -,L4"- //9.8525 (rolog)E -- . . 50/ 928 (,)a 0.54.2 7 /.4 + .9/ 94 (cnlg) E L 11.909 987'
1)40:arcal5
. 91 sin) D.14' Aaarc'1" =.92 i
D2.3872 2s 3.1 (m+') "9.81347 D 2.3873 z 3 5.11(o'+m) 198136(2/
()6.5445 2 tC4 3 2. 9500 sc4
[E for s - (A X)3
(8) rr E fo J) (K)"
(4) for AX +( F -A1
206 7 (4) for Ixl+ IF. .Aa 52 n92
'Total (8 3~ 3/. 7635 toa -j8 I. 2806COMPUTED8BY DATE . CHECKED BY DATE 'NOTE: Fur logs a to3., ornit teris. below hesvy black line NoT in
iR.4.S. -AMS MAY 56 W. c. A. -AM S MAY 56 heavyhsoldityper udrlined
PROJECT C - 6 8 NEW
in either latitude or longitude. This methodmeets third-order requirements when the trianglesides do not exceed 25,000 meters (approx).
c. Examples shown are for the Northern Hemi-sphere, west of Greenwich, and the following
variations should be noted:(1) For computations in the Southern Hemi-
sphere, the sign of the cosine is reversed.(2) For computations in the Eastern Hemi-
sphere, the sign of the sine is reversed.(3) Aa is applied according to the following
rules:
(a) In the Northern Hemisphere, if a is
less than 1800, Aa is minus; if a isgreater than 180°, Aa is plus.
fib) In the Southern Hemisphere, if a is
less than 1800, Aa is plus; if a isgreater than 1800, Aa is minus.
68. Direct Position Computation, Natural Func-
tion Solution
a. The formulas used in the solution of the
problem are as follows:
y2 =yo+[y+ (yfa10)]-Va+K (Th2
/ A"[Q1 6b 10-7)]H-+[Ix' ( V(a)) "10-7]
sin 4±sin 01-f-cos A0O
A solution, using natural functions, of a first-order
position computation is shown in figure 118.The solution is made using the Clarke 1866
Spheroid, therefore all factors are taken from
USC&GS Sp. Pub. 241. For the International
Spheroid, USC&GS G-58 would be used to obtain
the necessary factors. The station of unknown
position is Hayford, numbered 1, and the known
stations are Pioche and Burger, numbered 2 and
3, respectively. The azimuths between the known
stations are entered on the form, and the second
and third angles in the triangle computation are
applied to these azimuths to get the azimuths
from the known stations to the unknown station.
The distances (s) are taken from the triangle
computation (if the triangle computation was
logarithmic, enter both the log distance and the
PRJCPOSITIN CIUPUTATIN Fi*t DER TRIANGULATION (Far ceat bg mdiaclbscaIpAtad)
LOCATION ORAIAINDATE
NVevada OGNAON A 4S.. Lie. 20/a~y66a_
2 OjpCjjE To 3
RRE 326ZLIA a3 JRE TpjpC~g !4.M5qq
2d L & .±8 1L _M , 6 3d4 - Z8 L.534a2To 1/AYOR) 3 a5 3a~ B_3 UaER o
1 4YC~ 421
180 00 00.00 180 00 w 00.00
a' 1jAp~ To 2 2I1HE /7 /s a' 1 o3'"3 r
I - , ~~~~First Angle of Triangle ~:-A P R o'U Q R ?3
' 9 27.46 1 )-'/ / 3' 27. 1
/ 9b=(y/10, 000)- 2/.) . jf b=(y10,000) 2 ./ )
a n a o . _ - f l i ax
c r a . MMCs a8 1 9 3 0 8A 0 z + / .7 Q 5 9 9 C O s a +0 4 3 7 9 5 0 2 2e # 1 5 / 9 5 0M 8 7 6
zssin a f 0~72.~i H 0.s 04232 6sin a + 154 954 /76 H 0.040263424y.=-sOa - 4677 Hx'=(approx.A") * 4135.8 428 y=-80"11 2.7402 Hx'=(approx.X) 119. 6 25
s-x10O2
Arc-ain V Y aY 2 Arc-sin=V (Va)/055/000) ff-S32 cor = 15 238.0 n=(z'/10.000)2 2302.89O09 cor + 15 2.y cor.=+fa p66. AV yecor.=+fa /895.3 ~- 46 611,8.3825
+ 1.5916.o2&L753.823 sin* f v6,4- M y 4,132-43269$a sin 0.60634257
- 14.Rg82d~ .9033~ 2757.812 an' 0,5S9 703393y )L1. + cos"* /f ?73125 yi 40059. 6 74, A86 1 + cosA /*91930
y'2 40, 32g.033 -°a" (approx.) + 2507. 76/ y2 4. OSp 329,030 . 7 -a' (approx.) + A-3V+F()
.3681, 484 7
K (Va/1,000)2
+ 0.035 + 257x1 (Va/1,000)2
+ 0.16 + 369/. i65COMPUTED BY DAECHECKED BY w AENOTE Foa under 8,000 meters omit terms under the heavy blerk line not in7 a .C. R D A 1 1y R ev ol yeor underlined.
DA, FE571923
Figure 118. Position computation, first-order, natural functions (DA Form 1923).
179
distance on the form). Extract sin a and cos afrom the tables (eight-place natural functionsmust be used to meet the accuracy requirements).
Next, multiply the length by the proper functionsto determine x and y. The x value takes the
algebraic sign of sine a, while the y value takes thealgebraic sign opposite that of cosine a. The
algebraic signs of sine and cosine are explained inthe log computation. Next, compute the b value
by moving the decimal point in y four places tothe left and squaring the result. Multiply b by
/f to find the x correction factor. f is found atthe bottom of the page in USC&GS Sp. Pub. 241,using the latitude of the known station as the
argument. The minus sign before the correction
factor means x is reduced numerically. The
correction factor is in units of the seventh place
of decimals; therefore, in the example on the
left-hand side, x would be multiplied by 1.0-
0.00008843 or 0.99991157 to get x'. (For other
methods of finding x', see page 81 of USC&GSSp. Pub. 241.) Now use the value of x' to
compute a in the same manner as the computation
for b, taking a to 4 decimal places. Multiply f
by a for the y correction factor. The plus signindicates y must be increased numerically. The
correction factor is in units of the seventh place
of decimals, so in the example, y is multiplied by
1.0+0.00008660 or 1.00008660 to get y'. yo is
taken from Sp. Pub. 241 under meridional arcs
(meters) using the latitude of the known station
as the argument. y' is applied to yo to get the yl
value. Now using yl as the argument, the value
for V and K are found. Note the correction for
the second differences of V. The correction is in
units of the fifth decimal place. (No space is
provided on the form for K.) Multiply V by a
and K by (Va/1,000)2 for the two corrections to
be applied to yl to get y2. The value of y2 should
check on both sides of the form. The Va cor-
rection is always minus and the K(Va/1,000)2
correction is always plus. Use y2 to interpolate
for the latitude (4') of the unknown station.
After 0' is determined, A4 is taken out to tenths
of seconds and entered on the form. Using 0' as
the argument, the value of H is interpolated in
the table. (Note the correction for second
difference in H.) Multiply H by x' for an
approximate AX". Next, compute the arc-sine
correction factor which is X$5 of the product of
V and Va. The plus sign indicates the approxi-
mate AX" is increased numerically. Like the
180
other corrective factor, the arc-sine factor is inunits of the seventh place of decimals. Apply
the arc-sine correction to get AX", which is appliedto the longitude (A) of the known station to getthe longitude (X') of the unknown station. The
longitude should check on both sides of the form.From a table of natural functions, interpolate
the cosine of A4. The sines of 4 and 4' can beobtained from Sp. Pub. 241. Add 1 to the cosine
of A4 before entering on the form. Sin 0 andsin 4' are added and the sum divided by the
1+cos A4 value. Multiply this value by AX" toget the approximate value of -Aa". The small
correction F(AX") 3 must be computed and added
numerically to -Da". The argument for F is4,,. The F factor as tabulated contains the factor
10-12. To effect this factor, move the decimal
point in AX" four places to the left and then cube
the result. It is sufficient to take AX" to the
nearest second for the computation of the cor-
rection. Notice that this correction is not the
factor type such as the x and y corrections. In
this case, the approximate - a" is numerically
increased by the quantity F(AX") 3 . Apply the
final -Da" to the appropriate azimuth (a) to
obtain the back azimuth (a'). On the computa-
tion form, the sum of the azimuth from 1 to 2
plus the first angle in the triangle should equal
the azimuth from 1 to 3. In this computation,emphasis is put on the fact that the plus or minus
before a corrective term indicates, respectively, a
numerical increase or decrease of the term to be
corrected, irrespective of the algebraic sign of the
term.b. Figure 119 is a solution of a third-order
position computation. For lines under 8,000
meters in length, none of the terms involving the
use of f or b has a significant effect. As both
lines here are well under this 8,000-meter limit,no corrective terms are needed. Seven decimal
places are sufficient for sin a and cos a. Both
x' and y' may be taken the same as x and y,respectively. V need be taken out only to the
same number of significant figures as a or perhaps
one more. In the example shown, V may be
interpolated by inspection to the number of
figures needed. The arc-sine correction is not
needed, and AX" is the product of x' and H.
Five-place sines are sufficient for the computa-tion of -Aa".
c. Examples shown are for the Northern Hemi-
sphere, west of Greenwich, and the following
variations should be noted:
(1) For computations in the Southern Hemi-sphere, the sign of the cosine is reversed.
(2) For computations in the Eastern Hemi-sphere, the sign of the sine is reversed.
(3) Aa is applied according to the following
rules :(a) In the Northern Hemisphere, if a is
less than 1800, Aa is minus; if a isgreater than 1800, Aa is plus.
(b) In the Southern Hemisphere, if a isless than 1800, Aa is plus; if a is greater
than 1800, Aa is minus.
69. Direct Position Computation, Natural Func-tion Solution (Any Spheroid)
The formulas used in the solution of the problemare those listed in paragraph 67a. Tables forthe A, B, C, D, E, and F factors for all majorspheroids were prepared by the Army Map Serviceand are published under the TM 5-241-series(i.e., TM 5-241-18, Latitude functions: Clarke
1866 spheroid).
70. Inverse Position Computation, Logarithmica. Figure 120 is a logarithmic solution of the
inverse position computation on DA Form 1924(Inverse Position Computation). The positionsof the stations, arbitrarily numbered 1 and 2,are entered at the top of the form. Next, A0,
Ak'iA05 (seconds), A, 2 and AX (seconds)
are computed, completing the top block of the
form. Log A'0 and log A are extracted from
the tables, to seven decimal places. The arc-sin
corrections are interpolated from table XXVI
in TM 5-236, or from USC&GS Sp. Pub. 8, page15, and subtracted from log A'0 and log A giving
log Ao1. Log cos -A and log cos 0km are computed.
¢,m is used as the argument to find log A' and logB in USC&GS Sp. Pub. 8. The cologs are com-
puted and entered as colog Bmn and colog Am,.
Log A'k1 , log cos -2X' and colog B.m are added to
PROJECT 56 POSITION COMPUTATION, 71 ird ORDER TRIANGULATION (For cakat macdas caylulsu)5-6886(7M 5-237)
LOCATION GergaORGAN IZATJ ON A D$ nc ATE
a 2 ATOD To 3 /'IAQY 2?is~.i a 3 fqy To 2 A7-o MA/B14 .31 2.3-m2d L &o I 2L 3d L -//2 20 / .'5
°a. .5 . 1 a - .3.8
10 00 00.00 180 00 00.00
a' rr rAro 54 ~qg To 2 ATbo / 9 .49 a'
1LJT.i.SE plO To 3 MAY 186 II30.0-First Angle of Triangle 4 25 S$i
_27 12. * /12 v 3/ 2 4g~3 MARY 8/ /18 A12,351
+ .2 'I .MPF/. X,/ 6* 2 4)k2~rr s ' 8/ /816.36b(1000 =lo5492(g8a= 24526 )b(l /10.000)2 ) '°* b=(Y/10,000)2
sine a x46/ cor-fb sine a 0.x07&Z3L 2 2
Cosa.393 X, C( ca +6. 9941" 7 S" (JRMQ. 746319- 3 29s sin3aH O0743g x=sglsia + Za9 4 H
,13049
Y7=-S cosn a -32.71 Hx' =(approx. A) Y= -Cs sa - 1473 Hx' =(approx.Ah)
a=(z'/1O,000)2
' Arct in=+VA 15)Jt(e1,002 0 90 Acri=V(Amr-15 15('1.0) p ~ A~BnV(a
y' cor. =e+fa -' ,o.99cor. _+fa Axe 784
Y. .34&, 77A, 777 sin "" Y 4.pq.9 In
-' .3,, 429.71I sin ' -1. 748a. 73 sin +"
Y , 8,35-0 1+co°+Y1+
o +.48 ,63Va - 0 ___ osin Va - 0. 002 or-a-i sin4.
Y2 4 O,34 3 -°a" (approxt.) Y2 3. 478, 344 - -ha" (approx.)
V 4.7,9 + F(°)"')3
V +.0 +F (°A")3
K (Va/1,000)2+ - a" -5734 x (Vail,000)2+ - °a + 3.75COMPUTED BY DATE2 £ CNCE YA DT OE oe ne , atro/ oeado b ev lc a o e
28 /& CHECKE BY ,~ Si DAT 3J heavy bold type or underusned esoi em ne h ev lc lentl
DA , B7 1923'
Figure 119. Position computation, third-order, natural functions (DA Form 1923).
181
PROJECT 23'105 jINVERSE POSITION COMPUTATION
LOCATION
Oreg9onIORGAN IZATI ON
AX, cos '~ A« - A X S it(«+-2\ -A«,co s, Cos (a± V 2 O5 -A«=AX sin ¢,, sec 2 +F(AX)'S
1 Sf '! A,,2/ 23
in which log AX=Iog (X'-X) -correction for arc to sini; log Ao 1=log (0'-0) --correction for arc to sin; and log s~log s,+
correction for arc to sin.
NAME OF STATIONS
1. * 4S .59 OQ0.715~ SPENCER X I23" 05' 4.1.2482. '4- 30 38.293 PETER~SON X' 122 58 05.537
A(=-0 +31 AX~R________
*~ 244 14 .41.504 _____
A0, (secs.) +j jf '7 5'7f9 AX (secs.) ;5.________ 7______
logA# 3.2 78 9 o A . 68 2951cor. arc-sin - 1.5 cor. arc- sin-
log A4 1 3. 2.78 9qg log AX, 2. (6 854plog cos- Z ggc 9 9? 7 log cos4,., g. 9 5 it111
colog B. I.48q 4114Z colog A,. 1409(oppst in lga i
log s1 cos «+-2 4,747 (6720 s ign t tA4) log - 4,p 4 p 4-004
log AX 2.r58 ( 6 3 log AX 7g log tan «± 2 12 g7 916.237 1253log sin 4,m log F at-if logF L47 40at-log sec- - a log b 584 log sin «+-~ 9202
log a 2. 502f36,4n logcos «+- - 193 630a -317.98 logsl .4,774 0490b 0.00 cor. arc-sin+
- A«(secs.) 1 311.8 log s 47 74 050
, ggS _59436.15 m.-~ - 2 .~ gcg NOTE.-For log sup to 4.0 and for A0, or AX (or both)
Aa up to 3', omit all terms below the heavy line excepta_}. I 10those printed (in whole or in part) in heavytyeo.69 typ ftLb'1 't Ithose underscored, if using logarithms to 7 decimala~lto) ia~45 ~places.
A« + 5 1 i. 9180
a''(2to 1) 50 q(6COMPUTED BY DATE CHECKED BY DATE
leGs ov4W.cZIJ. 17AiOV 64
DA l FERB57 1924Figure 120. Inverse position computation, logarithmic (DA Form 1924).
182
obtain the log [81 cos (a+±Owhich is opposite
in sign to A4. Log AX1, log cos m., and colog
Am are added to obtain log [si sin (a+-)] which
takes the sign of AX. Subtract log [s, cos (a+t)a
from log s sin(a+ 2a to obtain log tan a+--)•
(a+ ) 1is extracted from the tables using the
log tan (a+ ), and this value is used to inter-
polate for log sin a+ 2 and log cos a+ ).
Log s, is obtained by subtracting log sin (a+\2
from [s sin a(+--), 2 and checked by subtracting
log cos (a+2) from log s cos a+ The
arc-sin correction is added to log s, to get log s, and
s is extracted from the tables. Log AX, log sin
A44) , and log sec 2 are added, the sum being log
a, which has the same sign as AX. 3 log AX and
log F are added, the sum being log b, which also
has the same sign as AX. The argument for F
is 4. The values for a and b are extracted from
the tables and added to obtain - Aa" (seconds).
Then 2a is found and applied to (a+-
giving a (1 to 2). As and 1800 are applied to
a (1 to 2) to obtain a' (2 to 1).
b. Figure 121 is a computation of lower-order
accuracy made by following the instructions in
the note at the bottom of the form.
Note. For log s up to 4.0 and for AO or AX (or both) up
to 3', omit all terms below the heavy line except those
printed (in whole or in part) in heavy type or those
underscored, if using logarithms to seven decimal places.
c. Examples shown are for the Northern
Hemisphere, west of Greenwich, and the following
variations should be noted:
(1) For computations in the Southern Hem-
isphere, the sign of the cosine is reversed.
(2) For computations in the Eastern Hem-
isphere, the sign of the sine is reversed.
(3) Aa is applied according to the following
rules:
(a) In the Northern Hemisphere, if a is
less than 1800, Aa is minus; if a is
greater than 1800, aa is plus.
(b) In the Southern Hemisphere, if a is
less than 1800, Aa is plus; if a is
greater than 1800, Da is minus.
71. Inverse Position Computation, Natural
Function Solution
a. Figure 122 is an example of the inverse
position computations performed with natural
functions. The positions of the stations, arbi-
trarily numbered 1 and 2, are entered on the form,and a4 and AX are computed. 4 is used to find
yo (meridional are in meters), and sin 4 in
USC&GS Sp. Pub. 241, and 4' is used to find
y,, sin 4', and H. A is changed to seconds, and
the arc-sin correction factor is obtained from the
formula; correction factor equals 0.39174
(0 sin 4') This factor is in units of the
seventh decimal place, and is added to 1.0 before
being used. AAX is divided by the correction
factor to find Hx', which is divided by H to
obtain x'. Now compute a and fa. Use y2 as
the argument to interpolate K, and an approxi-
mate V, which is used to compute an approximate2Va
Va. The correction K 1000) is found and
subtracted from y2 to obtain the approximate y2.
Va is added to the approximate y2 to give an
approximate yl, which is used to interpolate a new
V. A new Va correction is computed and the
correct yl is found. y' is the difference between
yo and yi. y' is divided by the fa correction
factor to find y. (Remember the correction
factor is in units of the seventh decimal place and
must be added to 1.0 before being used.) b is
found and the correction factor for x is computed.
x' is divided by the factor to obtain x. (This
factor must be subtracted from 1.0 before being
used.) The distance (s) is computed by the
Pythagorean theorem,
Divide x and y by s to find sin a and cos a. Ex-
tract a from the tables of natural functions. Aa
is computed by the same method used in the
position computation, and the back azimuth is
computed and entered on the form.
b. Figure 123 is an example of an inverse
position computation for lines less than 8,000
meters. All correction factors are dropped and
x' is used as x, and y' as y. This computation
is sufficiently accurate for third-order surveys.
DA Form 1923 is used for the computation.
183
PROJECT IINVERSE POSITION COMPUTATION5-~8~ I(TMf 5-237)
LOCATION
ORGANIZATION
Ae.In./Aa\Acs; I 'Ac, A¢, AX
sl sin f AX, 2 O )= s, cos l a+- 2-)= 2O~ --Aa=AX sin 4%, sec 2+fF(X)'
in which log AX,=log (a'-a) -correction for arc to sin; log Ao,=log (gy'-0) --correction for arc to sin; and log s=log s,+{correction for arc to sin.
NAME OF STATIONS
0 7 IOC 0.ma2
1. *' X, __________________
2.1 4?
* a
L2 =*-w 3L 26 39,7910 _____ ______
A# (secs.) -AX, (sees.) - 109. 2Lg
log A0 2- 0466551 log AX -2.04/0728
cor. arc - sin - cor. arc - sin -
log A#, 2O4Silog AX., 2.102log cos -j 10060000 lo *s-0 9. 310238colog 13. 4,45 ~colog A., o s cs a(opoit i A
2sign to A¢) log s, sin~ -3.42770log Is, cos a+-j Z }
log AX 3 log AX log tan a- 9 927590
lo i .l g Fa } 2lo-
- 3 4log sec-2 - - log b -log sin a+ 8 / 7 5log a A5492log cos a+-~ 2) 2 A5
_-1__________a_-___3_ log s,3.~S44
b-cor. arc-sin +
- A (secs) - 7342 log a .s544
Aa- 6 44 92.566rm.NOTE.-For log s up to 4.0 and for A# or AX (or both)up to 3', omit all terms below the heavy line except
Aa~~~/ 245 /6+ 7 those underscored, i fo using logarithmshtov7 decimal
_1 5toepitd(na( o2
1 4 4,41 lcs
19 4 lcs
a' (2
to1) 0 451COMPUTED BY DATE CHECKED BY DATE
CcrAa . C. aw.7flS 1 65
DA I OS51924
Figure 121. Inverse position computation, logarithmic, third-order (DA Form 1924).
184
c. Examples shown, are for the Northern Hemi-
sphere, west of Greenwich, and the following
variations should be noted:
(1) For computations in the Southern Hemi-
sphere, the sign of the cosine is reversed.
(2) For computations in the Eastern Hemi-
sphere, the sign of the sine is reversed.
(3) Aa is applied according to the following
rules:
(a) In the Northern Hemisphere, if a is
less than 1800, Da is minus; if a is
greater than 1800, Da is plus.
(b) In the Southern Hemisphere, if a is
less than 1800, Da is plus; if a is
greater than 1800, Da is minus.
72. Inverse Position Computation (Long Line)
a. A noniterative solution of the inverse prob-
lem (fig. 124) based on ilelmert's successive
approximation method was developed at the
Army Map Service by Emanuel Sodano. In
this method it is assumed that A the reduced
difference of longitude, is equal to AX+I-X, whereX is an unknown quantity. The computation
procedure consists of computing the value of X
directly, then determining Al. Using Al in the
Sodano equations, the azimuths (a and a') andthe geodetic distance (5) are determined.
b. The procedure and formulas for computation
are as follows:(1) The known data are listed on DA -Form
2858, Inverse Position Computation (So-
dano Method).,01=Latitude of west point
0 2 =Latitude of east pointAX=Difference of longitude, always posi-
tive
(2) The parametric latitudes, 01 and /32,
are computed and the sines and cosines
obtained (preferably from 10-place
tables).
tan ,6i= 1-e2 tan ti
tan 132= 1-e 2 tan 02
PROJECT 8-86POSITION ______________ ORDER TRIANGULATION (For calcuinlig nmim cm* t iic) IWS T327
LOCATION NdORGANIZATION, ___
a 2 3 a 3 To 2
2d1 & 3d1 & -
a 2To 1 //dyIOIed 34 059...03.63 a 3 To 1
_____________ - 4l 4782A
180 00 00.00 180 00 00.00
a1Y/Er To 2 p
1q 2/4 /7 /5.8/ a 1 To 3
Fis Anl of Tranl . " o . o . "____________________________
* 7 590. 102 2' 1Ce a11 D: 32 3
3927746 11 ,, .~ // ?.3* I)4m _& 5
0' 9294 42.2986 b (y/10,000)2 215.494 _____________________)
sin a35 5 Xoor.=-fb 8.3sin a x cor.=-fb 4 ,7325csa +x
oal
x./307 +102. 719 6/0 C~
xssnaH 04232 ZssiaH _/027S64Y = -acose a 4 - 9 '~ x'= (approx. &") 139 Y4 = 7-s cos a Hx' =(approx. Al')
4,7,182AcsnV(a2Ar-i+VV)
a=x/0 0.13 o +1 ==/0 o 529y cor.=_+fa 89660 ae + 4i'M. 42 icor. _±+a __,
40 42 05,753 8 23 yo in ysi 0.557A ,
Y, - a ^ in0 sin*
0,5g703393 ya 4, 9
ain1 csA . 9 32_T Y
+Csq
Va - 1A -A 6 sin .0+asin +co 0 , rsn0 Va - in+ sAin or sin 0
Y .0 1, .99 -Ad' (approx.) 50,6 2-Ae" (approx.)
V S.A8 + F (w~)3
53 V + F (A?') 3
K (Va/1,000)2+ 0. 035 - 2507. 8/5 K (Va/1,000)
2+-
COMPUTED BY t DATE CHCE YDATE~ NOTE: For, souder 8,005mtr1 mttrm ne h evybakt o ICed C23K BY jr ) C. A" IS heavy bold type or underlined. m trsode h ev lakhentI
D A t FE 7923
Figure 122. Inverse position computation using natural functions (DA Form 19938).
185
e2= (eccentricity)2a -2
(3) The approximate spherical distance (0)
between the two points is determined,and converted to radians (1 degree=0.0174 5329 2520 radians).
cos a=sin 01 sin t32 -+cos 01~ cos 02 cos <A
sin 6 = 4cos (31 cos R32 sin2
(2)+in2 (123211
or
sin 0 =1- cost 0
(4) The constant terms k1 through kg6 usedin the solution of X are computed,using the dimensions of the referencespheroid or taken from table XXI,appendix III.
k-16N_-e'2
k116e2N2-+-e2' where N=e
16 sin 1"k e2(16e2N 2+e' 2)
16e 2N2+ e' 2
k3= e/2
_2e'2
k416e 2N2+e' 2
_16e2N2
k51e 2N2+ e/2
7"16e2N2.
(5) The variable terms, A through G, usedin the solution of X are computed andtabulated on the form:
PROJECT S_8' POSITION COMPUTATION, 3 ORDER TRIANBULATION (For calculating machine computation)5-68eE IVfSE (TM 5-237)
LOCATION ORGANIZATION A 15, Ic DATE 8My1!
a 2 To 3a 3 To 2
2d Z & + 3d L & _-
a tW To 1Lll I4 A 5 a 3 Tol1 ' -
180 00 00.00 130{ 00 00.00
__ 1 b// % To 2 1/914j 44.8 a' 1 To 31 _
First Angle of Triangle
* /.734/2. Atwood " 8/ 20 0 ?_ 455 3
31 125 . leSa eo~ / /86/9. 36 *'1 I
O '" .3/ ;26 lo g "-00)2) o lo10,000)2 ) h=cyiooi
- .4467030 .- COlS aim 2'Csa
x=sain o (same as x.' H O o78643if x=s sin o H
Y=-S COma (54m as .j Hx'=(approx. AW) Y=-s COma Hx'=(approx. AX"')
2 Arc-sin V (Va) 2 Are-sin V (Va)=(/1,0)cr =+ 1-- a=(x'/1,000) car =+ 15
. 14 o.+aYcr= f x
Y. 3- 481i 773.1177 si~ o .2677 3" Yof si
y' 4.67 sin 0' 3'f sin o'. l.O3 { o 0Y 1
I +cosAo0
Va - sn + in a' r sin . Va -sin 0 + sin 0'orsno
1~~ + Co or si + cs A
~' 7..4 Y267 -Ae0 (approx.) Y2-Aa" (apprax.)
V4. 769 + F (Ac)3
V +f F (Ax,")3
H (Va/I'000)2
+ FZ. -57 34 K (Va/I,000)2+-Aa
COMPUTED BY DA-TE . CHECKED BY £T NOTE For aunder 8,000 meters omit terms under thr teravy black line not in
1FEB Ml
Figure 1923. Inverse position computation for short lines (DA Form 19923).
186
PROJECT LOCATION
64148 AFA -ASA INVERSE POSITION COMPUTATION (SODANO METHOD)
STATION (Westward) 1STATION (Eastward) 2
DATUM
AL(;FFL,/920) MO (NIP,/1930) WORLo /155
00.000
X2/06 00 00. ooo E
Cos 8057/8 67342 /
SIN 99836 3499S"
C+ .09. 29/22
(C + D)51NB
-l . 08964 (oC
Al = AX +x
TAN63597 02343
TAN-2
.00000 00000
/06
SIN AN
.96126 /6959
CosA&
- . 27563 7395588 86 43 17.95062 ARC[ A
6 is-Lq7 R?77 .,.,,~i B
TN.36o274 47453
TAN 0
99663 29966SIN r
*34100 26695SIN M2. 7059/ 33541
a .2407/ 83382
.64109.99270
d 15787 RAD~IA B 410 16 F o29 7629K1
+B c[ .00675 47028 23- . 09654 76o151 /
19 S 16.70600
44 564 12.16752
COS 1
.94006 2327/COS 02
7898/9 72
b66584 445'S
El - .19244 45306
E + F ~- .16551 67683GI 2.29467 4739
23.4 9 09 - --
(KI +;B) 9 RADIANSn 3970/75 /3
l/06 1 3.62305
TAN # 2 COS 01 - COSA l SIN 0SINN 1
COT #, COSAI - COS 02 TAN 01SIN Al
COS 60
055oS9 74 28/ 9SIN
2 No
.58 986 /2195
6338. 3124 9SIN 2 8o
1/00 2. 74685
0 0 A~
86 50 29.58/78 AC
(E + F G
- . 37,980 7/
COS A I
-. 27877 5/848ICOT a/07455 96393
COT a'
- , 47245 22892
SIN 80
99848 09829
90/15/567 0984 RAIN .163 6325/
J 72155 0/899
C052
2a
.575)25 63486AX. = DIFFERENCE IN LONGITUDE AND IS ALWAYS POSITIVE.
TAN =1-e2 TAN 4
a =5SN t1SIN 0b=Cos 01 COS 0
2
C9 8a+b cos AX
Cos B,=a+b COS0l
Cos 20
7G/08
842
89299Cos 4a
/5851 26272J = K B51N
4
X SEC. OF ARC
673. 62305SINA l
96035 63200a 0 i'
222 56 30.0356
IIS~ /7 18.5987H. 80
9'6 44596 .1811 SIN 8, Cos2
4816. 692JSIN 2 60 c.os 40
.0/4METERS
96 494 12.859Cos 4c =2COSZ2U ~I
AS AX E= -aKS SIN2
°~= 1 s-Cos 2a a9SI 9 IN9 SIN2 0 0
~B=A
2 A F=CKe H =b 9 + SIN
2 ,0 (K 7 -K 9 SIN
20)
c= O OB OG= - I = SIN2
g (K-K 10 SIN2
0. a IN
K3 SIN 9' 0 0CLOCKWISE FROM SOUTH
D=-aK 4 X= + D+®I)A SIN2OO2SIN Bu COS B a IN
S=H B,+ISIN
CONSTANTS FROM TABLE
SPHEROID
INT~ERNATIONA L
b, (Semi-Minor Axis)
6356911. 946 M.
V0.99663 2996.KI
237. 23882 /8K
2
0.34/88 82393KS
4.98652 0650
K0,40108 12630Ke
0.79945 93685Ke
3. 98652 0650
K9
K/3.64991 726K
1 0
/8.129988 265SIN 1
0.00000 48481 36811
0.01745 32925 2 RADIANS
90Cos2a -J51N2 9
0COS4u
QUADRANT III OR IV FOR COT a + OR -
RESPECTIVELY'.
QUADRANT I OR II FOR COT a' + OR -
120000 0o.~oo N
"45 00 00.000 N
1000 00
COT a =
ICOMPUTED BY DATE i CHECK ED BY TPATER.4. Z-mdt - AMS JAN. 64 IG. T.nmca - AMS I IAN. 4
DA FORM 2858, 1 OCT 64
Figure 124. Inverse position computation (Sodiano method) (DA Form ,2858).
i
cos 1 30 cos 02 sin AXsin 0
B=A 2
cos 0-B cos 0kC3
D=-1k 4 sin #1 sin 2
E=-k5 sin 01 sin 132
F=Ck
02
G 00 in radians
(6) X and Al are now computed.
x" (JI~+B)0+ (C+D) sin 0+ (E+F)G]A
0 in radians Al=AX+X
(7) The azimuth (a) and back azimuth (a')
of the line are computed. The azimuthis referenced to south.
tan 12 cos #1-cos Al sin #1
sin Al
cot a' sin 132 cos Al-cos 132 tan 1sin Al
When: cot a is positive, a is between180°-270°; cot a is negative, a is between
270 --360'; cot a' is positive, a' is between
0o°90o; cot a' is negative, a' is between
90o°180o.
(8) The true spherical distance (0°) between
the two points is computed, and con-
verted to radians. The formulae in (3)
above are used substituting Al for AX.
cos 0o=sin 1i sin 1 2 ±cos 1 cos 12 cos Al
(9) The constants terms used in the solution
of the geodetic distance (S) are computedor taken from table XXI, appendix III.
/2e'
k7= b b°=semi-minor axis (meters)
__be'4
k=
128
3boe4,4
693b4 =6k8
4e'
o=b16 -
(10) The variable terms used in the solution
of the geodetic distance (S) are computed.
sin2 a1 (cos 9i cos 12 sin Al)2
sin 20°=2 sin 0° cos 00
cos 20=2 sin 13 sin 12- 00sin2 13
cos 4r=2 cos2 20-1
I=k7 sin2 p°-klo sin4 g3
J= ks sin4 00
H=b0 +k 7 sin2 g3°-ke sin4 l3°
(11) The geodetic distance (S) is determined.
Smeters=Ho+I sin 00 cos 2a
-J sin 20° cos 4a0° is in radians.
c. For short lines or reduced accuracy on longlines, X becomes A0 " F (Flattening), and all terms
in sin4 /3° are omitted. Use cofunction of tan 1 or
cot a when their values are too large.
d. Computations using the above formulas
should yield results with a maximum distance
error of one-half meter and azimuth error less than
0"2. For shorter lines, the error is much less.
Section IV. ADJUSTMENT OF A TRIANGULATION NET
73. Methods
Two methods of adjustment, the directionmethod and the angle method, are explained inthis section. The azimuth, latitude, and longitudeequations normally used in the direction methodare explained in paragraph 75.
188
74. Direction Method
The most rigid method of triangulation adjust-
ment used for all first- and second-order triangu-
lation in which corrections to observed angles are
used to make lengths, latitudes, longitudes, and
azimuths agree with fixed data, is known as the
direction method. In this method each angle isconsidered to be made up of two directions, andcorrections are made to each direction individuallyto correct the angle as a whole. The correctionsto the directions are ordinarily designated by theletter v and a subscript number such as vl, v2.The value for each v is determined by a solutionof certain conditions which must be satisfied tomake all the elements of the net compatiblewith each other and with previously fixed data.The conditions are stated as equations and theseequations are then solved by the method ofleast-squares. The Doolittle method of solutionwill be used throughout this section.
a. The conditions that may have to be satisfiedare-
(1) Triangle closure. The three angles of atriangle must total 1800 plus the sphericalexcess (unless the computations aremade on a plane grid system wherethere is no spherical excess).
(2) Side agreement. The length of the com-mon side of adjacent triangles must bethe same, no matter which of twoadjacent triangles is used to computethe length.
(3) Length agreement. The length of a line Bas computed through a series of trianglesfrom the fixed length of line A, mustagree within certain limits with the fixedlength of line B as determined by a basemeasurement or an adjustment.
(4) Azimuth closure. The azimuth of a lineC-D as carried through a series of tri-angles from the fixed azimuth of a lineA-B must agree with the fixed azimuthof line C-D as determined by an astro-nomic observation or a previous adjust-ment.
(5) Latitude and longitude closure. The lati-tude and longitude of a point B as com-puted through a series of triangles froma fixed position A must agree with the
fixed latitude and longitude of point Bas determined by an astronomic observa-tion or a previous adjustment.
b. When stated in equation form, these condi-tions are known as angle, side, length, azimuth,latitude, and longitude equations. All of theseconditions are not necessarily present in everytriangulation problem. Figure 125 was chosenas a typical problem found in extending first- andsecond-order triangulation. This problem will
Red
Black
Figure 125. Sketch of triangulation net for direction method.
be treated in several different ways and theadvantages and disadvantages of each methodwill be pointed out.
c. In the arc under consideration here, thereare five angle equations, two side equations, andone length equation to be satisfied. The numberof equations should be carefully checked to insurethat all conditions are met and no identicalequations are included. The formulas for de-termining the number of angle and side equationsare as follows:
Number of angle equations = n' -s'+ 1
Number of side equations = n - 2s+ 3
in which n is the total number of lines, n' is thenumber of lines sighted over in both directions, s isthe total number of stations, and s' is the numberof occupied stations. Total numbers include fixedlines and stations. Any lines or triangles fixed byprevious adjustments may require additionalconditions. The number of angle and side equa-tions may also be determined by building up thefigure point by point. The rules are that at eachpoint the number of angle equations is one lessthan the number of full lines (observed at both
189
ends) between the point and previously considered
points, and the number of side equations is two less
than the total number of lines from the point to
previously considered points. Fixed points are
previously considered at one end of the arc only.
In our problem-
n= 11, n'= 10, s=6, s'=6
Number of angle equations= 10 - 6+1 = 5
Number of side equations = 11-12+3 = 2
Point-by-point determination starting from line
Hicks-Lincoln-
Station
Burdell-------- ----------
Red-
Nic__
Black_
No. angle No. side
1 0
2 1
0 02 1
5 2
This checks the number of equations found by
using the formulas. On long or complicated nets,this check should always be made. Since
there are two fixed lengths in this problem, a
length equation is included. No azimuth equation
is necessary in this case because the two fixed
azimuths radiate from a common station, thus
fixing an angle at that station. The fixed angle
at Hicks from Lincoln to Black must be recognized
and the list of directions altered to maintain the
fixed angle. No latitude and longitude equation is
necessary since the fixed positions are connected
by the fixed lines.
d. To start an adjustment, a sketch of the net
is drawn showing all lines observed, with lines
observed in one direction only being shown solid
halfway and dashed halfway (fig. 125). The line
from Red to Nic illustrates a line observed only
in the direction Red to Nic. Fixed lines and
positions are distinguished. Position, lengths,and azimuths may be fixed by previous adjust-
ments, astronomic observations, or measured
bases. The directions are numbered clockwise
around each station, starting with the first clock-
wise line in the figure. This system of numbering
facilitates designation of the angles. No numbers
are placed on the fixed lines. Theoretically, the
observations made over the fixed lines should
receive their share of any error to be distributed,but in practice, application of this theory can lead
to complications. In this problem, for instance,applying a correction to the observations on the
fixed lines from Hicks to Lincoln and from Hicks
to Black would lead to the condition that the two
corrections would have to add to zero to prevent
any disturbance of the fixed angle Lincoln-Hicks-
Black. Since the corrections to each direction are
relatively small, it would be advisable to eliminate
the corrections to the fixed directions altogether,and thereby eliminate the possibility of disturbing
any fixed data. The numbers on the lines are used
as subscripts to denote the correction (v) which is
to be applied to the observed directions. Generally,the v is dropped and the subscript alone is written
in parentheses such as v1 written as (1), v2 as (2),and so on. If these subscripts are written on the
list of directions (fig. 126), it greatly simplifies
applying the final adopted correction (v) after the
solution of the condition equations is made.
e. The triangles are written out on DA Form
1918 (fig. 127) and the observed angles (differences
in directions), with the appropriate v symbols in
the left hand column, are entered. The directions
used to obtain these angles are the directions re-
duced for eccentricity and sea level. In this
problem, there is no correction necessary for either
of these factors. Using the list of directions com-
plete with its v's, the correct angles and symbols
are obtained directly. As an example, take the
list of directions for station Burdell. The angle
Hicks-Burdell-Lincoln is-
Station Direction
Lincoln -------------- 0 00' 00."0 +vs
Hicks-------------- 3370 10' 32."9 +v4
After subtracting- -_ -_ 22°
49' 27."1 - v4+v5
The angle 22° 49' 27."1 is entered in the observed
angle column of the triangle computation sheet and
the symbol for the angle [- (4) + (5)] is written in
the left hand column. The symbols for the con-
cluded angles at Nic in triangles Nic-Black-Red
and Nic-Hicks-Red are obtained by changing the
signs on all the other v's in the triangle and apply-
ing them to the concluded angle. The concluded
angle itself is inclosed in parentheses.
f. The total of the three angles in a triangle
should be 180 ° plus the spherical excess (E). The
computation of the spherical excess is explained
in paragraph 57. The difference between the
sum of the observed angles and 180°0 + is the
error of closure of the triangle.
g. Now the condition equations can be chosen
for adjusting the net (fig. 128.). In this problem,three angle equations and one side equation are
necessary to fix the quadrilateral Lincoln-Burdell-
Red-Hicks, and two angle equations and one side
equation will fix the quadrilateral Hicks-Red-
190
PROJECT /-243 ORGANIZATION LIST OF DIRECTIONS7 "(TM 5.237)
LOCATION STATION
Co/iform BLACK (USCt6S)OBSERVER IINST. (TYPE) (NO.) DAB s
Cps. Pf s,v/t kW( '5?460 0247 .
OBSERVED STATION OB0 SERED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJSTED
REDUCTION REDUCCTION' WITH ZERO REITIAL DRCIN
1/CS (scG) 0 00 00.00 - - 0 00 00.00
,E5&w (9'E,,:r, go31 - -
LOCATION CfSTATION____________ B&rC#4.V 1 ns)
OBSERVER INST. (TYPE) (NO.) DAT
4t.1. E iS.T~fi K4dE *67460 ._____ ______
OBERE SAIO BSRVDLUETIN ECCENRIC SEA LEVEL CORRECTED DIRECTION ARJUSTED
OSRESATO ODEEDDECIE REDUCTION REDUCTION' WITH ZERO INITIAL DIRECTION'
LICL 11 UAt as 0 00 00.00 _____-- 0 00 00.00 0
ado (29!%&,js 254 4&.L2. - -_ ___
LOCATION STATION
Ca__________ f_______ iCKS (u!caOBSERVER INST. (TYPE) (NO.)DT
Cp , E.5R$ S~~/ ' 57 0 ~ ___Z __ _ _ . SA5
OBSERVED STATION OBDERVED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJCSTEDREDUCTION REDUCTION' WITHI ZERO INITIAL DIRECTION'
LINCN (uSC a~ 0 00 00.00 ____ 0 00 00.00 0.
RL. acs)_ __ - ad .54.6 j
LOCATION 1STATIONC4 /;irna j L/V'COIN (05c*")
OBSERVER INST. (TYPE) (NO.) DAT
CP/. p E. Se /t dtE 4S74o0 ?.2_ 40,___ SSi
OBSERVED STATION OBNERVED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ARJESTEDREDUCTION REDCCTION' WITH ZERO INITIAL DIRECTION'
#/A aLk.. 0 00 00.00 - - 0 00 00.00., -
Bore/el '- (~&Ws) 25L0S296.___ ___ ____
LOCATIONC 4 ~r~ STATION A 2 gs
OBSERVER INST. (TYPE) (O)DATEC4. 'a. S, ,ni6 (NO.) 16746O 2 Aug US
OBSERVED STATION ORNERCED DIRECTION ECCENTRIC SEA LETEL CORRECTED DIRECTION ADJUSTEDREDUCTION REDUCTION' WITH ZERO INITIAL DIRECTION'
O / if if if S / if ! i
8Lcc (s*s.0 00 00.00 - - 0 00 00.00 8Jric* 5 osRGS) -5 Q/ _____. 6
LOCATION 1 STA TION
Ca/', na ed Of~'~ _____
OBSERVER INST. (TYPE) (NO.) DATE
C7,/1 P E. ," J K.E 'o 7SO .2s A4.q MOBSERED STTION RNERTD DIRCTION ECCENTRIC SEA LEVEL CORRECTED DSRECTISNI.. ARSEOBEVE TTIN OBEVD IECIN REDCCTION REDECTION' WITH ZERO INITIAL DIEECUTON
/T if if 0 .n if I if '
Ib/cA-s (oC dG) 0 00 00.00 - - 0 00 00.00 _____
Sici ,~ G -)3 4 - 42 02.0
A ~ ~ )A4 17 /40BLAC (O G-SJ - -.32 9 M83z,
Figure 126. List of directions for net adjustment (DA Form 1917).
191
PRJC 1-243 DAECOMPUTATION OF TRIANGLES
LOCATION CaionaORGANIZATION 2 4n~
SPHERICAL SPHERICAL PLANS FUNCTION:STATION OsasE ANGLE CoRRSCTIoN. ANGLE EXCESS ANGLE LOGA IrIE
m. /J4
#51 lBUpdp/ 22 49 2Zi 427 .1.
+2 .Ofi5$ 0-0 56.0 t .%&9 87782.33-/ 3 180 143J L4 g7/~
__ _ 1-3 -_ _ _ _ _ _ .g690
1- 2pjdl.IC S _ _ _ _ _ _ __ _
______53.7 fJL .Q..Q4II :0~t
2-3 WIK -ICON___ 3. 68755O4
1 2e 304 9 /2,~~ 97e4o
3 L INCOLN 57 48 45.3 A.5 A&8 0.0 A 9. 92753420__1-3 Rd NCL2±Z35W4
1-2 Red-. H/CA' __3 9070305Z
__________ ~ 5/ 2 4 &2.QQI0.
2 -3 ic'e 4.7 ei___ 0624
11 #12 2 wicxs 42 22 11.0 40.iL 1.LJ3 9.2434546 Bun/el 42 2 31.7 4-Zh 33.& Q.L -1. S,2%.6Sf62
1-3Ro -8rdl/ _ SQ8L44
1-2 e/ HCS 3. 90 70386,
____3__ 5.7 ,F5 .S 0.2 0. /8/3 6 _ 1.
f1410 1 Re ~ 71. 4 L Q-~L~ 95531~5
-1+2 2 -0NOL - 02 6. ZL 5I o. g*- 9. 44
-3+5 h8 n/.I1___ 65 1L54±18~ 121i 9.l? 9,579829/
_1
-2 Red' -LINCLN _____ ____ 3. 9F7935c965
__________6.2 #4.0 00..2 102'10.I7Wi (: .247 /COMPUTED Y~-DT -DT"SDBYDT CH-ECKED BY DAc.'L26'E '
DA.1 FEBl 57191 8
Figure 127. Computation of triangles for net adjustment (DA Form 1918).
192
PROJEC DATECOMPUTATION OF TRIANGLES1-24 19 4;955(TM 5-237)
LOCATION ORGANIZATION
Cal/fornia 2______ 29- Eng-s.
SPHEICA SPERICL PANEFUNCTION:STATION OBSERVED ANGLE CORRECTION. SEICLPHRAL LNELOGAmsH
m:= .47/
___ 2-38LCj IK _ _ _ _ _ ____..ppgl
1Rd38 50 27.2 -1LL 6SA 0.& 25,6 9. 797345d 2 BLCk 5L 5Z 43.3, ±4 .4L4 o0.0 4. Z± V31
-12 3 HIc~es 89 II48.3 -L2 ALL1 QtL AZ0 5.9i7241-3 Red - HIC~ ______ 3973
1-2 Red-$LACK ______ 4./02
I 2 -3BLcMcs388' /
-16,47 1NAic 45 0/ 074 fi 05 ~ OS98933
-13 3 PicKs 41 32-. 0.0 07 &4M9
__1-3 NIC - WICs 3 9584,3 75
__._ -2 i 0.AC 3 6, 2 9.6/s /
2-3BAK-Rd4/083
1,Is1 ic 426 0. I78 9 ZLa415 2E BACK 38 41 26.6 -~L 1L5 Q0 2. 5 i X,795f474a-6,,7 3 Reel .36 52 /88 ±L 201Q02L 18ZZ
_____ _____00,/ QQLQJ cl. I .Qj. 6,2*~ -9.0/41
2-3 WCs-e _____ 008~
2-1 1 Mc .'9 -1.4- 0Z4 Q.0 07.A d19-4Ms9L-2L12 .44 52 01/1#L 6 QL 06 984853
- 8 3~t ___ 75 42 46.0 ±t ~jQL.A2 .965~___
1-3 N -Rd__3. 82056754
1--_ 2 Ai -WICKS _____ _3.95643762
COMPUTED BY D3 A $6 CH~ECKED BY.C4 0.13142 S
Figure 127-Continued.
193757-381 0 - 65 - 13
rt) 0= -4.0 - i +2 -c3) <s;-(! +Wo
a- =
-9 - *( )* L _ _
0 = -6.5 3-r >-8t) 9 -0o 0 2-( --8'- O)- 1L(/J
- 0 . - -t ___-7__ __
5id F eau t ons
(0= o.I-nera S 6) -L.ra)2.3 -z31i o5000t* 74
L n- pe -/it o i
Figure 128. Condition equations for net adjustment.
Nic-Black. A length equation is needed to holdthe fixed lengths Hicks-Lincoln and Hicks-Black
constant.
(1) Generally, the triangles with the largestangles are used for the angle equations,and the triangles with the small angles
are used for the side equations. Al-though four angle equations could bewritten from the four triangles, one ofthe four would be a combination of theother three and any solution whichsatisfies three of the equations auto-
matically satisfies the fourth. The sum
of the v's designating the angles in atriangle must equal the correction neededto make the sum of the observed anglesequal 180°0+. For example, in thetriangle Red-Lincoln-Burdell:
-(1)+(2) - (3) + (5) - (9) + (10) =-4.0or 0= -4.0-(1)+(2)-(3)+(5)-(9)+
(10)
which is equation number 1 in thisproblem. Number the equations in
ascending order of v's to keep thecorrelate equations from spreading toofar apart.
(2) Side equations are now set up on DAForm 1926 (Side/Length Equations) toinsure the condition of the side agreement
194
previously stated (fig. 129). Thesimplest way to set up this condition is toselect 1 station as a pole and write theproduct of the ratios of the lines runningto that pole and equate the resultingexpression to 1. By the law of sines, thesines of the angles opposite the sides canreplace the sides. Replacing naturalsines by logarithms reduces the ex-pression to one which can be solved byaddition and subtraction. In this
problem, the side equation for thequadrilateral Lincoln-Burdell-Red-Hicks
was written with the pole at Hicks.
Therefore, the expression for the ratioof lines is-
Hicks-Lincoln Hicks-Burdell
Hicks-Burdell Hicks-Red
Hicks-Red
X Hicks-Lincoln
Substituting the sines of the angles op-
posite these sides and using the symboldesignation of the angles as they appearon the triangle sheet, the expression be-comes:
sine (-4+5)sine (-8+-10)sine (-1) sine (-3+4)
sine (-2) -1
sine (--8+9)
Replacing sines by log sines:
log sine (-4+5)+log sine (-8+10)
+log sine (-2)-log sine (-1)
-log sine (-3+4)
-log sine (-8+9) =0
The items are entered on the side
equations form under the appropriate
heading. The column headed "Tab.
Diff." contains the difference oflogarithms per 1" change in angle. The
tabular difference is usually expressed
in units of the sixth place of the loga-rithm. Notice that the sign of the
tabular difference for angles over 90° is
minus. Subtract the sum of the right hand
log sine column from the sum of the left
hand log sine column and express thedifference in units of the sixth decimal.
PROJECT 1-243 SIDE/ EQUATIONS
LOCATION
C a/i for,,aORGAN IZATION
DATE____ 29 Fngf ___ _____ __ _Y_
SYMBOL ANGLE Loa. SINE TAB. Dir,. SYMIBOL ANGLE Lo(;. SINE TAB. Diy
____ Po AR E ICKS~ _ __
--4 *5 22.AE 7. 9.588725/-9 +~5.00 -1 /W L~~09 L L9778.3,57 --.
+ 1LQ 95OJ2 094n OAE 2L0i 2&A 9985/O +1-,9-2.31
-2 SZA&A859 g27 q95 -8S- 9 3o 4-59.7 1,0.3z
__ __±5L4A1593 -,
-+ ~ ~ ~ ~ ~ ~ ~ -- 6941+i~~L 50~+LL
_ _+i L 0- 5 s -z +A23(3 _ _ _-.+500 -0
_
PoleBLACK
-7+&352. J38aL... -12 .89 9IL3 i, sz2. +0.07B -,6 6-7f/4-I5L 4 64 a 4 ~ L 5 -' 66 28 . 9. 77t1 /d 2-8
-13 44 1A9 41A432 ±zJ5+ -/6 1-17 AiQ5 Z f40u1 6467 f
+ a2.~_ _ _ __ _ __
+ 15(5)6)OI~ 2.10(17 _
____ +2.295 (6.266 -0S5125 +0.26 +000~2/
COMPUTED BY DATE CHECKED BY DATE
D AI ,F1926
Figure 129. Computations of side equations (DA Form. 1926).
195
This difference will be the constant term
of the side equation. Observe that if thesum of the right hand column is largerthan the sum of the left hand column, thesign of the difference is minus. Thetabular differences are the coefficients ofthe v's shown in the symbol column. Toform the equation, the coefficients of each
v are obtained by adding algebraicallythe coefficients for each v from bothsides of the equation. The signs of the
tabular differences on the right hand sideof the equation are reversed for this
operation. For example, the first side
equation in this problem is formed asfollows:
0= +0.15+(+0.69) (-1)+(+1.32) (-2)
+(-2.31) (-3)+[(+5.0) (-4)+(-2.31)
(+4)]+(+5.00) (+5)+[-(0.19) (-8)+(- 3.55) (- 8)]+(- 3.55) (+9) + (-0.19)(+10)
Remember that the numbers in the sym-bol column are only the subscripts of thecorresponding v's. A simpler method ofwriting the angles in their correct col-umns is to write the angles opposite theside started from in the left hand column,and the angle opposite the side going toin the right hand column. In this equa-tion, the rule would work as follows withthe pole at Hicks:
Starting from side:
Hicks-Lincoln ______
Going to side:
Hicks-Burdell .....
Starting from side:
Hicks-Burdell -......Going to side:
Hicks-Red__________
Starting from side:
Hicks-Red _________
Going to side:
Hicks-Lincoln ......
Angleopposite
-4+5
- 1 Right
-8+10 Left
-3+4 Right
- 2 Left
-8+9
Use the smallest angles possible in theside equations, since a change in a smallangle has more effect on its sine than thesame change in a large angle would haveon its sine. It may be necessary todivide a side equation by a constant, ifthe coefficients or constant term of theequation are large in comparison withthose of the other equations. -The sec-ond side equation in this problem was
divided by 10. When dividing an equa-tion by a constant, care must be takennot to forget to divide the constantterm of the equation as well as thecoefficients.
(3) Length equations are written in the samemanner as side equations with the addi-tion of the logarithms of the fixed lengths
(fig. 130). The log of the length fromwhich calculations are started is writtenon the right, and the log of the length onwhich calculations end is written on theleft. In precise work, the logarithms
should be corrected for the difference in
are and sine (known as the arc-sine
correction). A table for this correction
may be found in TM 5-236.h. The explanation for the solution of condition
equations by the Doolittle method may be foundin paragraph 65, and in USC&GS Sp. Pubs. 28 and
138. By use of the accumulative features of themodern calculating machine, the forward solution
of the normal equations can be fitted onto a pre-arranged sheet which is illustrated in the problem(fig. 131). The written-backward solution or solu-tion of C's is also eliminated by use of the calcu-lating machine. When the calculated C's aresubstituted in the normal equations, the equations
should equal minus rl. This check will prove the
numerical value of the C's. The diagonal termsmust be included when checking the C's by this
method. After the v's are computed from thecorrelate equations, they are applied to the angles
and directions.
i. The corrected spherical angles on the trianglecomputation sheets are reduced to plane angles bysubtracting % of the spherical excess from each
angle. If the spherical excess is not evenly divis-
ible by 3, apply the odd amount to the angle
nearest 900. The triangle sides are now computedby the sine law which, on DA Form 1918, meansthat log of the length 2-3 minus log sin angle 1plus log sin angle 2 equals log length 1-3, and
similarly log length 2-3 minus log sin angle 1 pluslog sin angle 3 equals log length 1-2. Check thelength of sides appearing in two or more trianglesfor agreement.
j. After computing the adjusted triangles and
entering the final adjusted seconds on the list ofdirections, the geographic positions are computed(fig. 132). DA Forms 1922 and 1923 may beused for the computation of the geographic posi-tions. These are triangulation position computa-
196
Figure 130. Computation of length equations (DA Form 1926).
tion forms for logarithmic and machine solution,respectively. The forms can be modified for short
lines by omitting certain correction factors, making
them suitable for both first- and third-order tri-
angulation. Paragraph 67 gives an example of
both first- and third-order position computation
by logarithms. USC&GS Sp. Pub. 200 is used for
logarithmic computation on the International
Ellipsoid. For machine computation of geo-
graphic positions, USC&GS Sp. Pub. 241 (Clarke
Spheroid) and USC&GS G-58 (International
Ellipsoid) are necessary.
k. Although azimuth, latitude, and longitude
equations were not used in this example, they are
necessities in some problems and their use should
be mentioned. An azimuth equation is required
whenever two or more fixed azimuths occur in an
arc or net of triangulation, unless the fixed azi-
muths all radiate from a common point as in the
example given in this manual. When two or more
stations of fixed position occur in an arc of tri-
angulation and are not connected by a line, it is
necessary to relate the fixed positions by latitude
and longitude equations.
75. Azimuth, Latitude, and Longitude Equa-tions as Used in the Direction Method
Since neither azimuth nor latitude and longitude
equations were required in the example as origi-
nally solved, the problem has been modified
slightly, only for the sake of illustrating an azimuth
and a latitude and longitude equation. The modi-
fied problem is shown in figure 133 and consists of
fixing the positions of stations Black, Nic, Lincoln,
and Burdell, and the azimuths of lines Black-Nic
and Burdell-Lincoln. The fixed data is taken
from the adjustment of the original figure by the
direction method. The observed angles are from
the original list of directions.
a. The purpose of an azimuth equation is to find
a means of correcting observed directions or angles,so that the angles can be used to carry azimuths
through a network of triangulation between fixed
azimuths without a residual error.
197
PROJECT I-4 * LENGTH EQUATION'S4 (TM 5-237)
LOCATION
Cai0oria___ORGANIZATION DATE
2___ 2 -Egrs. __ _ _ _ __ _ _ __ __ _ IAug 55SYMBOL ANGLE Lao. SINE TAB. Dirr. SYMBOL ANGLE LoO. SINq TAB. Dist.
-5
_ -7 ~ I~6~--8 * 9 3do-u.- z 5910a031BM20 +15 -2 5Z1A&A 2zS225 44132.414 _4 74- 9%S8QZ12 #AL-65 -7+4 '~ .i ri32di8~A
_____ ________ 3.4i24582 _
Q0zOA /.32(.2)2. ( - ( (0.~(
O!2" z /3()*. uP16 0 16 (
-J _ - - _ _z 6 -2ffi M? 2..2 ... iL - - - 1..LL -. 0.132. -1Ja2 . - -3. 2
- i --- - ± 31 -,09 -2.2
.E~~-,8 -'+0 za5. ___ O. -~f1.76-03. 2 a
-O 1 -___ ______ -5 -Q3-1___ L L.*LZ21 5L
-1 - - -_ _ __ _ 1Q.2L ____ -JLL L22 27
16 -i -021 -. -1 -/ __t~ +
2 a .- / +o Z2 15& -.Z8± -L,2L282 77L 1
-i -021 - - aa. -Z2.. z i.* AQfLL
- - -__ - - tO -0,22. -OA.OQ +
- - 6 - -- -_ i 5 +~% j222 6,6, -4 A L -0,8162
___E m w wad Solkio Equti
/ 2 3 .... ~6 7 & L..- -2 +a + ~ - .z.2 40.. 1-
S 2 2 - - 42 - 0,223 -4.0ilI
_. ±1 -42 - -2 _37 -,258 ±O,89 -8 229_I3 113 - -AJ6h733 +0.J646 -iO2. -7.1?0
_+ - - tQ~?2_ _ - -1,55i ' - 258 + 0,6I6 - -LL62
_ +2 ~ - A - -1.6O -0,6160 +0.7&0+3za 6,19
_ _ _4 -LO ±64 -72 -l
+___ - -Z06Z250.243.8-LQ62.______ -0... - + +,? 681 -L 541
_ _ - +4 ±
_ _+ - +0A4 +4
C______ - +o ~~~ - -Qd.. 2
_ +t40- Q+QQ,21 2L
_ ~ 2Z5 ~2. +. -1804
I _ _tO-23o43 -LO3
_ __-~ ~4& 2
Figure 131. Solution of condition equations.
198
PJET/_243 POSITION CO MATIO, 2a- 9M A TRI NULATIS (For sahu~uI m. uCptAU.)
LOCATION ORGANIZATION DAT /9I Au M5
a 2 14 To 8 Live.L aI 8 J__'I~L To 2 kcs I~ Q IZS
2'"L & +49om~~L. S& _______________34.
a* To 1 "L _Q Z3 a 8To 1Rr~l 22L.5M°a __________ _a __________ * 53.45-
lf 00 00.00 13i0 00 00.00
±' 1Br/I To 2140Ck5 I q__ a s. To
3LNCL 0 3 ,6416.00"ds VintAngle of Triangle 22 49 . = AJQ*
p'/10,000)2 4 .15ss 01 /8 /110,000)2 .HM
4ma Q,.15,592045 -1183 4291 0" a 02397 940 3e 2o.046 7
x- ina 7,93, 6223 H 0. 04/Q 07 759 . -2L49 H 0.04077T9y. - cosa ~~g6 o Hx'-(approx.eA.) 43.97 Y--smoa -2 t.u Hx'-(approx. A),') -"377 Z~
$-z/O0) An-ala V (Va) z.aoV()
pYceor.=+fa 11.4 i4~ Mr yo.=+(a 69 -O78
Ye 4 221 A&?. 97 do 0. 6Z7441/A Y' 4,25797.1J21 do 0.61795179
Y' f I 84.06J sin' 0.61 767017 le da"' O~77"a 4232.63 1+ oeA*M /. 99999gg1 7
1 4223 7M 2+4.02499999
Va - 8. ° ni + si °' ~ j Va - 5. 20+ °o '0uao1 + cos 0. _____+ ________
Y7 4 2 . 5 2 1 -A a " a o . $- g (approx.) - 2 8 8 52 Va q2 . , I +4 9 F-(Xh V+ (app78~ - 3 .
K (Va/1,000)8+ -er -298.85 K (Va/1,000)2
+ Z.= -2 45
mOPTD s s. °A~t CNEKE ult r W . C. ij. oTls s n :bl Fvv umkr0,rSomIM Malt .m m do b. im~ thmn sNmmlhr
DA FE;1923
Figure 132. Position computations (net adjustment) (DA Form 19P23).
b. The procedure in setting 'up the equation is to
start with a fixed azimuth and, by successive use of
observed angles, compute the azimuth of the nextfixed azimuth. The difference between the com-
puted and fixed azimuth thus obtained is the con-stant term of the azimuth equation. The sum ofthe corrections to all the observed angles used inobtaining the computed value of the fixed azimuthmust numerically equal the constant term. Whenthe adjustment is made by the direction method,the corrections are the v's applied to the observeddirections.' When computations are in the geo-graphic coordinate system, the forward and back
azimuths of lines do not differ by exactly 1800,but by the amount known as geodetic convergence.It is to obtain the convergence that preliminaryposition computations are made for the lines
through which the azimuth is carried. For thisexample, the convergence can be taken from the,position computation made in the adjustment ofof the original figure.
c. The numerical example is as follows:Fixed azimuth Nic-Black.. 123°21'40"~.99Observed angle at Nic___ (+51) +45 01 07 .4
Azimuth Nic-Hicks -- 168 22 48 .39Convergence at Hicks --- - 00 46 .36
180 00 00 .00
Azimuth Hicks-Nic -- 348 22 02 .03Observed angle at Hicks_ (- 56+ 57) -44 52 05 .1
Azimuth Hicks-Red__ 303 29 56 .93Convergence at Red.- --- + 2 50 .50
180 00 00 .00
199
POET1-243 POSITION COMPUTATION, 21 ORDER TRIANGULATION (For calculatiug mchne cwonud)(7M 5-37)
LOCATION Calit'ornia ORGANIZATION "2 J L f DATECaiona1
-1 To 3 ,L/NcrOM' 2/2 00 L3 a 3 1ICL To 2 Wcs 3
2dL. d +& 29 ij0L 3d1j &5 i2 8.
a2 HCs To 1e 203 7~58.7 a 3 ,LINyCoL To 1 32e j84 0~
Aa#'02 50.50 da __ _ _ __ _ _ 0/ 4508180 00 00.00 180 00 00.00
a' 1'Red To 2H/K /Ll' 23 32 47.24 -a' 1To
3 LINCOeN 1S 4 It 3.J.f -8.2070 First Angle of Triangle TO 42 >!f a 8:2070
.387 4.6.2. 29ICKS b/24 3 /ON AI = 6073. 7- AA - 04 -36.256 s= 9 -
05m' O 21.652 1 "' 22 38 53.110 *' 38 05 21.652 1 Red- /22 38 53.110A lga= 2 3.90703857 (Ig=
sin~ ;er=-hA a -0x cor.=2b 3O833GL 2 0-8/7 -043493404 2 3.24cos p5512X82,~ i . 6 7 3 2 .0427 32n
xs sifl a -673203 H p. 041035236 x=s sna - 414Z 45 2 H 0./353Y= -sco.sa -45781 Hx'= (approx. Ah) 27. 255T7 Y =-cosna -8586,46536 Hx'= (approx. Ax)-70 ~4-4 5 , 1a=( ="I1O ,O0O)2 A r c sin + V (V ) a= (x "/1 000)2 A rc-sin -V (Va)
___________5 0,1720 cor = 5 0.43yers~a 3.72 ~ - -276.4256 ILyor.=+}fa 1.41-
Y. 4 221667-97T si 0. 61744118- Yo 4 225 7g7. /21 in 0.6/795/79Y' -445in78# 0. 6/66 q?5j le -858g. 655 si' of 0. 6168M44535Y
10 1+co "9 4271--41 +csA
sin 027).9 1+A sin .999g975 sin 027,o4 1+sin~97 2. 18 1+cos Ad ~0,6764 s - 1/.055 1 o m .140
Y2 27094 -Aue (approx.) - Y'2 4 27294 -Aa' (approx.) 17,46 05-O82V 6.13444 + F(A t,)
3 ~- V 6,13644 +F(A,)
K (VaI1,000)2+ - A5 -170. 49,6 K (Va/1,000)
2+ 4- -Aa" -1 O5 0 2
COMPUTED BY DAT 4, S r CHCE wYoi . . DATE 7s,9.1 h aOTEbot tyFor unde 50(5) atefs omst trms under the heavy blaek lise not is
DA F s_1 923
Figure 133-Continued.
Azimuth Red-Hicks___Observed angle at Red_
Azimuth Red-Lincoln.-Convergence at Lincoln
Azimuth Lincoln-Red -
Observed angle at Lincoln_
Computed azimuth Lincoln-Burdell -- - - - - -
Fixed azimuth Lincoln-Bur-
dell - - - - - - - -
123 32 47 .43(-61+62) +30 41 59 .7
154 14 47 .13- 01 45 .08
180 00 00 .00
334.13 02 .05(+66) -50 2046 .1
283 52 15 .95
283 52 22 .55
Constant term (computedminus fixed) -
Azimuth equation :
- 6".60
0 = - 6.60± (51) +f (56) - (57) - (61) + (62) - (66)
(1) The azimuth equation is formed by settingminus the constant term equal to the sumof the corrections to the angles. Trans-ferring all terms to one side of the equa-
tion and setting the equation equal to
zero gives the form shown.
(2) A source of possible error occurs whenever
an observed angle is subtracted to obtain
an azimuth. The error frequently made is
neglecting to change the signs of the cor-
rections for that angle when the equation
is written. In the example, the observed
angle at Hicks is designated (-56 +57),but the angle is subtracted to obtain the
azimuth from Hicks to Red so the desig-
nation is written in the equation as
200
POET123-POSITION COMPUTATION, 2 ORDER TRIANGULATION (For calAaig madse comups afro)
a 2
&ACK To 3CK 212 40 / Z~ a 3 AI~' To 2 BLACK _J2-4L 45
2d L & +3d~ & ZL 4.a
2 RLACK To 1 Ni ~ l 2.L a 3n~- To
1 Nic 3 22 05. /4°a +#02 /4.1 '/ Ac + 0 46. 36-
180 00 00.00 180 00 00.00
a' I 1 To 2BLACK 12 21 4Q±. 9 ' To
3 //CKS 168 22 ,51.30 'e.is nl fTinl 50 /. +827
-t-- 8. 207 Fis Ani of Triangle
2.3 04077RLACK n' /,22 35186"3 7~629 IK 12243 22. 366I S.Z s= 6350.335 I l- - 03 37.626 I= I 07,356 I - 01 /5/46
122'/f 2 / .220 3' 02 57569 A/ic 'a' /22 42 14.220(Iogas= 2 3 80279661 lga 8. 68 4
C/521 b=(Y/10,000) 0.122 .) 04 48. 6 b'=(Y/10,000)2 0. Ni
siny a x cor.=-2f si bcr. -f08 3 2 0.SOI am -0,20/62338 icr=-f 3.250
cos a #0. 54937453 Ye B5061 _8 COs a cf 01794633 -' ,a. 'xnsasin a -536.1R91 H 0. 04101358S x=ssin a -1832. 22.34 H 04OA8
03a 3488. 723. Hx'=(approx. AX"~) -2/7.6258 .y=-a Cosa -,8470. 732. Hx'=(approx. °X" - 75.14602 Are-sin+V (VA) 2 Ar-sin+V (Va)
a =(x'/ 10,000) cor = +mr 15 0. 71 _ a= (3e10,000) .0cor 15 0.08Y cor.=+fa -2/7.A26 y cor.=+fa A.2 -2766 7,4
&~426 257.470 sinl 0.6/6-77 74 y,' 4 2216 ~7. 975 si 0. 617441M.3,-. 488.7Z13 in' 0. 663.,i%.L Ysi - 8? . 730 0,191433963
Y1 4 212 768. 757 1 + ens A¢ Y 9995~ 1 422 767. 24,5 1 + ens A .9gg?0sin + sin ',, Va - sin + sn m
Va - /1.72 ~+cos AO m0.6 / 0. 206 1 +cos A* .6,3971YZ 4,212 767. 03d -Aa" (approx.) -/3.7 Y 2 767 's -Ai' (approx.) -4.3
V6.12763 +F()3 l 6,12763 +F(Wih)3
-
K (V a/1,000)2
+ -A-/3, 179 K (Va/l,000)2+ - ha 46.35
COrMPUTEDBY iADATE' A1f CHECKED BY C. A. DATE.., SSJ hev N oTE: tyor under S1etS eters omit terms under the heavy blaeh lise not in
DA, FORM71923
Figure 132-Continued.
- (-56+--57) or (+}-56 -57). The same
situation arises with the angle at Lincoln.
d. The discussion of latitude and longitude equa-
tions in this manual will be limited to a description
and a short example problem. For a complete
development of the formulas, see USC&GS Sp.
Pub. 28.
e. The example is shown in figure 133. The fixed
data is taken from the adjustment of the original
figure by the direction method, as it was for the
azimuth equation. The fixed data in this case is
the positions of stations Black, Nic, Lincoln, and
Burdell, and thereby the lengths and azimuths of
the lines Black-Nic and Lincoln-Burdell
Jf. The first step in forming the latitude and longi-tude equations is the setup and computation of
preliminary triangles. A single chain of triangles
is used for preliminary triangle and position com-
putations, and it should be the R1, or strongest
chain (fig. 134). One angle in each triangle is
concluded and the other two angles are observed
angles. The observed angles are usually the
distance angles, that is, the angles used to compute
the unknown sides in the triangle. The concluded
angle is known as the azimuth angle. The excep-
tion to the rule that the azimuth angle is always
concluded occurs when one of the distance angles
is not observed. In this case, the observed azi-
mnuth angle must be used and the unobserved
distance angle concluded. The important point to
remember when using concluded angles is that the
designation of the correction to the angle is the
sum of the corrections (with their signs changed)
of the other two angles in the triangle.
201
PROJEC POSITION COMPUTATION, 2 ORDER TRIANGULATION (For calculatig ma cbins comptation)
LOCATION CaionaORGANIZATION, DATE
a 2 To 3 ~ a 3 Red To 2 YIsKe 1243 .c-
2d L & +i 34Z & 5 0.a 2 /4~S To 1 i 382~~4 ~ e o1NcA L~J
180______________ _ 00 00.00 180 (1 000
a' 1
To 2 2 S A& .2 51 M a' 1 To 3 7 Z 2f' 8.2070 First Angle of Triangle 459 25 074 f - 8 .207/
d1&- 07 46,2592 I4CK "/2 3323 *L 3a
Ad b ~~~( /1 2 3. 958437.2 d(os= 2.i "Sb= 46.69 b=(y/b=,00/) ~ A0' 0202408,b=( /102000) 0.197
-0. 2016339 32x cor. =-2fb sin5 a +0. Z4120493 x cor.=-Zfb 0.80 -acos a
x' ~ ~~-1832. 22 2 cosa #06 17f 3et4X'.9
9 6 ~ i aH: ssnaH0
4 0A AX~sn 32~A H a140.85 ~ i 4003 H9 p. p414/pA
7SOa-8072/ H=(prx') 71460 Y=-s cosa -44.94 Hx'=(approx. l A')0 1 j 0
-721a=('IlO ,OO)2 -0.0336 mr 05) 08 0 24 a=( :'/1, 00)
2 ' Areosin.±V (V )
0. a15. 0y____r.=+fa________ kcor.=+fa 1.97
* 2 1 67. 9757 4401 6141B I20.l . 0~p
-o 82 " 7 3 2R i n" d e 4 2 Z 2 9 4 2s nd0 1 4 U '- ' 0 ,4 9 4 4 4 0 .8 9 6" '0 . 1 6 3 2
Yl 421267 1 I+cos A4 1. 99999902 Y'1 421 +6/ lcos Ad
Va si i 'Va - sin d + sin1 n+ cs 111 0. 1.473 + } osA
Y2~ 4 212 767. OZ ,5 -~Ac (approx.) -435 Y2. 4 242 767 -4 -la'° (approx.) , 124.007V 4,/2763 + F(AA')
3 -V 6. 1276? + F (a)
3
K (Va/1,000)2
+ - -46.35 K (Va/1,000)2+ - 124",0' 7
COMPUTED BY ! ',.,.., DAT CHSECIKED BY DATE IOT: or ae8,15meters smit terms unlder lbs heauy black line not inR Vw7 .~w .268 esybdtype or underlard
DA' F .,1923
Figure 132-Continued.
g. In addition to the correction symbols on the
angles, each angle is also designated by A, B, or C.
The azimuth angle is always called C, while the
length angle adjacent to the known side in the tri-
angle is called A, and the length angle opposite the
known side is called B. The data from the pre-
liminary triangles is used to compute the prelimi-
nary geographic positions. The combined data
from the triangles and position computations isthen used on DA Form 1927 (Latitude and Longi-
tude Adjustment) to form. the latitude and longi-
tude equations.h. The triangles used in the preliminary compu-
tations are shown in figure 134. The A, B, and C
angles are labeled on the sketch for use in the lati-
tude and longitude equations. The preliminary
triangle computations are shown in figure 135.
202
Notice that the C angle could not be concluded in
triangle IRed-Nic-Hlicks, because the distance
angle A was not observed. This is the case whenthe azimuth angle C must be used as observed.
The triangle computation can be made using either
natural or logarithmic functions. In this example,
the computations were made using natural func-
tions, as the position computations were also made
using natural functions. Some argument could be
raised that a complete logarithmic computation
could be more efficient, since the tabular differences
for 1" for the log sines of the A and B angles are
required for the latitude and longitude equations,
but this is actually a minor point and the choice of
logs or natural functions should be left to the
individual computer.
Lincoln BurdeliBurdell
Reds8 Red
/ /\? /
Black //
Nic
Figure 133. Sketch of triangulation net (modified problem).
i. Using the angles and distances from the pre-liminary triangle computations, the preliminaryposition computations are made (fig. 136). Notice
that both sides of the position computation form
check the position of station 1, even though no
formal adjustment was made of the triangles.The check is possible because of the arbitrary
adjustment made by concluding one angle in each
triangle.
j. On completion of the preliminary triangle and
position computations, all the necessary data is
available for forming the latitude and longitude
equations on DA Form 1927 (fig. 137). On the
form, the following entries are made from the
available data:
(1) In the right hand, upper corner of the
form in the spaces 0,, and X,, the com-
puted latitude and longitude (in degrees,minutes, and decimals of minutes) of thefixed end station.
(2) In the left hand column headed "Station,"
the names of the stations at which the C
angles are recorded. It is convenient for
later use, if a notation is made beside each
station as to whether the C angle at thestation is used in a clockwise or counter-
Blaocl
Figure 134. Sketch of net (R1 arc).
clockwise direction. A plus (+--) sign in-dicates a clockwise and a minus (-) sign
a counterclockwise C angle.(3) The computed positions of the stations in
the left-hand column in the columnsheaded 0, (latitude) and X, (longitude)in degrees, minutes, and decimals of
minutes.
(4) In the columns headed A, B, and C, the
correction symbols for the A, B, and C
angles. The symbols for A, B, and C canbe taken directly from the triangle com-
putation sheet when C is plus, but when
C is minus the symbols for C on the
triangle sheet must be entered with op-
posite sign on DA Form 1927 by the
following rules:
(a) When C is plus, the combination of the
symbols for A, B, and C should be zero.
(b) When C is minus, the combination of
the symbols for A and B should equal
the symbols for C.(5) The tabular differences for 1" for the log
sines of angles A and B in columns
headed +-A and --5B. The tabular
difference for the log sine of angles over
203
Lincoln
PROJECT -DATE
1-243 14 Auy 55 COMPUTATION OF TRIANGLESPre/i IAEP (TM 5-237)
LOCATION C~ionoORGANIZATION29E
SPHERICAL SPHERICAL PLANE FUNCTION:STATION OBSERVED ANGLE CORRECTION. ALE XCS ANE eATRL
___2 Nc-BLACK ___ ____ ____
- 1 HICAs -4 494. 1.9 QL~41.9 06867ms'2 N/Ic (45 ol01 . - 3 0~.o 08.3 .77309
- 3 L~i 90 39 09.9 - . 0i .1 oi 0 11g5i
1-3 loc&s -" BIK ______6428.231
1-2Au wex Nc 07 0
B - lj 1 75 42 46.0 - 6.o 0.1 5. .99?6*~, 24'ic(2E0~o) 0140.0 O9.o0 0.8609/2(6
-s43Aj 44 52 . 05.1 0 05, 0.70547681-3Rd-ICS__-8l0AL
1-2 Red 15.494____ _____ ~~0.1 _ _ _ _ _ _ _ _ _
2- ReacK- 8073 064
8-"46711 LIlcoLN 57 48 453 -00.1 0 A5-3.'Q O83a7+u- 2 Red (342o 4 - 08.6 4& 08.6 0.510 7
A -o-O3 HCA'S 9 29 06.2 - 062 4L Q6.L 0. 99966413___1-3LIoL-CS 4 Ad__1-2 LIM OL N -Red ______ __ ____ __ S52
_____________ ~~ ~ ~ ~ 953 _______ ___0/___ _____
Figure 155. Preliminary computation of triangles (DA Form 1918).
90° is minus. The tabular difference forthe B angle is entered on DA Form 1927with opposite algebraic sign than asgiven in the tables; therefore, if the tabu-lar difference in the log tables is plus forthe log sine of a B angle less than 900,the tabular difference entered on DAForm 1927 in the -S column will beminus.
(6) The computed value of the position of thefixed end station and the fixed value ofthe position of the fixed station. Thecomputed value is designated c,, and X,,and the fixed value is 0', )4. Bothvalues are entered in degrees, minutes, andseconds.
(7) The quantities a1 and a2 from the table atthe right hand side of the form, using the
204
PROJECT OlINCUU ui. E HAOLrIIFrclmaigmci.cutl)/ '243 (ForeN caicJltIDP Andwl ASTfnT(,EiatN5-37
LOCATION a/ 0 ,. ORGANIZATION WbL/E Y,,. JDATEA _
a 2 Afc To 32S AL 40.M 2 a 8 BLACk' To 2Ni, I.16L
* li 2M"492 To 1i/c' L 432. 3 ALACK To I 2L2_ AO I6AL
__a _____4___._____ Aa + 0/ 27.92
180 00 00.00 180 00 00.00
a,1 To 2
M 22 O Z93 d 1
1AxS TO 3 BLACK 132 41 44.83
15072 First Angle of Triangle 44 If 4J. 8 f e a20/* . ,
_3_2 2. *7 MC 2- _La_02 '10 a" A) 7. a- 6428.237 '°' - 22.476
' 0 4 .5 7 1 / ' 1 2 2 4 3 2 .3 7 0 V' 3 8 0 7 4 6 .2 5 7 1 I 2 2 4 3 2 9 .3 7 0 4 8 ,-2
/r 0 0)07 , 2 S 1 0 0)60
2 3
*i ° xer 0.04z -2b 3.250 sin a -0..5398/967 x cor.=_-2fb 1. 20CSa -O 0-gsAW& Z" + 1830. 3047 Cosa zo-41780~ 9 ' - 34 7o. 8A4
zssin a /8,30. 3f73 H 0,04105841t z=s sin a - 347o. pg H p 4089
Y=-s Oosa f a901. 0692 Hx'=(approx. A),") 4475,15 Y=g _=s cosa b, Hx'.65 H= (approx-4.47Al3
a=2/0002Are-sin,,+V (Va) a=(s"/1o 0oo)2
Arcsin=+V (Va0.0335 cor 15 p 0,124 cor 15 0.0
Y cor.=+ta 0.27 °>" 7S.150 Yc or.=+fa 0 ~ c -142.47Y. 4227707- sin b 0. 6/6 33963 Yo 4 / .270 sin - ./67
Y 4 8 90.069 'Sin b' 0. /743e ,I 5411 -16 sin b' 0674Y1 4,221 648. 106 1 + cos A* 1-929902gp Yi 422/ 68 .63 1 + cns °b I- 9irj9qqg4
Va- 026 sin b + sin b' Va - sin b + sin b+cosA* 0~".6/689070 0.740 +enA 0/K0S
Y2 A22LZ 6b . -d (approx.) Y2. 85 2/~7 c -°ds (approx.) . .1 431Vr+F()) ,43 (17
K (Va/1,000)2+ -~ - Ad,' '446.359 K (Va/1,000)
2+ -- A -87923
COMPITI BY OAL~,~ .,.. HECIED Y NTE: ee nede Rn mtro emit teem, under the bevyp binek line net in
/G
[
eybl yeo
I FEB A 923
Figure 136. Preliminary position computations (DA Form 1923).
computed value of on, from the top of thepage as the argument.
k. Now that all the data has been entered fromthe preliminary triangle and position computa-tions, the computation of the coefficients for thelatitude and longitude equations can be completedon the form. The columns headed " O-O'," and"An-X," need no explanation except the reminderto watch the algebraic signs of the quantities.The quantities in the columns headed "LatitudeEquation" and "Longitude Equation" are simplythe products of the quantities in the columnsalready filled out. Watch the algebraic signs.
1. In forming the latitude equation, the quanti-ties in the column headed "(On- O) A" are thecoefficients of the symbols in column A, thequantities in the "(On-0,) (-SB)" column arethe coefficients of the symbols in column B, the
quantities in "(X,,- k)al" column are the coeffi-cients of the symbols in column C. The constant
troftelttdeqaini7282 n-nm. The longitude equation is formed using thequantities in the column headed (X-X)A"as
the coefficients of the symbols in column A, the
quantities in the column headed " (An- Ac) (- 8B) "
as the coefficients of the symbols in column B,and the quantities in the column headed
"(OOca2"as the coefficient of the symbols incolumn C. The constant term of the longitudeequation is 7238.24 (An- An') . To help clarifythe formation of the equations and the coefficients,the latitude equation in the example will be writtenout in full before any of the coefficients of the
symbols are collected into the final form.
n. The latitude equation in full is-
205
PROJECTM~~I!,*CE hI~9!AW (For calculating machine computation) -
a2 To 3 - r ~ T i
a2To 1 4 ze2 a 3cA. To 1.a
Aa ___________ ¢02 04.01 Ac ____________
180 00 00.00 180 00 00.00
a' I To 2 Nic47 0Z,30~ a' 1Rd To 3 WICKS ~ 1~.43.f =8..77 First Angle of Triangle 75 42 46.0 fa 8.2070
* m l y~ 2. Al' 122 4214.2.2 * 3s 074Z7 ICKS a 122- 4 29370Ia661/S 494 I A -0 21.106 "=8073-064 41 -04 3%.2S6
m' 05 2165 1 a' 5:4 *'I '34
A lg=2AI0s=02 24,082 (~ =yi,0) og0 44f2'b=(y/10,000)2 97 02 24 0.0(Iogo
sina -xor= bsna x cor=-0. 740 7qqp03~ cr=- 0,808 dna 0-8338 93 or..8___
C 067727 Zr _ e -90 cos a z05r22 X 5G732Q51
z -ina H 004/0 359315 X=asin a -6G732. o605 H 0. 0409,535Y=43 8042sa Hx' =(approx. AA") -20 Y= 7-s coa 744578 Hx' =(appro. AA
5
2 Arc-sin V(Va) 2Arc-n V (Va)a=(3 e/1,000) 0. 2402 -cor =+ 0.6 a=(x/10OOO) 0. 45 2 cor +- 1/14y cor.- +fa I.? Ae 6,o ycor.=+fa 3.72 AK -274 .254
Y. 42/2 767. 037 sina* 0./633 963 Yo 4221 667. 913 sin m 0.61744117, 4 443 -S05 a i m' (p 89 Y, - 445575 si'~' p66p
Y14/720 4 1 + COS Am I 9 11 99976 Y1 4 2/7 2/2. , 16 1 + con Ab* 9? ? 9 97.Vs - sin m+ sin '* Va - sin #+ sine
1 + cos A 0.46616 .2 1 + cos Am
Y'2 427pg.. -Ae (approx.) Y2 ~ ~ ~ / 3 -ace (approx.)' 10,9V 634 + F (Ae~)
3 - V + F (44 + )F
K (Va/1,000)2
+ -124.' K (Va/1,000)2+ - a -170-49,6
COMPUTED BY w. 1/... DATE CHECKED BY /DATE INOTE: Fer s under 8,000 meters emit terms under the heavy bluek line not In.234a .57. £. C Qdewndnu 2 d~. heavy held type or undrlined.
DA ; Fa 71923.
Figure 136-Continued.
0= (7238.24) (+{-0.004)-}+ (-0.21) (-52)
+ (-15.14) (- 57) -- (-15.14) (58)
+} (-0.86) (52) + (-0.86) (57) + (-0.88) (-58)
* (2.79) (56) + (2.79) (-57)-}+ (2.79) (59)
+} (2.79) (-61)+ (-1.18) (-59)+} (- 1.18)(61)
+ (-2.94) (56)+ (-2.94) (-57)
+ (-0.28) (-54)+I (-0.28) (56)
+1 (-6.12) (-66)+ (-6.12) (67)-I- (4.70) (54)
± (4.70) (-56)+ (4.70) (66) + (4.70) (-67)
Combining coefficients of the same symbols with
regard to algebraic sign, such as (+0.21-0.86)
(52) and (+0.28+{4.70) (54), and rearranging in
206
ascending order of the symbols,becomes-
the equation
0= +28.95296-0.65(52) +4.98(54) -5.13(56)
+ 14.43 (57) -14.28 (58) + 3.97 (59) - 3.97 (61)
+ -10.82 (66) -10.82 (67).
This equation would probably be divided by 10
when used in a least-squares solution. Whendividing an equation by a constant, care mustbe taken, not to lose any significant numbers by
rounding off.
76. Adjustment of Triangulation by the AngleMethod
a. The figure previously adjusted by the rigid
direction method will now be adjusted by the
I~OC~lO CO? PUATIO, ODER RIAOULAION(For calculating machine computation)
PROJECT o £W3~ 3 1.85 a
a4 04 (Iog = T3O 3 ) 3' To 23.a h=(y/lOOOO)2
S77/
2d i-.3456 Xt +10. 4f 2 066 m 3d3 /35 xcr=f 0 0 6.
Ca, 2 006 ? go4/o42.mc i 65 3 14 o 3 a 3 87/4 -256/. To99037 -6A~alm -'4 01450 66 H pj q~gzsAa -2f/ 0/~ H .5
0" 4 2/7 209 a82 1' 0. 53.114s O 4 .2 .3 122 0/4 1 7
Vat --0 870047 0.5 In01 + 'Y672%1 ~oi40679,0
6/30+(log ) a- (/535 +Faxe
K 04fOO2
b~ =- (y/10,070 2 1Va(y/10,00+)- ) -655/
DA~~~0 71ES87 2Figur 16-Cotinued
angle method (fig. 138).. As the name implies,this method of adjustment solves the most prob-
able angle, and individual directions are not
corrected.
b. The angle method involves the solution of a
series of triangles. Consequently, 1 diagonal in
each quadrilateral must be omitted. Figure 139
illustrates the net with 2 diagonals omitted.
The stations are numbered starting from a fixed
line and each station added is given a consecutive
number. Thus, building up the figure point by
point and numbering each station as it is added,Burdell is number 1, Red number 2, Nic number
3, and Black number 4. Notice that Black is
numbered, even though it is a fixed station. The
angles in each triangle are lettered. a, b, and c.
:Letter b is always assigned to the angle opposite
the side of known length. Letter a is assigned
to the angle adjacent to the known side and
opposite the side through which the length is to
be carried. Letter c is assigned to the remaining
angle in the triangle which is also the angle used
to carry the azimuth through the figure. Com-
bining the letters with the number of the new
point in the triangle completes the designation of
the angles. For example, the angles in triangle
Red-Ilicks-Burdell will all carry the number 2
since Red is the new point in the triangle. The
angle at Red will be letter b because it is opposite
side Hicks-Burdell which is known from the
computation of the first triangle. The length is
to be carried through side Hicks-Red; therefore,
the angle at Burdell is lettered a. The remaining
angle is at Hicks and is lettered c.
c. The angles are taken from the list of directions
(fig. 140) exactly as in the direction method of
207
PROJECT LOCATION -LATITUDE AND (OTUD AD-U3MEN
1-243 (TMS-297)I,
ORGANIZATION FROM
TO.BLAE-K Nac BASE L4aLN-Rps.BASEDATE 22A ___- ___ __
TAIO(Endc
i. n~ of ]t +8uat ion Long eqution 1t eqato B 2ng. equation4 Lat. equatbon c L.ong. o0quat1on TABLES
OAIN (, ) A (,- )84( ,.-Oe) -a,,) (11.-91.)(-b) 1Xn -)aG (+11 or-L) (O,,-4,c)a2 0
C~ ~~~~ 2. 12.. 2..SZ 0 2 9~___
2 1.2 20
- I2 2.04 2. 14
}5 -7IL 74 2.02 2.10
1ll 2.011 2.210_ _ _ _-il Z _ _ 2.030 2.221
____0 __________ _______2 ~i.A 2-93..±. 00(a11 0.oi -. 1 .. i -2I t 5-5 120 2.2
___10 2 2.9 2.2__t_ 2.08 2.22
1 2 2.02 2.11
_ _ _ _ _1Z4 .08 2.02
17 2.04 2.2024 2.43 2.420
____ 8_______________ _______ 00j____ )1,____ 29 1 4 .35_ 2.42 2.29
19_______________ _____4
2.00 2.424Y1 -420 1.72 2.47
_________22 _____ _ 47 .4 2.4264CZz12... 7Z3A,.O 'A4i 22 1.47 2.44
403* 2 2 .42 2.34+_ 95___ 26...L.25) 3.5~±.J).~f.±..56)4 1.0 2.272 2.27 2.112
46 1.85 2.19
4 2 2.44 2.27
46 2.22 2.441
a 2.271 2.49B24 2.24 2.2
448 2.22 2.674.06 51) 1. 165 ) '99 1.22 4. 0
42 2.072 . 8U
44 1.52 2.247 .42 2.24
44 1.14 2.47
72 1.427 3.44
72 1.42 4.91
DA FOR 1927Figure ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 6 137. 4op.to8o0aiuean ogtd eutos(A om1 )
Lincoln Burdel~l
Red
Black
NicFigure 138. Sketch of triangulation net.
757-381 0 - 65 - 1420 209
Lincoln Burdellai
4b I
Block ks
cNiFigure~~~~~~~~ 13.Secho eR(igetrage)
210
oROECT ORGANIZATION LIST OF DIRECTIONS- 3 ~(TM 52?
LOCATION SATION
Ca/ fornia aLACK (U15Cd6s)OBSERVER INST. (TYPE) (NO.) DATE
OBERE SAIO BSRVDDIETIN ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDOSRE STTOOBEVDDECIN REDUCTION REDUCTION' WITS ZERO INITIAL. DIRECTION'
/-/K uea) 0 00 00.00 ___ __ 0 00 00.00 0.
Red 295'4fn" L5 L3T ____ ____ 4.7L
Ni (.&'i, ,,,j ? 2O39 ____ - 10.2
LOCATION - ATION
Californlia idel //(90nOBSERVER INST. (TYPE) (NO.). DATE
C01. P. S~nd57L 1 0OBSERVED STATION OBSEEVED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTED
REDUCTION REDUCTION' WITS ZERO INITIAL DIRECTION'
Real z Ene,)29448 .2 ___ -- 54
A/CA-s (u C#6 / 3J~ 3Z - - 28.8
LOCATION STATION
OBSEVERCah~rn~ _______________ HICKS (csc-.sJOBEVRINST. (TYPE) (NO.) DATE
OBSERVED STATION -ORSERVED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDREDUCTION REDCCTION' WITS ZERO INITIAL DIRECTION'
-G'o~ q Sc)c 0 00 00.00 -- -- 0 00 00.00 5.
R edS~' Y 426 5 .2 - - ~O IL
A/ic ( Gnye 136 2/ 11.3 _ - _____ .I6." Jr- /Red 6y prf-awso
AZACK +uc~ ___-2/________S .
LOCATION STATION
Clfri _________ L.JNCOLA( 4, GsOBSERVER INST. (TYPE) (NO.) DATE
OBEREDSTTON OBERE DRETIN ECCENTRIC SEA LEVEL CORRECTED DIRECTION -ADJUSTEDOBEREDSTTON OREEE DRETIN REDUCTION REDUCTION' WITH ZERO INITIAL DIEECTION'
O / if / It 0 5 ViV
P/ks + 0 00 00.00 - ~ - 0 00 00.00
Real(9Pos)2L.5Q28. ______
LOCATION- STATION
Caliorna ___________ A/i (2 ~E qsAOBSERVER INST. (TYPE) (NO.) DATE
Cpi PP Smi4i0OBSERVED STATION OBSERVED DIRECTION I .E ENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTED
EUCTION REDCCTION' WITS ZERO IITIAL DIRECTION'
0 if If I O / 'n /
BLACK (99C#45) 0 0 00.00 --r . - 0 00 00.00 0.
LOCATIONSTIO
OBSERVER INST. (TYPE) (NO.) DATE
OBSERVED STATION OBSERVED DIRECTION ECCENTRIC SEA LEVEL CORRECTED DIRECTION ADJUSTEDREDUCTION REDUCTI WITS ZERO INITIAL DIRECTION'
RIic*s (OscsG 0 00 00.00 - - 0. 00 00.00
LINCOLN f6as) 4 52 -
Figure 140. List of directions for angle method (DA Form 1917).
211
adjustment. The triangles are entered on DAForm 1918 (fig. 141) as explained in the direction
method. The symbols in the left-hand columnare entered beside the corresponding angles asshown on the sketch. Notice that the angle at
Hicks from Lincoln to Black is fixed, and the list ofdirections must be corrected to hold the fixed
angle. Only four of the eight triangles in thecomplete net are written because the two diagonalswere omitted. The missing triangles will be com-
puted after the adjustment is completed. Since
the net now consists of only four simple triangles,there will be only four angle equations to close
these triangles, and one additional equation topreserve the fixed angle at Hicks (fig. 142). Thislatter equation is merely a statement that thesum of the four corrections to the four angles atHicks must equal zero. There are no side equa-tions, and only one length equation (fig. 143) as inthe direction nethod.
d. The designation of the angle will also desig-nate the correction to that angle. As in the direc-tion method, the sum of the corrections to theangles in a triangle must equal the triangle clo-sures. For example, in the triangle Burdell-Hicks-Lincoln, (la) + (lb) + (lc) + 6.4, or writtenin the usual form, 0=-6.4+(la)+(lb)+(lc).
Equation 5 fixes the angle at Hicks. The lengthequation is formed as in the direction method.The six equations are now solved by the methodof correlates (fig. 144) as were the equations in thedirection method. The final adopted v's in thiscase are the corrections to be applied to the angles.The angles are corrected on the triangle computa-tion sheet, reduced to plane angles, and the lengthssolved. Although the solution of the conditionequations is much simpler in the angle method ofadjustment than in the direction method, theapplication of the corrections to the list of direc-tions is more complicated.
e. Examples of the method of correcting thedirections are-
(1) Station Lincoln.Observed Preliminary
Station direction seconds
Hicks----.----00'00"0 00"0Burdell - - 251 50 28. 6 25. 6Red--------3021114.7
Triangle 1 angle Burdell 108°09'34!4
to Hicks
At Hicks correction is 0'0
At Burdell correction is- 3'0
Total-3!0
Average correction-iV 5
Finalseconds
01'.' 5
27.1
(2) Station Burdell.
Station
Lincoln - - -
Red- - _
Hicks_ - _- .
Observeddirection
000'00':0294 48 01. 2
337 10 32. 9
Triangle 1, angle Hicks toLincoln
Triangle 2, angle Red toHicks
Angle Red to Lincoln
At Lincoln correction is
At Red correction is
At Hicks correction is
Total
Average correction
(3) Station Hicks.
Station Obse
Lincoln-------------00
Burdell___________ 49
Red--------------91
Nic -------------- 136
Black ------------ 180
rved directiol
00' 00''
00 55.
29 06.
21 11.
40 54.
Triangle 1, angle LincolnBurdell
Triangle 2, angle Burdell
Red
Triangle 3, angle Red
Nic
Triangle 4, angle Nic
Black
At Lincoln correction is
At Burdell correction is
At Red correction is
At Nic correction is
At Black correction is
Preliminary Finalseconds seconds
00"0 57'7
56.2 53.9
31.1 28.8
220 49' 28'9 (A)
42 22 34.9
65 12 03.8 (B)W10
-5. 0
-1. 8
- 6'8
-2!3
Preliminary Finaln seconds seconds
0 00:0 58"9
2 56. 8 55.7
2 08.6 07.5
3 12.7 11.6
5 54.5 53.4
to 490 00' 56'8 (A)
to 42 28, 11.8
91 29 08.6 (B)
to 44 52 04.1
136 21 12.7 (C)to 44 19 41.8
180 40 545 (D)0'0
+ 1.6+2.4+1.4
0.0
Total +5" 4
Average correction +1!1
(4) Station Red.
Station
Hicks -- __-____
Lincoln---_____
Burdell-_--_-_ -
Nic--------------
Black--_____-
Observed direction
00 00 00:'0
30 41 59.7
95 09 11.0
284 17 14.0
321 09 32.8
Preliminaryseconds
00'.'0
13. 5
13. 2
Triangle 2, angle Hicks to 950 09'Burdell
Triangle 3, angle Nic to 75 42
Hicks
At Hicks correction is 0"0
At Burdell correction is +2.5At Nic correction is -0. 8
Total +17
Average correction +0' 6
Finalseconds
00!6
14. 1
13. 8
13:5 (A)
46. 8 (B)
212
PROJECT DATE
1-243 20 Au3 55 COMPUTATION OF TRIANGLES(TM 5-337)
LOCATI ON CQionaORGAN IZATION 2Onr
SPHEICA SPERICL PANEFUNCTION:STATION OBSERVED ANGLE CORRECTION. SPEANL SPERCSL PANE LOOAIH
___2-3 3~7 5
1-1 +udl 24 7 L& 2&.9 0.O 2&91 2,589734I92 WcS49 00 55 L % S ~~ 8787
SLIjeoLM /08 093. ±30 14 O.K 34.3 9.97781/571-3 8ukU~ON3.9767005
1-_ 2Q~apj-iK 4. 07662783
__ __ _ __ 7 4-6.4 0.1 _ _ /_ _ _ _
2-3 __ _ _ _ _ ___4. 07622k 1
2 W95 09 11.0 442.5 /3 Q.'--1. ~ .998,24070
2 WICK 42a 28. 11.0 AOL&. 1 ?.. JL58294373 Bqurt/elI 42- 2 31.7 ±32 4 t 32%88511-3 2dSrll___ .0725
1-2 Red - HICKS ___3.90704529
537 f . 0.2 _ _ _ _ _ _ _
2-3 .? 42
361 Nvic 5 .5 *r0 40.2 0iL2 O.0 02L 9. 93994
2 I4ICKS 44 52 05.1 -LIO±oo~ 0.L 9.8484806 7
3eat 75* 42 46.0 fO&A. 0L. A±L7 77863558
1-2 _/ - IK _____ 3. 95844203
- ~~0.1 _ _ _ _ _ _ _
___2-3 ____3.54203
Ak 1 8LC 0390. 0.2 IO-L 0.L 110.0 .59421L8L8
2_ 2 gC : S ~ 44 19 43.2 -1.4 AL.& 0. A.81 98443
3_ .45C 0,/ 0 40. 08.2. 0.0 0. 9a .8.51-3 p -LCK NiC__
_ 3.80280349
__ 1-2 BLC- FlCS______ ____3.80809877
__ _ _ __ _ _00.5 -0.4 __ _-0.1 __ _
COMPUTED BY DATE CHECKED BY DATE
DA I FEB571918
Figure 141. Computation of triangles for angle method (DA Form 1918).
213
- + +
.3a - 0. ___
Figure 142. Condition equationsf
(5) Station Black.
Station
Hicks---------
Red------------
Nic------------
Observed direction
00 00". 00"0
51 57 43.3
90 39 09.9
Triangle 4, angle Hicks to
Nic
At Hicks correction is
At Nic correction is
Total
Average correction
Preliminary FinalStation Observed direction seconds seconds
Black--------------00 00' 00"0 00"0 00"0
Hicks-------------45 01 07.4 08. 2 08. 2
J. The preliminary seconds are obtained by
using the adjusted spherical angle from the triangle
computation sheet. A direction which was not
used to obtain an angle for a triangle used in the
adjustment will not enter into this computation.
These directions will be corrected later. The
__ _ _- difference between the observed direction
and the preliminary seconds is listed for each
direction at which it occurs. The sum of theseor angle method. differences is divided by the number of directions
involved (including the initial direction), giving
the average correction per direction. The sign ofPreliminary Final the difference must be such as to change the ob-
seconds seconds
00'!0 00'[ served direction to the preliminary seconds. The
- - --- average correction is applied to the preliminary
10. 1 10. 2 seconds to obtain the final seconds.
900 39' 10':1 (A) g. The final computation in the adjustment is
the solution for the lengths of the diagonals0"0 omitted from the net when the adjustment was
+0. 2 begun. This computation is best performed on
+0"2 DA Form 1919 (Triangle Computation Using
+W"1 Two Sides and Included Angle) (fig. 145). After
PROJECT S/LENGTH EQUATIONS1-243 (TM 5-237)LOCATION
calirorniaORGANIZATION DATE
29E 9rs. 2oAug ssr
SYMBOL ANOL LOG. SINE TAB. Dirr. SYMBOL ANGLE Lo. SINE TAB. Di.
81 1 1-4bZCKS8L.ACK -e09 ACC-LNO 3. -54
1& 249. OBZL /a 108 09 31.4 3 S31TLL -0.6?
_L_ _ _ 9.8286511
0h -. aJ.5.0 +1.25 75 46.02 9.- ( )5L 2+i~L _
-- / . 28_ _______ _
Figure 143. Length equation for angle method (DA Form 1926).
214
I (6) Station Nic.
- 2 ~ ~r
-# +L -1 1- - C.
-- AL -A -#42/ -42Z Qff
-/ -56 -6. -- 3-23 +242 2 .83 1±
-~+ +- IL. - --1 i .L ....
4/~ ~~2/ +0/ +_____ J 2. - 0l. ...c..
_ ormALd 1h~as
- - -L -6.ZL Q 1JL 00
C- -0. ±L3 f21 +O~ f2 Q2
-0,=- Q,2-167-8Q5- - - -+ZI -fi.2 3 +0.4 +~2% 2.2
- -- ~ L ~ ~ L~
_ ¢ -. 5 f iM-2 1
_ -- 2
Figure 144L Souto ofeqaios
215
PROJECT d TRIANGLE COMPUTATION USING TWO SIDES AND INCLUDED ANGLE1-243 j(TM 5-237)
LOCATI ON ORGANIZATION 2qDEng,
btan (45°-+q-0) (Call longer side a). tan z (A 0-B 0 )= tan 'p tan Z (A, +Bp): =sn-D
C. 91 2'? 08.6 Log a 3 4 j90ogSph. excess ~Lg ~ ~ j 4 LgiC
SLog ta (456g)7550j4q45 Log ai C0.1
CD 45 44 2.5 (4,50±') 2-)o58sl~Log .b ____
.6690°-j~p- j(Ap+Bp) 44 15 25.75 0 . 3 .2 Log sph. ex.
I (A,,-Bp,) 13 33 3.56 Log tan 0 g* 39350083 Sphi. excess 0. ISum=AD 57 485 41.31 Log tan I (Ap+BD) 9,8733Diff=BD 304 21 Log tan yl (A 0-Bp) 9. 382 23920 ___ ___
CD '1/ 29 e'j.q, [/ivaN (Sketch)
Log a 3.90704529 b
Log sin CD gg~g
Colog sin A0 C Red
Log c 3-9793643_____ _______________CHECK COMPUTATION
No. STATION SPHERICAL ANGLE SPHLRICAL PLANE LGR't~
2-3 HCs -LINCOLN 6S8Z5501
1 Rd 30 42 022 2.. 006AOo
2 141CKS g, 2g 086 0.1 085q
3 LINCOLN 57 48 493 493 q.25348 71-3 &Ed - LINCN .179 6YL
__ 1-.2 Red _ 14ICKS___ A142.
2,3LINCOLN -BunadeII ____ E 70605
___1Red 4 27 11.3 0.1i 1L. g53j55
___2 LINCOL.1N 50 20 45.1 4 45.1 9.8644030
au 6w-ell 65 /2 03.8 a., 1oT%5f8Q.
1-2 Re - LINCOLN ____3 736451____ ____ ____ __ _ ____ ____ 0.2 _ _ _ _ _ _ _
*The subscripts s and p on this form refer to spherical and plane angles respectively.
COMPUTED BY nn DATE4 CHECKED BY DATE 4
DA , FES571919
Figure 145. Solution of triangles for angle method (DA Form 1919).
216
PROJECT TRAGECMUAINUSING TWO SIDES AND INCLUDEDANL1-243 I RAGECMUAIN (TM 5-237)ANL
LOCATION ORGANIZATION 2 . DATE
Ca/i> (r-AiQ 91 tr./A p5
C btan (450 +0)) (Call longer side a) tan a (A 0 -B)=tan tan (A +B) z
CsnAl]
5 9o. C o
9 L g 3 .1 4
p
.ecs.
C5 j~ tj Log a 3 gpp41 Log simC
CD gi 45.8 Log tan (45'04) 0. pgg461 Log a
-_____ 44 35 52.9 (40t)7~ 5 ob ___
445 8(5+)° %SLgb1 900- C5 (A+Bj5 ) A5 2.4 .1 . 06 28 5 fi Log sph. ex.
I (Ap-Bp,) 3034.33 Log tan 9.0547118 Sphi. excess 00
Sum=A5 Lo a A + p .0 60 3 51 5 7 49 63 Dif B 8 .0 2 -7Log tan (A ,5 +B) Q .q j_
__ _
C, 89 //45 Red' (Sketch)l
Log a Q.op5? CLog sin C5 gg 724 a
Colog sin A5 O3841,0626Log c "/gCk B LACK~
CHECK COMPUTATION
No. STATION SPHERICAL ANGLE SPERCAL PANE LOGARITHM
2-3 .3.6080 gg,
___1 Red 36 50 24.6 O.O 2446 9-2M37J4U
28LACK~ 5/ 57 49.6 11.0 .6 1796L493 /WkS 8'? /1 S.I Q~~ ~i21-3 Re 4CK __________ ___19W4516
___1-2 ked-RLACK 4OOEi
2-3 3,80 8a3491 Pd36 52 22.2, 0Q-f 22.2. 5.2z81BO92 N-c 04 26 I 74 0-1 JJ,3 ~8Q6259
3 LACK 38 41 205 20 20.5 91 44761-.3 Red-SLACK 4LQ16ML&
___1-2 Red-Nic __ _
*The subscripts s and p on this form refer to spherical and plane angles respectively.
COMPUTED BY DATE 'CHECKED BY DTA 1-f s w. C. A. 2A7
DAIFORM 1919
Figure 145-Continued.
217
these computations are made, any missing anglescan be obtained by combining angles fixed by theadjustment and computed angles. For example,computing the triangle Red-Hicks-Lincoln toobtain the diagonal Lincoln-Red, angle C atHicks is found by adding the adjusted angles (1c)and (2c). The side Hicks-Lincoln is the originalfixed length, and the side Hicks-Red is taken fromtriangle 2 of the adjustment. Solution of thetriangle on DA Form 1919 gives the length of thediagonal Lincoln-Red plus the angles at Lincolnand Red. The check computation on DA Form1919 is made by solving the triangle Red-Hicks-Lincoln by the sine law using the computed anglesat Red and Lincoln. The second triangle in the
check is Red-Lincoln-Burdell, the fourth trianglein the quadrilateral. In this triangle the lengthLincoln Burdell is taken from triangle 1 of theadjustment. The angle at Burdell is the sum ofangles (lb) and (2a) of the adjustment. Theangle at Red is found by subtracting the computedangle Lincoln-Red-Hicks from the adjusted angle(2b). Similarly, the angle at Lincoln is obtainedby subtracting the computed angle Red-Lincoln-Hicks from the adjusted angle (1a).
h. The list of directions can now be completedby adding the angles from the check computationsto the directions previously corrected.
i. Position computations are now made as in thedirection method, using the adjusted angles andlengths (fig. -146).
PROJECT 1-243 POSITION COMPUTATION, (T 5RDER TRIANUILATO (For cdalmidcin couptaw)
LOCATION ORGANIZATION OT
a 2 To 3 aJ~0I 3/ . ~ _ ~LAt To 2 cks I .. ,A24Z A 4 161 3dj & _
a 2 ' To 1~JI Z~O 4a 14 3 LIN=N To 1 8 ~. .zS04 ___________ * '"__ ________ Q53.45
180 00 00.00 180 00 00.00
1l;a~/ To 2 ICS8/ 06 47.co a'lR1 .4 To 3LICL 035 i;a
10 d i'8 First Angle of Triangle 22 49 Y5.y 1
4629AikS~ 122 43- 29. + , j3 ICL 12A_43611929,454 - 147 03 na-s ~
* ' 22 35 25.43 *' 38 08 4639-22 2.3vi. 8 6 9 lQ bogyI0,000) 2 4.7628 )/
(togse= 2 3-76700S.e-cOb(1000 ,52sin 0.977o2 x cor. _-fb sin a -0. x7 2 xcor. _-Z 02/
C 0. ISA91612 Z11783,7568 C a39(.s a921.73
z~s sina _ 1783,77 'H 0.0417=5 sin a - 2"13 H
4a-3.~ 7,32s p3~ x'=(approx.A") ~ 3q Y=-s Coa -2272 .42 H'aprx.i6j
a=(zd/10,000)2 .38 Arc-sin-+V (Va) 2 =xIO.0) Arc-sio+V (V) 2.y cor. _+fa °a.4 -__ e--8.~3 yor. =+fa °--37.7
Y. - 2216,47 +7 sin *.4nA Yu si 5. 2
__1_______________+Icos, °oI s, Yi. I 2 2-( 7 1+cs
Y2 4 2 / 7L -Ace (approx.) -28.3 Y2 4223 S19. 4Z1 i -Asd (approx.) -243.453V (0 49 + F(°a')
3
V 6.19 + F(A)a
K (Vanl,000)2
+ - Ae -2_aj K (Va/1,000) 2+ - Z. -233,453
COMPUTED BY A DAT ao7 . CHECKED BY DT NOTE: For r tinder 8,00 meters omit terms under the heavy black tie not in
D A s£4)7 .S ' , . ( A DA 31fq 4 . 5 heavy hold type or underlined.
DA 1FORM 6193
Figure 146. Position computations for angle method (DA Form 1928).
218
PROJECT ' POSITION COMPUTATION, 3 ORDER TRIANGULATION (For calculating machine computation)1-243 (TM 5-237)
LOCATION ORGANIZATION DATE
__ Ca/,fornie .__ E9t~.~.iA1X
a2 ICS To, 3 212OL a 3 _~LICL To 2 HIck-s 01 32 SL6.82d Z & + i 3d~ & Qt - 48 49,3a 2To 1 Ree 3 ?,9 a 3 i-iot To 1 /es-341LO~
__a _02__0.___a__+# / 4-5.08180 00 00.00 180 00 00.00
a' 1 Red To 2 HCk 13 3 S0. 44 ' To 3 LICON 154/4 152.43iF5 .2070 First Angle of Triangle 30 12 02.2 {-8.2 070
*3 4292. A C' 122 43 29P36 -u 39 / 008 3 LIA/IYIN A122 41_ 43.3o5.3 74,5 =6173/9,2 AN 04 9~ 535W A' d) - 02 50,1 98
3O' 21. 1 A'd /2 53.1/0 *' 05 2/1648 1 A'd S23.107(loga 2 3.A9oN704S24 ~ (log a= 2.7363
02 41b=(yI10,000)2 0,19 04 38.53 h=(y/10,000)2 ) 737
2i o._-f 0.817 sin a -0-434 93623 xcr -2f 3.024Cos a ¢0.sJ967 -6732,1210 cos a f 0. 90046126 x -414 75328
XS S Ufl7 a~ -i a 721/ H x.4I39s. Xsin a- 4147-3341 H 0.04035936-y= -s Co. a .Y=-s5co/a Hx'=(approx. AA
5) ~.7 59 Y=-s cos a -82586 7(/ Hx'= (approx. 7)
a=x/0OO2
Arc-sin+V (Va) a=xIO002 Arc-sin=+V (Vaa(/1.0) 0.4532 cor =+ 15 J14 ~ 0.12 =x/000 o 15 0.43
Y u . f .7 1 wy c r = f 1 4 N 1 0 18-2 6 29 4 2 2 / 6 6 7 9 7 S s n ~ 0 6 1 7 -4 4 I 8 ~ Y o 4 2 2 7 9 7 1 2 l s nf l 0. 6 17 9 -f/ 7 g
yo - 4 4-a. a 3 sin 0' .616 .954 Y - -8586.76 sin 0'
Y'1 4 2/7 212.opA2 1 + cos °¢ /. 9g997-1 l 4' 4217 2/0.358 t+ cos ° /.ggggggpg
a-2. 781 si * sin o'f or Va si 0.5 ~ + sin or sin o.0I74209Y2 4 / 0.3/ -°a (approx.) I70A98 Y2 4 2/7 209. 303 -°a (approx.) - 05. 084V 6. 1-344 + F (°>')
3 V 6./364-4 + F (°a)
3 -
K< (Va/1,000)2
+ - °a" -170-.9 K (Va/l,000)2
+ - - Aa' - /05.084COMPUTED 8Y IAe. I DATEAf j CHECKED BY 4.C 'DATE -SS N OTE: For aunder 8,000 meters omit terms under the heavy black line not In
Ji . A 1 .3(C..384 9 V heavy hottype or underlined.
°aDA 923
Figure 146-Continued.
219
PROJECT 1-243 I MSMTON COPUTATN)U, 3 ORDER TRIANGOULATION (For cidevilathg macw camputa tm)
LOCATIONo/ ORGANIZATION P9E1r.DATE PA so ,
__ ofornga 7?Egs /?u5,,___a
2RL To 3/q~ 21 01/.9 3 U~c&s To
2 BLACK 32'.L &t +q IjQj 34Z - 4 11.. A4..a
2 BLAcKe To 1 pjjc 30 It -29, L. 3H ~~ To 1Pi 2~ 04.°a 02 14.18 °a ___________ __ ,) 3
180 00 00.00 ISO 00 00.00
a' I Vijc To 2RLAC 123 2/ 142.14? a' 1Ii To
3/~C' /68 22 50.40First Angle of Triangle 45 / 08.2 " ~6.20 70
~V[4O72.RLAcK I 122L 455 14 *M J_-24.S I& 2 2S7 350 4-35 - 03 37628 ?0Z -5 - 15,4
3'.8 02 57.566 1 i' !22 / .2/ *'3 02 57566 1i ~ '/2 42 /4.21 7°m ( =2ao80549 [jiog= 2 3,54 )3
(I'=0.122 .) 04 4969 h=y/00
-QL a 8357302 x c 0 Vor.=-jib sin" a 0 2066 x cor.= 0-fb 3.250C~ s a 0 .5 4 9 3 7 9 3 9 'tC
s a -t -5 3 0 6 . 25 /9 ..: =s sin7
4 6 2 0 a-/8
32=.sn2 8H9253
6 .
- 3488. 798/ Y-so Hx'= (approx. Aha) -276284 Y=-so" p9, ,a Hx'.=(approx. &I 38,741a(/0002Aesn+ V)a('1,0
r-i=V(a
Ye 4 21625Z 70 sin 0. Q6 7)74 Ye 4 22/667. 974S sin 0,617441 18Y3 - in f Y' _.I39~ -8 00. 813 .Ini, 0. 6633962
Y1 4 1 ~./ 1. + cos A# / ggggggp Y1 4 2/2 767. /62 1 + Cos a*/ 9go,____i__+_i_1 r in. V sin* + .In' o i .-1.726 1 +cos A4 O5fn06S5".73V 0. 20m 1 + cos a Q
Y2 4 2/2 766. 94 -°d (approx.) -/34/A IN 2 421,2 766 . -9546 -°d (approx.)
V6.12743 + F (A,'3 V 6. 12 7A3 + F(A)3
K (Va/1,000)2+ - /34.18 K (VA/1,000)2+ - -4635
COMPfYrED BY DATE CHECKED BY DATE NOTE: Forea under 8,000 metern omit ters under the besyp blk Um~ not InAw(.C . .73A" f beev bold type or underlined.
DA FORM
Figure 146-Continued.
220
Section V. SPECIAL PROBLEMS
77. Descriptiona. Many of the special problems encountered in
triangulation are connected with the location of
additional points which are not normally con-
sidered to be part of the main scheme of triangu-
lation. The methods most commonly used for
locating additional points are intersection, three
point (resection), inaccessible base, and special
angle.b. In these four methods, at least some of the
directions have not been measured. At least one
of the angles in some of the triangles must be
concluded (deduced from known information) and
the best results will be obtained if the concluded
value approaches as closely as possible the probablemeasured value. In many cases, this may be
accomplished by subtracting the sum of the two
measured angles from 180 ° plus the spherical
excess for the triangle. Each figure must be
studied thoroughly before making a decision on
this matter.
78. Intersection Station Problem
a. Intersection is the process of locating a
point by observed directions from two or more
stations of known position. This method is used
in locating additional control points or prominent
objects during the process of completing a survey.
b. In figure 147, stations Jones, Howard, and
Kelley form the triangle for which the directions
and lengths were previously determined. Station
Spire is the intersected point. Directions have
been measured from all three established stations
to station Spire.
c. In this problem, there is one geometric con-
dition to be met. The three adjusted directions
must intersect at a common point, in this case,station Spire. A condition equation in the form
of a length equation or a side equation with the
pole at Spire may be used. In the sample adjust-
ment, a length equation is used. The center line,which is the common side of two of the concluded
triangles, is designated direction number 3; the
line from the known station on the right is desig-
nated direction number 2; and, the line from theknown station on the left is designated direction
number 1. This numbering system is arbitrary.
d. On figure 148 is shown the length equation
and its solution. The directions designated have
been previously explained, the concluded anglecarries the combined designation of the other two
angles in the triangle, but with opposite sign. The
length equation is set up from fixed line JONES-KELLEY to fixed line KELLEY-HOWARD.The equation is solved and the v's are computed.
e. Figure 149 illustrates the computation of thetriangles.
SPIRE
/ \
. KELLEY
Figure 147. Diagram of intersection station problem.
221
PROJECT TESTENGT 53EQUATIONS
LOCATION
C. A.__ _ _ _
ORGAN IZATION DATE
___ TO. ________ _OCT. /962
SYMBOL ANGLE LOO. SINE TAB. DiFF. SYMBOL ANGLE LOG. SINE TAB. DIE?.
JON S -/<ELL Y KEL EY -HOW RD__
/3 9/6.3 7 m 4.143.53742 192 5 5 .8 4 3 M 4.28456254 __
-/I 73/1201.37 9.98105778,+6.64. +1 -3 46 32 01.45 9,8608047.3+2.00-2+3 86 13 09.38 9.99905381 +0. 14 +2 72 01 30.00 9.97826785 +0.68
ARC SINE - 34 ARC SINE - 66___
_____4.12364867 4.12363446___
_____ ____ ____ 3446 __
_____ ____ ____ + 1421 _ _
O +14.2/ - .64(1) -0.82(2) +2.14 (3) _
D IR~ECTION COEFFICIENT COEF /CIENr V ___
-__ / 2.6 4 6. 9696 +3.o7 __
2 -o.82 0.6724 +o.95 __
3 +2.14 __ 4. 5796 - 2.49 __
____ _ _ ____ __ ____ __ ___12. 2216
________SOILUTIONJ OF NORMA 5 __
____~~ ____ +~ 14. 21 + 12-2216C__
_____ _ ______12.2 /6c = - 14.2/ __
____ ___ ___ ___ ___ c= -/.!6270 _ _
COMPUTED BY DATE 1 CHECKED BY .DATE
J.LL~ot - A-M S FE ( G. T7A-u - A MS5 F .(B. S. GOVERNMENT PRINTING OFFICE :1957 0-420724Ua1 EB 57192
Figure 148. Least squares adjustment of intersection station.
222
PROJECT TETDATE
TET c /96 2 CMUAINOF TRIANGLES
LOCATION OGNZTOC. A ORGNIZAION7.0.
STATION OaauaVRD ANGLN CORESCTION. ANGML EXIC~AL PANL om RITHWA
2-3 _ _ _ _ _ _ _ ___ _22137.620,1
1 KELLEY 82 01 /8.70___________2 JONES 59 28 29.32 Fl XEP _ _ _____
3 HOWARDP 38 30 /2.66 __ _______
1-3 __ __19255.843.
1-2 __ __139/(6737
______00.68 -00o.68 -
2-3 -__ ___ _ 22.137.620
(+1-2) 1 SPIRE (132 45 10.83) + 2.12 12.95 -0.08 /2.87 .734 2801/6
+2 2 HO1WARD .33 3) /734 + 0,95 18.29 -0.07 /8.22 .552 253/7
-1 3 JONES /3 43 32.05 -3.07 28.98 -0,07128.91 .237256911-3 146 __ 4_ /69. 736
1-2 _____7/52. 996
______________00.22 00.00 60.22 00.22 00.00 _______
__2-3 1_ __ 9255.843
(-2+3) 1 SPIRE -(86 /3 09.38) -3.44 0594 -0.1I 0583 .997 82257+2 2 HOWARD' 72 0/ 30.00 +0.95 30.95 -0./i 30.84 .951. 19252
-3 3 KELLEY 2/ 45 2o.93" +2.49 123.44 -0. 1/ 23.33 .370 662491-3 1_________ ____ __ 8355, 28.3
1-2 __ _ _ _ __ _ _ _ _ _ 7/52.994
_____________00.33 00.00 60.33 00.33100.00 ______
_ _2-3 1__ _ _ _ __ _ __ _ 39/6.737
(+1-3) 1 SPIRE (46 32 01.45) +5.56 0701 -0.19 06.82 .725 79746+3 2 K<ELL EY 60 /556775 -2,49. 55.26 -0.19 55607 .868 3312 7
-/ 3 JONES 73 12-01.37 -3.07 583 -p12 58. .967 3/685-
i_ -3 _____ _ 6649.-739
__1-2 1________ 8355.984
__________ 00. 57 100.00 60.57 00.57 00 .00 _____
OMPUTED Y DATE, I CHECKIFDGY DATEJ a Io.-AS Fs6 G-:7- -ASE:6
FORM 91DA FB 1
U. S OV0NN2W01T PGTI6OUPICE: 1W7o-42060
Figure 149. Computation of triangles for intersection station.
223
(1) Observed angles extracted from the lists
of directions are entered in the observed
angle column. The corrections entered
in the correction column, and the ob-
served angles are added algebraically to
determine the adjusted spherical angles.
These angles are next reduced to plane
angles by subtracting the spherical ex-
cess. Final lengths are computed using
the adjusted plane angles. The double
determination of all three lengths pro-
vides a valuable check on the adjustment.
The lengths should agree within the
limits of the computed precision of the
angles.
(2) The fixed triangle (KELLEY-JONES-
HOWARD) is shown for the starting
lengths and also to provide a check on
the spherical excess. No corrections are
applied to the angles in the fixed triangle.
Lin
Hicks
Hicks 1192
co
79. Three-Point Problema. A method sometimes used for supplemental
or mapping control in an area where primary con-
trol exists is the three-point (resection) problem.
In this method, the unknown station is occupied,and angles are turned to three or more known sta-tions. Four known stations should be used if
possible, as this will give a check on the work.
b. For purposes of illustration, a portion of the
sample triangulation problem will be used for the
sample three-point problem solution (fig. 150).
The lengths and angles in the triangle Burdell-
Hicks-Lincoln are those obtained from the adjust-
ment by the direction method. Angles A and B
(at Red) are the unadjusted angles as taken from
the list of directions.
c. DA Form 1930, Special Angle Computation
(fig. 151), provides a simple, rapid method for this
computation as case 1. In this sample computa-
tion, angle A is 30041'59'.'7, angle B is 64027 '
In
0o
.B 9l
,m "'" Burdell
RedFigure 150. Sketch, three-point problem.
224
PROJECT SPECIAL ANGLE COMPUTATION1-241(TM 5-237)
LOCATI ON DATE
ORGAN IZATI ON CASE USED
2? aEngr. ________ 1.® 2. E 3. 1:1
b
a G"bbB I N X Y> a ,% b
A q /"F rG
- vx
EA0BC
D As
\xa
Case 1 GCase 2 Case 3sin x _b sin A _tn« A B sin x = i i i assinux b sinAsin C tan
sin y_ a s- in sin y sin B sin D sin FsiyasnBsnDTHREE-POINT PROBLEM INACCESSIBLE BASE PROBLEM SPECIAL ANGLE PROBLEM
(Case 1: 180O-.2 (A+B+G) = 0
2 (xy) = Case 2: 2 (C+D) = 7 2 3
Case 3: 270Oo- z (A+ B+{-C+--D+-G) =___________Leave blanks below here for values not involved in the CASE used.
log b 3. 9766n9409 log a
log sin A 9.706'03120 log sin B 9531&log sin C log sin D
log sin E log sin F
* OSum 368472529 *OSum
-- ® - , -089/ -
log tan a 0.04/856/9 log tan a
a 47 45 24. 285a450 02 45 24.28.5 a-45________
log tanI (x+y) log tani(x+y)
log tan (a-450) 8. 68260V4 log tan (a-.450) ____________
Sum-log tanj(x-y) .38214Surn=log tani(y-x) ___________
R~X-Y) G3 pj /77 I(Y-X) 0 / P
Z(x+y) ?8 20 3735 iRy+X) ________
X______ 91 28 J.os Y_____________
a is an auxiliary angle needed only for the, computation: it is always between 450 and 900
* Where Q is greater than ® use only the left side of the form below here, and vice versa.
COMPUTED BY~e ~ DATE A jCHECKED BY DT
D ,FORM 13DA1FES 5713
Figure 151. Computation of three-point problem (DA Form 1930).
757-381 0 - 65 - 15 225
11"3, angle G is 108 009'34"3, side a is 4870.241
meters, and side b is 9477.507 meters. Subtract
Y of the sum of angles A, B, and G from 1800° .
(As the sum of angles A, B, and G nears 1800, theaccuracy of the solution decreases. If the sum is180 ° there is no solution.) This is 2(x+y).Enter log b, log sin A, log a, and log sin B on theform where indicated. Log sin C, log sin D, log E,and log sin F do not enter the computation for
case 1. Add log b and log sin A to obtain sum 0.Add log a and log sin B to obtain sum @. Since
0 is larger than ®, use the left side of the formfor the rest of the computation. Subtract ®
from ( to find log tan a. From a table oflogarithms of trigonometric functions, find a.Subtract 450 from a and place the log tan of
the resulting angle in the appropriate blank.Add the log tan of Y(x+y) (if over 900 tan isnegative) (sign is important) and the log tan of(a-450 ) to find the log tan Y(x-y). From tables,find the angle 32(x-y). The desired angle x isnow obtained by adding the angle 32(x+y) to theangle 2(x-y). Angle y is obtained by subtractingthe angle (x-y) from the angle 32(x+y). All
the angles in the figure are now known or can beobtained by subtracting the fixed angles at Hicks
and Burdell from the computed angles x and y
respectively. Note that an error in the compu-
tation of the angle Y(x-y) [ (y- x) on right side
of form] will cause compensating errors in the
angles x and y which will make the sums of the
angles in the figure check, but the solution will be
incorrect.
d. The same three-point problem previously
solved by logarithms on DA Form 1930 is now
solved on DA Form 1920. This solution utilizes
natural trigonometric functions throughout. Re-
ferring to the schematic diagram (fig. 152), the
formula used to solve the problem is:
b sin A sin [360°- (A+B+C)]tan x=b sin A cos [360-- (A+B+C)]+a sin B
in which a and b are the known lengths, C is the
known angle, and A and B are the observed angles.
Angle y equals [360 ° - (A+B+C+x)]. For use on
this particular computation form, the formula for
tan x is modified to read as follows:
sin [360--(A+B+C)]tan x-
cos [3 60°--(A+B+C)]+a/sin A
b/sin B
Lincoln
Hicks
Figure 152. Schematic diagram, three-point problem.
The reason for this change is to obtain the ratios
a/sin A and b/sin B, which are used later to solvethe triangles. In solving the formula, care mustbe taken to use the correct algebraic sign for the
sine and cosine of [360°-(A+B+C)]. This is
important. The algebraic sign of tan x depends on
the signs of these functions. The algebraic sign oftan x determines the quadrant in which the angle xlies. The cosine of angles between 90 ° and 2700
is negative. Negative tangents indicate angles be-
tween 900 and 1800 and between 2700 and 3600.
e. In the numerical example, stations Hicks,Lincoln, and Burdell are fixed in position, thereby
fixing the lengths Hicks-Lincoln and Lincoln-
Burdell and the angle at Lincoln. The angles atRed of Hicks-Red-Lincoln and Lincoln-Red-Bur-
dell are observed. The triangles to be solved are
Red-Lincoln-Burdell and Red-Hicks-Lincoln. The
triangles are written on the computation sheet
(fig. 153), using the first two sections of the form.The remainder of the form is used for the addi-
tional computations required to solve the problem.
The observed angles at Red are entered;
226
PROJECT
TRIANGLE COMPUTATION (FOR CALCULATING MACHINE)(TM 5-237)
LOCATION ORGANIZATION DATE
SYMBOL STATION OBSERVED ANGLE CORR'N SPNER'L SPHER'L PLANEANGLE EXCESS ANGLE SINE DISTANCE SIDE
1 64 27 11 __ 2-3
2 1-3
.old3 )S 12 /9.61-2.
D=Ratio, side/sine
1 2-3
2 2-3
3 1-2
D= Ratio,, side/sine
C I a/sin Abs B/ Ca .t 2-3
Gie2 30 anges 15. Bid Given 2 side and incude anl& 4*l. '3
3 11 sin B1 sin ass ) CHECKED BY 1-2 DATE
n 2 de9 6 DRatio; sidesine
1 2-3
2 1-3
3 1-2
D=Ratio, side/sine
Case I a/sin A=b/in' B~c/sin C Case III tan A-a sin B/c--a cos B COMPUTED BY DATE
Given : 3 angles, 1 side Given : 2 sides and included angle w" C. awmer*~ J3 d.5 S
Case II sinl B=b sin A/a CsIVcsA2s-)/e-s=2(bc) CHECKED BY ecc DATE S
eGiven: 2 ssdek''gild an angle oppoosite Given: 3 sides
DA 1 F 57 1 920
Figure 15. Computation of three-point problem (DA Form 190.)
64'27'11"3, angle A in the triangle Red-Lincoln-
Burdell; 30o41'59"17, angle B in the triangle Red-
Hicks-Lincoln. The known lengths are Lincoln-
Burdell, 9477.507 meters (side a) and Hicks-
Lincoln, 4870.241 meters (side b). The sines of
the known angles are entered on the form and the
ratios, side/sine=D, are found for each triangle.
For the triangle Red-Lincoln-Burdell, this ratio is
9477.5070.90223288 equals 10504.502. For the
triangle Red-Hicks-Lincoln, the ratio is 4870.241
-0.51054167 equals 9539.360.
J. The computation now shifts to the third sec-
tion of the form. The fixed angle at Lincoln,108o09'34".3 is entered and the sum of A4-B+Cis found to be 203°18'45"3 (64027'11'.3±30041'
59"7-x108°09'34"3). This sum is subtracted
from 360° and the answer 156°41'14".7, entered on
the form. The sine and cosine of this angle areentered on the form as shown in the example (sine
equals +0.39574720, cosine equals -0.91835949).
Notice the minus sign on the cosine because the
angle 360°- (A+B+C) is between 900 and 2700.The ratio of triangle 1 (Red-Lincoln-Burdell) is
divided by the ratio of triangle 2 (Red-Hicks-
Lincoln), i.e., 10504.502---9539.360, and the result
1.10117471 is added algebraically to the cosine of
360°-(-A+B+C) which is -0.91835949. The
result of this algebraic addition is +0.18281522.
The sine of 3600 - (A+B+ C) which is
+ 0.39574720, is divided by the result of the addi-tion + 0.18281522, to obtain the tangent of x which
equals +2.16473880. The plus sign on the tan-
gent indicates the angle is between 00 and 900 or
180° and 270°. Since angle x must be less than
360- (A+B+C), which is 156°41'14"!7, the
angle in this example must be between 00 and 90 .
The angle corresponding to the tangent of
+2.16473880 is found to be 65'12'19N'6. Fromf
the schematic diagram, the angle x is seen to be
the angle at Burdell in the triangle Red-Lincoln-
227
Burdell. The angle x, 65012'19"6, can now be
entered in the first triangle, and the third angle of
the triangle concluded to be 50020'29'.1. Angle y
is found by subtracting angle x from the angle
3600-(A+B+C), i.e., 156041'14"7 minus 65012 '
19'6 equals angle y, 91028'55.1. Again from the
schematic diagram, angle y is seen to be the angleat Hicks in the triangle Red-Hicks-Lincoln, and
can be entered in the second triangle. The con-
cluding angle in the second triangle is found to be
57°49'052. The two triangles can now be solved
by the usual method. The common side Red-
Lincoln in the two triangles should check, and the
sum of the two concluded angles should equal the
fixed angle C. In this example, the length of the
common side as solved in the two triangles is
9536.169 and 9536.169, which checks within the
accuracy of the computation. The sum of the
concluded angles is 108 009'34"3, which checks the
fixed angle at Lincoln.
g. If the new point lies near the circle passingthrough the 3 fixed points, the solution is weak.If the new point lies on the circle or A+B+C==
1800, the solution is indefinite.
Hicks
80. Inaccessible Base Computationa. The same quadrilateral selected from the sam-
ple triangulation net for the three-point problem
solution will again be used, but this time it will be
solved as an inaccessible base problem (fig. 154).
This problem deals with the solution of two un-
known angles in a quadrilateral which are desig-nated x and y. This situation arises when two
stations of unknown position, and unknown
azimuth and distance between them, are mutually
intervisible, and from which the stations at each
end of a line of known length and azimuth can be
observed. In the example, Red and Burdell
are the unknown stations. Hicks and Lincoln
designate the ends of the line of known length
and azimuth. All the angles at Red and Bur-
dell are known by observation. The angles
Red-Hicks-Burdell and Red-Lincoln-Burdell
are concluded in their respective triangles.Burdell-Hicks-Lincoln is the unknown angle x, andHicks-Lincoln-Red is the unknown angle y.
b. The solution to this problem is best performedon DA Form 1930 as case 2 (fig. 155). Angles Aand F on the form are the concluded angles at
Lincoln
7
N7
Figure 154. Sketch, inaccessible-base problem.
BurdellRed
228
PROJECT SPECIAL ANGLE COMPUTATION1-24 (TMf 5-237)
LOCATION DATE
Ca/lornia 2AqSORGAN IZATION~ CASE USED
2__ __ _ __ _ __ __ _ Q. Z.~ s.
b
a G, b G B 77b
yX "' _ a , b
X1
A B
QE A DO
DC
Case 1 11GCase 2 Case 3
sin x =b sin A tn ABa sin x .sin Asin Csin E _ assin x b sin Asin C =tansiy~ aimRsny sin B sin D sin~tf~~~n sn si D
THREE-POINT PROBLEM INACCESSIBLE BASE PROBLEM SPECIAL ANGLE PROBLEM
(Case 1: 180°-z (A+B+G) = o p
1 x Y Case 2: 53(C24) 1.5
Case 3: 270O- Z (A-+B+C+T)+G) .Leave blanks below here for values not involved in the CASE used.
1og b log a
log sin A *824475log sin B 97832
log sin C g93 85log sin D &*8d'51oIlog sin E 9. -588 7251? log sin F g 886 44668
* S u m 9 3 7 3 4 9 1 0 9 * 0 S u vm- --
9 .3 7 j 9 1 0 9
log tan a ________________log tan a 6 o938oa0 P n .
_____ _______________ _____ 48 16 02.436
a-45O a-45°_ 03 16 02.436
log tanj(x~y) log tanj(x+y) . T33 8log tan (a-450) ____________log tan (a-45°) I 8. - 56 5432 7
Sum=log tanj4~c-y) Sumn=log tanj(y-x) 8 8975V~x-V) 0 " Vy-x) 04 0 23 5.4
i(x+y) i(y+x) -53 24 51."5X y 4 39Y __ _ _ _ __ _ _ _ ___ ___ __ ___ ___ __
a is an auxiliary angle needed only for the. computation: it is always between 450 and 900
* Where ®Q is greater than ® use only the left side of the form below here, and vice versa.
COMPUTED BY DATE~. 1 CHECKED BY 4C DAT~
DA.,FE FSM57930Figure 155. Computation of inaccessible-base problem (DA Form 1930).
229
Hicks and Lincoln, respectively. Angles B and Care the observed angles at Red, and angles D andE are the observed angles at Burdell. Find 2the sum of angles C +D and enter on the formas )(x+y). From the table of logarithms oftrigonometric functions, find the log sines of theangles and enter them in the designated spaces.Log a and log b do not enter this computation.Add log sin A, plus log sin C, plus log sin E, toobtain sum 0. Add log sin B, plus log sin D,plus log sin F, to obtain sum 0. Subtract thesmaller sum from the larger, in this case ( from
Q. The left side of the form is now ignored for
the remainder of the computation. Subtractingsum from sum ( gives log tan a. In thetable of logarithms of trigonometric functions, findthe angle (a) corresponding to log tan a. From
this angle subtract 450. From this table, find log
tan 3(C+D) and log tan (a-45°). Add these
two logs to obtain log tan ) (y-x) [this sum is log
tan 32(x-y) on left side of form]. Find the anglecorresponding to this log tan which will be angle
2 (y-x). To find angle y, add the angle Y(y-x),which was just obtained, to the angle % (y+x),which was computed previously as %(C+D).Angle x is found by subtracting the angle 3 (y-x)from the angle 3 (y+x). This completes thecomputation for the unknown angles. The tri-
Black a .3.8080991 Hicks
angles can now be computed by the usual method
employing the law of sines.c. It is emphasized that this solution may check
within itself, as was the case with the solution ofthe three-point problem, but still be incorrect.An error in the angle Y (y-x) [or 3~(x-y)] willmake compensating errors in angles x and y, andthe sum of the angles in the triangles or in thequadrilateral will still check.
81. Special Angle Computation
a. A sample figure (fig. 156) for solution by case3, DA Form 1930 (Special Angle Computation)
(fig. 157) was taken from the sample triangulationnet. Again the problem is to solve two unknownangles, this time in a five-sided figure. This typeof figure is not as common as the three-point orinaccessible base problems, but the occasion mayarise where this figure can be observed and theothers cannot.
b. As in the inaccessible base problem, twostations are intervisible, but no distance can beobtained between them. From each of these twostations, only two of the three available fixedstations can be seen. The sketch (fig. 156)illustrates the fixed lines Hicks-Black and Hicks-Lincoln and the fixed angle (G) at Hicks. Theangles at Red and two angles at Burdell are
b
3.68755045 Lincoln
IGC'oo0
10
"
,1
,.
RedFigure 156. Sketch, special angle problem.
Burdell
230
-"\
LOATO ATE
D c
Cas 1 Cas 2 as 3
xiy1' syn sisinsn YiysTHEEPONTPRBEMINCCSSBE AS PQBE SPCA ANL RBE
Lev lnsblwhrefrvle1o nole nteCS sd
Iog 367B.slg 3pplog inA9 7~3 723 lg snB g9984/E
log z b sinE __tan__ A__sin___siAsCinE__ long sin i AsnC1
log tn £ og tn ~ 0 08/85 7
lo t(x+y) lo Case2: 1x(CD))
log tan A 40 __________ log tan B~5O 98 9 7g
Slog tanj-) log tan 0y-x)57
lo aj(x-y) lo tsj(y) 20,7605 4.2
I(X+Y) 4(y+x) 80 03 44.2
y __ _ _ _ __ _ _ _ __ _ __ _ __ __7 _ _
a is an auxiliary angle needed only for the. computation: it is always between 450 and 908
*Where ® is greater than Q use only the left side of the form below here, and vice versa.
COMPUTED BY Q'.DATE J CHECKED BY DATE
W ?3Ay SS ll. C'. A.
IAFORM Q2E S
Figure .157. Computation of special angle problem (DA Form 1930).
231
observed. The lines Red-Lincoln or Burdell-
Black cannot be observed, so the inaccessible base
solution is not applicable. Angle Hicks-Black-Red
is the unknown angle x, and angle Hicks-Lincoln-
Burdell is the unknown angle y. The angles are
lettered A, B, C, D, and G as shown on the sketch.The fixed length Hicks-Black is designated by theletter a, and the fixed length Hicks-Lincoln by theletter b. Solve for the angle 3 (x+y) for case 3 onDA Form 1930 as 270°--(A+B+C+D+G).
Enter the logarithms for the lengths a and b and
the log sines of angles A, B, C, and D in theirdesignated places on the form. Log sin E and logsin F do not enter this computation, and thesespaces are left blank. Log b plus log sin A plus logsin C equals sum 0. Log a plus log sin B plus log
sin D equals sum@. Subtract the smaller sum fromthe larger sum, in this case sum ( is subtracted
from sum @. From this point on, the left side ofthe form is not used. The result of this subtrac-tion is log tan a. a is an auxiliary angle neededonly for the computation. Using log tan a, findangle a from a table of logarithms of trigonometric
functions. Subtract 45 from a and find the log
tan of this new angle. Find the log tan of the
angle % (x-+y). The sum of log tan 2 (x+y) plus
log tan (a-45 ) equals log tan 2 (y-x). From the
tables obtain the angle 32(y-x). Add the angles
h (y-x) and 2 (y+x) to obtain angle y. Subtract
angle M(y-x) from angle 2(y+x) to obtain angle x.
c. This completes the computation on this
form. Now two angles in each triangle are known
and all of the triangles can be computed by the
sine law, starting with the triangles containing the
fixed lengths a and b.
Section VI. SHORE-SHIP TRIANGULATION
82. Description
The shore-ship method of triangulation estab-lishes distances between shore-traverse stationsthat cannot be measured directly. The formulaused is-
b' b sin A sin (A'+C')sin A' sin (A+C)
Where b is a known distance, b' is the unknowndistance, and angles A, C, A', and C' are theobserved angles at the shore stations. Figure 158illustrates the problem.
83. Computation
The computation is performed in four steps.The procedure is: first, abstract the directions
Ship
/ I \
Length Unknown Length
Figure 158. Sketch, shore-ship triangulation.
from the field books; second, perform the lengthcomputation to find the values for log X, where-
Xsin A sin (A'+C')= sin A' sin (A+C) '
third, compute the probable error to establish the
rejection limit for values of log X, and final log X;
and fourth, using the final log X, compute log b',and interpolate the length of b' from the tables.A correction will be applied to this length when
the error in the traverse distances is determined
by a check on another known' length (base).
a. Certain specifications must be followed toinsure third-order results in this method. These
are-
(1) Angles will be abstracted and computed
to the nearest 1".
(2) Logarithms for the individual computa-
tions will be taken to five decimals, the
computations using meaned logarithms
to six decimals. The angular argument
for these logarithms can be rounded off
to the nearest 10" before entering the
tables.
(3) The rejection limit for log X will be deter-
mined by the rule: Reject any log X
whose residual exceeds three times the prob-
able error of a single measurement, when
the probable error is found by the follow-
232
2
/
2131.37 Beta
Figure 159. Sketch, complete traverse (shore-ship).
ing method: Find the mean of all log X's,
and find the residual for each. Compute
the probable error of a single measure-
ment by the following formula:
E=0.6745 yzn- 1
Where: E probable error of a single
observation.
vresidual or algebraic differ-
ence between the mean of a
set of log X's and the log X
for each pointing.
2;v 2=sum of the squares of the
residuals.
n=number of observations.
(4) The lengths between traverse stations are
adjusted by-
(a) Determining the discrepancy between
the computed and the fixed length of
the closing base.
(b) Dividing the sum of the lengths of the
traverse into the discrepancy to fur-
nish a correction factor.
(c) Multiplying the correction factor by
each length accumulatively to deter-
mine the correction for each course,
the final course having the whole
correction.
b. Figures 159, 160 (DA Form 1928, Abstract
of Angles, Shore-Ship), 161 (DA Form 1929,
Shore-Ship Length Computation), and 162 illus-
trate the computations performed to determine
the length of the course Beta-Gamma from the
fixed length Alpha-Beta. The corrections applied
to the length was computed from the complete
traverse (fig. 159) using the rule stated previously.
c. After the lengths have been computed, they
are used with the traverse angles to perform a
regular traverse computation.
i1% PROJECT ABSTRACT OF ANGLES, SHORESHIP//B B\ Moed(T 52V
/ 1 LOCTIONSHIP(NAME AND/OR NUMER)
A. ORGANIZATION SHIP POSITION NUMBER DATE2
A -A
RECORDED ANGLE SHORE-SHIP ANGLE SHORE ANGLE ANGLE C ANGLE.C SHORE-SHIP ANGLE
323 A35' N 055 490 O 22' 57" 32 23' 39'
-p55 .2/ 05 I /9 47 /8 59 22 1N4/ 43 18/17 "22 58 /1549 20.5
I/
43/14 /6 49* 24 4/ 14 051 2fl 7
to 219 38 /2- 08 20 .15
49 04 13 56 N28 /2 JO 34 /9 44I/4d 58 1/1 02 3/ 42 07 -04/9/
40 25 ' A? 46 06 00 9 /NSO .59 09 04 " 34 M (4'26 /9i452 29 07 3/ 375 V_ 0 S4 /8 52
559 42 38 0 00 43 /2A
58 04 0 56 a ~547 86 M S /5 40
59 43 W /7 a 42 24 5622 /5 M9
3240/ so 36.58T/0 II 45 09 5337 /451
053S4 26- u 48 38 ____ %.. 12 570)7 58 520 I~~J
COMPUTED BY DATEJ .04fs CHECKED BY 1DATE
DA, FORM ' 9FEB 57192
Figure 160. Abstract of directions for shore-ship triangulation (DA Form 192~8).
234
PAFORM 19ID FEB 071 92
ROJECT LOCATION ORGANIZATION oe I \RMT
A b Cc' '~ A' SECT. NR. FROM (STARTING POINT OF TRAVERSE) TO (ENDING POINT OF TRAVERSE) DATE
POSITION (SHIP) (/) (/)b. ) c ) / d ) / e ( / ) f ( ) )g ( )-h
angle at ( Aa~ ) 324.52 36 2L 0:A 3&JZ 17 36 6JL021 LaC angle at ( Seto ) gjS j5 41 , S1 7=2 l2 1M2 28 14
A+C &2 04-2.A 2J L.15 121 L. 2Z L2L.41L2 I2AL. /2L !a 44~Li.1A' angle at ( ) aaz.z2.2 1E 3JL1 .±IC' angle at ( Ase& d 2% Z&a8 54 97 & 05 97 ia~ 34j.QL.C
A'+}C' Li 16- 1L 4LO2 IN J6.I 42LL3'.2 dJ2 - lL 3J ILL.1 LL-2log sin A -2-..ZJS 9 .. . M -LL. LZZ188 11971&3L 9 YZ7/. I.ZZLQL
log sin (A'+C') 2!o22 9= &a~2 ~ a qt. ~ -2colog sin (A+C) aoz "AZL OOQL OODL. Z~f.oao2 22O6
colog sin A' Q7j Q77 QLL...2ZD.Log X 0,1-1 o. 051 0.030Q 52 ,52 -1
POSITION (SHIP) ) / i C/) k ) /) (/)m ( / II. ( /. ) o (t)pangle at ( AIpho2 3 0 4 A QOZ 3 3 0 11 .1M1 -2Q 'SfL 26J 4L 1 M s52
C' angle at ( )
A'agl t 3ILJ J. J62 3i25 1125 i2J 2L12 57 -3 2 JZC' angle at ( ) ~~~ ~6 1 Li ~ &L2
A'+C' LZLJ1LL LLLI L1LALLLILQL2 L.LIlog2 3 /9L &1917 /191 sin A9 11z Mz7zii 11. OA L2L. Id .32..
log sin (A+C
colog sin A' Q,214 0.ezw Q-7-6 226 0QI.28Q. 0,71 0.2
Mean Log X p Unadjusted length to BtGa mlog b .. 3R52Correction -1,328.2.2log b 3. A1286 Adjusted length Rao to ,0~
COMPUTED mY DATE CHECK p.Y* Shor at0AGro C. Qr /G . Adjusted SoeTraverse Angle at to 170 i 4
(TM 5-237)
Figure 161. Length computation (DA Form 1929).
L aI 2
2 318, -5 __
3 ~3O ±QZ 49W ~325 42 __
6~ 332, *±Q 8
8 32L 02
A 316 -0Z A9J0 38 11 25_JL 326 +3 __
12 12 -11 L2L _
13 3&6 -67 AE14 -3/9 --0 _
/E -32R +05 25
Sum Q95165 X _____ ____
Figure 162. Computation of rejection limit of log X.
235
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