cos A,
cos 8 cos t = cos z cos ip + sin z sin cos/, (31)
sin z cos A = — sin + cosd sin ycost, (32)
I= cososin/1. (33)
These are of the same general form as (26), (27), and (28). Applying the
same principle as before, we derive
n sin N = sin 3,
n cos N = cos d cos /,
sin z sin A = cos d sin /, (34)
sin z cos A = « sin (4
log"
9 '7^73 5
A
9.51018
sin^-AO
9.90608
B
0.12180
cos(y-A^)
9-77273
sin z sin A
9.90894,
sin : sin .1
9.90894,
sin z cos A
9-67344
sinzcosyl
9-67343
tun A
0.23SSO*
tan A
0.2355U
A
300° 10' 3 1"
A
300° 10' 29"
sin A
9.93676,
sin A
9- 93677«
cos A
9.70126
cos A
9.70126
sin z
9.97218
sin z
9.97217
cos*
9.54007
cosz
9.54008
z (from cos,:)
69° 42 '32"
z (from cosz)
69° 42-30"
sin z
9.97218 Ck.
sin z
9.97217 Ck
TRANSFORMATIONS 31
Example 3. What Is the right ascension of an object whose hour angle it I7h2im34!6,
when the sidereal time is 2ihi4ni52!8?
By equation (35)
0 = 2ibi4n52!8
/ = 17 21 34.6
«= 3 S3 18.2, Ans.
Example 4. What is the hour angle of an object whose right ascension is 8hi2m34!8,
when the sidereal time is
By equation (36)
0 = & 6m28'7
a = 8 12 34.8
t — « 53 53-9> Ans.
15. Transformation of azimuth and altitude into right ascension and
declination, or vice versa. — These transformations are effected by a combina-
tion of the results of Sections 12-14. For the direct transformation, deter-
mine / and 8 by (26)-(28) or (29), and then a by (35). For the reverse
calculate / by (36), and then A and z by (3i)-(33) or (34).
Example 5. What is the right ascension of the object whose coordinates, at the sidereal
lime I7h2ilni6*4, are those given in Example I?
The hour angle found in the solution of Example i by equations (26)-(28) is 4h47m46'4.
This, combined with ff = I7h2imi6!4 in accordance with equation (35^, gives for the required
right ascension I2ll33nl3o!o.
Example 6. At a place whose latitude is 38° 38' 53", what are the azimuth and zenith dis-
tances of an object whose right ascension and declination are 9k27rai4'2 and — 8° 31 '47", re-
spectively, the sidereal time being 5h46m56«o? .
By equation (36;, / = 2ohi9m4i«8. We have, further, t = — 8° 31 '47" and
cos* — - —£, (39)
cos 3 cos if
will usually give satisfactory results. In any case, (39) affords a valuable con-
trol upon the value of / given by (38). The numerator of the right member of
(39) is readily calculated by means of addition-subtraction logarithms.
17. Application of transformation formulae to the determination of
latitude, azimuth, and time. — It was shown in Section 4 that the solution of
the fundamental problems of practical astronomy requires the determination
of the position of the axis of the celestial sphere and the orientation of the
sphere as affected by the diurnal rotation. In practice this is accomplished
indirectly by observing the positions of various celestial bodies with respect to
the horizon, the observed data being combined with the known position of the
bodies on the sphere for the determination of the position of the sphere itself.
The means for effecting the coordinate transformation hereby implied are to
be found in the formulas of Sections 12-16.
Although the most advantageous determination of latitude, azimuth, and
time requires a modification of these formulas, it is, nevertheless, easy to see
that the solution of the various problems is within our grasp, and that the
TRANSFORMA TIONS— GENERAL DISCUSSION
Example 7. For a place whose latitude Is 3S°56'si", find the hour angle, azimuth, and
parallactic angle of an object east of the meridian whose declination and zenith distance are
— 8° 16' 14" and 54° 16' 12", respectively.
Equations (37) are used for the solution, which is given below in the column on the left.
If only the hour angle were required, equations (38) or (39) would be used. As an illustra-
tion of the application of these formula;, the problem is also solved on this assumption. The
first ten lines of the computation for (38), being the same as that for (37), are omitted. The
remainder of the calculation for (38) occupies the upper part of the right-hand column. The
solution by (39) is in the lower part of this column. The object is rather too near the merid-
ian for the satisfactory use of equation (39), although it happens that the resulting value of
the hour angle agrees well with that from (37) and (38).
b
c
2i
s
s - a
s-6
i - c
sin (s - a)
sin (i-*)
sin (s -c)
cosec s
log /T2
log A"
tan %t
cot Yi A
tan % q
tan )4 t cot % A tan
K cosec s
—8° 1 6' 14"
38 5° 51
54 16 '2
98 16 14
5i 3 9
2°3 35 35
101 47 48 Ck.
47 3" 36
3 3i 34
50 44 39
9.86782
8.78890
9.88892
0.00927
8-55491
sin (s-6)
•In (i-c)
cosec (s-a)
cosec s
tan* y« t
8.78890
9.88892
0.13218
0.00927
8.81927
9.40964,
165° 35' 46"
33i ii 32
9.27746*
sin ,;
9-i5790«
9.40964,
sin y
9.79838
0.48856,
cos S
9-99546
9-38854,
COS if
9.89082
9.28674,
cos z
9.76639
9.28673, Ck.
sin r. sin v-
8.95628,
'65° 35' 46"
A
9.18989
162 o 48
B
0.06252
166 15 10
cos z — sin S sin y
9.82891
33i ii 32
COS S COS if
9.88628
324 i 36
cos /
9.94263
332 30 20
I
331° n' 34"
t
A
g
adaptation of the equations to any special case is only a matter of detail.
Consider for a moment either equations (26)-(28) or (3 1)-(33). Both groups
involve the five quantities A, z, /, 3, and ip\ but, since /=#-«, we may regard
them as functions of the six quantities A, z, a, 8, 183* the negative value of Table III be employed,
together with the value of Roo for l\\z following Jan. o.
If a somewhat greater uncertainty is permissible, the result may be more
expeditiously found by using 4m(i — 1/70) for III. If D be reckoned from the
nearest Jan. o as above, the corresponding error will not exceed 3".
Example 18. Find the right ascension of the mean sun for the epoch 1907, June 16
8h 2im 14-00 Columbia M. S. T.
By Equation (54)
^?o = Sh 34m 25*'0 (Eph. p. 93)
/. = 6h 9m i8'33 ll\L = i 0.67 (Eph. Table III)
Af=8 21 14.00 lllAf = i 22.34 (Eph. Table III)
R = 5 36 48. 1 1 Ans.
By Equation (56)
(D + M) = i67<)348 (Tables III and IV; ^?0o (1907) = i8h 36™ 0-5 (Table II)
4m (D + M) = 6697392 lllL = i 0.7
1/70 X4m (£> + M) = 9TS63 III(Z> -f- M) = 10 59 49.7
R= 5 3° 5'
Example 19. Find the right ascension of the mean sun for the epoch 1909, Sept. 21
26m 2 + A/) =— i77-?256 IIIL = i 0.7
1/70 X4mOD + M)=- 2.532 lll(D + M) = —2 54 43.4
0 = 8 10 55 Ans.
By equation (60)
i8h 37T7
M= 16 27.5
4m(i — 1/70) (Z> + .W) = — 2 54-7
# = 8 10. 5 Ans.
30. Given the sidereal time at any instant to find the corresponding
mean solar time. — We make use of equation (50), viz.
M=6 — R
Substituting as in Section 29 we have
M=O — RL — IUM
or
0 — XI. (61)
TRANSFORMATION OF SIDEREAL INTO MEAN SOLAR TIME 49
Multiplying equations (45) and (46), member by member, and dropping the
common factor 7m ft we find
(I + III)(I — II) = I
Combining this with (61) we find
M = d — RL — 11(0 — RL), (62)
where, as before,
RL = R0 + III/. (63)
Equations (63) and (62) solve the problem.
Equation (62) is susceptible of an interpretation similar to that given (58)
in the preceding section. Since 0 is the given sidereal time, and RL the sidereal
time of the preceding mean noon, 6 — RL is the sidereal interval that has elapsed
since noon. To find the equivalent mean time interval we must, in accordance
with equation (46), subtract from d — ^L the quantity 11(0 — RL). The right
member of (62) therefore expresses the number of mean solar hours, minutes,
and seconds that have elapsed since the preceding mean, noon, i.e. the mean
solar time corresponding to the given 0.
Example 21. Given, 1908, May 12, Columbia sidereal time ih7m 19*27, find the corre-
sponding central standard time.
By equations (62) and (63)
0 = jii jm 19*28
RL = 3 20 25.46
9 — RL = 21 46 53.82
•RL) = 3 34-«o (Eph. Table II)
M= 21 43 19.72
L = 9 18.33
C. S. T. = 9 52 38.05 A.M. May 13. Ans.
CHAPTER IV
INSTRUMENTS AND THEIR USE
31. Instruments used by the engineer. — The instruments employed by the
engineer for the determination of latitude, time and azimuth are the watch or
chronometer, the artificial horizon, and the engineer's transit or the sextant. The
following pages give a brief account of the theory of these instruments and a
statement of the methods to be followed in using them.
The use of both the engineer's transit and the sextant presupposes an under-
standing of the vernier. In consequence, the construction and theory of this at-
tachment is treated separately before the discussion of the transit and sextant
is undertaken.
TIMEPIECES
32. Historical. — Contrivances for the measurement of time have been used
since the beginning of civilization, but it was not until the end of the sixteenth
century that they reached the degree of perfection which made them of service
in astronomical observations. The pendulum seems first to have been used as a
means of governing the motion of a clock by Biirgi of the observatory of
Landgrave William IV at Cassel about 1580, though it is not certain that the
principle employed was that involved in the modern method of regulation. How-
ever this may be, the method now used was certainly suggested by Galileo about
1637; but Galileo was then near the end of his life, blind and enfeebled, and it
was not until some years later that his idea found material realization in a clock
constructed by his son Vincenzio. It remained for Huygens, however, the Dutch
physicist and astronomer, to rediscover the principle, and in 1657 give it an appli-
cation that attracted general attention. Some sixty years later Harrison and
Graham devised methods of pendulum compensation for changes of temperature,
which, with important modifications in the escapement mechanism introduced by
Graham in 1713, made the clock an instrument of precision. Since then its devel-
opment in design and construction has kept pace with that of other forms of
astronomical apparatus.
The pendulum clock must be mounted in a fixed position. It can not
be transported from place to place, and it does not, therefore, fulfill all the
requirements that may be demanded of a timepiece. By the beginning of the
eighteenth century the need of accurate portable timepieces had become pressing,
not so much for the work of the astronomer as for that of the navigator. The
most difficult thing in finding the position of a ship is the determination of longi-
tude. At that time no method was known capable of giving this with anything
more than the roughest approximation, although the question had been attacked
by the most capable minds of the two centuries immediately preceding. The
matter was of such importance that the governments of Spain, France, and the
Netherlands established large money prizes for a successful solution, and in 1714
that of Great Britain offered a reward of £20,000 for a method which would give
the longitude of a ship within half a degree. With an accurate portable timepiece,
50
TIMBI'lECES 51
which could be set to indicate the time of some standard meridian before begin-
ning a voyage, the solution would have been simple. Notwithstanding the stimu-
lus of reward no solution was forthcoming for many years. In 1735 Harrison
succeeded in constructing a chronometer which was compensated for changes of
temperature; and about 1760 one of his instruments was sent on a trial voyage
to Jamaica. Upon return its variation was found to be such as to bring the
values of the longitudes based on its readings within the permissible limit of
error.
The ideal timepiece, so far as uniformity is concerned, would be a. body moving
under the action of no forces, but in practice this can not be realized. The
modern timepiece of precision is a close approximation to something equivalent,
but falls short of the ideal. Thus far it has been impossible completely to nullify
the effect of certain influences which affect the uniformity of motion. Changes
in temperature, variations in barometric pressure, and the gradual thickening of
the oil lubricating the mechanism produce irregularities, even when the skill of
the designer and clockmaker is exercised to its utmost. No timepiece is perfect.
We can say only that some are better than others. Further, it is impossible
to set a timepiece with such exactness that it does not differ from the true time
by a quantity greater than the uncertainty with which the latter can be determined.
Thus it happens that a timepiece seldom if ever indicates the true time; and, in
general, no attempt is made to remove the error. The timepiece is started under
conditions as favorable as possible, and set to indicate approximately the true
time. It is then left to run as it will, the astronomer, in the meantime, directing
his attention to a precise determination of the amount and the rate of change of
the error. These being known, the true time at any instant is easily found.
33. Error and rate. — The error, or correction, of a timepiece is the quantity
which added algebraically to the indicated time gives the true lime. The error
of a timepiece which is slow is therefore positive. If the timepiece is fast the
algebraic sign of its correction is negative.
The error of a mean solar timepiece is denoted by the symbol J7"; of a
sidereal timepiece, by Jt). To designate the timepiece to which the correction .
refers subscripts may be added. Thus the error of a Fauth sidereal clock may
be indicated by Jtif; of a Negus mean time chronometer, by J7*N. Sometimes
it is convenient to use the number of the timepiece as subscript.
If 6' be the indicated sidereal time at a given instant, and JO the cor-
responding error of the timepiece, the true time of the instant will be
0 = 0' + JO'. (64)
The analogous formula for a mean solar timepiece is
T= T + JT'. (65)
The daily rate, or simply the rate, of a timepiece is the change in the error
during one day.
52 PRACTICAL ASTRONOMY
If the error of a timepiece increases algebraically, the rate is positive; if it
decreases, the rate is negative. The symbols fid and dT with appropriate sub-
scripts are used for the designation of the rates of sidereal and mean solar time-
pieces, respectively. The hourly rate,«>. the change during one hour, is some-
times more conveniently employed than the daily rate.
It is convenient, but in no wise important, that the rate of a timepiece should
be small. On the other hand, it is of the utmost consequence that the rate should
be constant; for the reliability of the instrument depends wholly upon the degree
to which this condition is fulfilled.
Generally it is impossible to determine by observation the error at the instant
for which the true time is required. We must therefore be able to calculate its
value for the instant in question from values previously observed. If the rate
is constant this can be done with precision; otherwise, the result will be affected
by an uncertainty which will be the greater, the longer is the interval separating
the epochs of the observed and the calculated errors.
If Jt} and Jt)' be values of the observed error for the epochs / and /', the
daily rate will be given by
in which /' — /must be expressed in days and fractions of a day. The rate having
thus been found, the error for any other epoch, t". may be calculated by the
formula
JQ" = J6' + 3d(t" — t') (67)
Example 22. The error of a sidereal clock was + 5™ 27561 on 1909, Feb. 3, at 6'.'4
sidereal time, and + 5™ 33510 on 1909, Feb. n, at 5^2; find the daily rate, and the correction on
Feb. 14 at 7^6 sidereal time.
We have J0 = + 5m 27-61, J0'= + 5m 33"°> and
/' — / = i id 5^2 — 3d 6^4 = 7d 22l'8 = 7<195.
Equation (66) then gives $0 = + 5H9/7-9S = -f 0:69, which is the required value of the rate.
To find the error for Feb. 14, 7*16, we have
t" — t' = i4d 7^6 — nd 5^2 = 3'! 2l'4 = 3
P.
M.
TV
4
3
1 6.
12
P.
M.
J7V
4
'3-
39
Arts.
Example 24. On 1907, Oct. 30, civil date, the Fauth sidereal clock of the Laws Observ-
atory read I4h 28™ 9575, when the Riggs clock, a central standard timepiece, indicated
oh ^m i7»oo P.M. The error of Riggs was +4:82; find the correction to the Fauth clock.
The reading of Riggs combined with its error by (65) gives the true C. S. T. From this
the Columbia M. S. T. is found by the second of (41). This converted into the correspond-
ing 0by (58) and compared with the reading of Fauth gives AOr-
In problems in which the given time is near noon, great care must be exercised in
determining the date for which J{0 is to be taken from the Ephemeris. In the present case,
the astronomical date for the goth meridian is Oct. 30, for the true C. S. T. shows that the
instant of mean noon had passed; but at Columbia mean noon had not yet arrived. Since
Ro is always to be taken from the Ephemeris for the precc ding local mean noon, the date to be
used is Oct. 29.
P.M.
P.M., Oct. 30, civil date
Oct. 29, astronomical
TK
o"
Sm 17500
AT*
+ 4.82
C. S. T.
o
S 21-82
L
9 18.33
Col. M. S.T.
23
56 3-49
/?L
«4
27 40.63
HIM
3 55-91
ft
14
27 40.03
Of
14
28 9-75
AOr
-29.72
AHS.
Example 25. When the error of a timepiece, a, is given and it is required to find the
corrections of two others b and c, the observations and reductions may be controlled by a
circular comparison, i. e. by comparing a and b, a and c, and 6 and c. The first comparison
leads to the error of b. The given error of a, and that calculated for b, may then be used to
reduce the second and third comparisons. Each of these leads to a value of the error of c and
the two results must agree within the uncertainty of the observations and calculations.
The Fauth sidereal clock, a Bond sidereal chronometer and the Gregg and Rupp central
standard clock of the Laws Observatory were compared in this manner on 1902, April 18.
The bracketed numbers are the results of the comparisons. Jg,. -= -f- im 14:3, find J#a and
#B ioh 8m 4550 1 TG*R 8h 26m 30-8 P.M. \ T\;»R 8h 28™ 25-0 P.M.
Br 10 7 27. i / yr 10 9 35.0 ) 0B 10 12 47.5
J0F 4 ' 14-3 J0F I 14-3 Jf*B - 3-6
B 10 8 41.4 .•< 10 10 49.3 H 10 12 43.9
A0e - 3.6 C. S. T. 8 35 4.0 C. S. T. 8 36 58.3
J7"c*K 48 33-2 JrG1R +S 33-3
CLOCK COMPARISONS 55
The second and third comparisons are reduced by the method used for Ex. 23. The
details of the conversion of 0 into C. S. T. are omitted. The two values of J7"c.»n present
a satisfactory agreement.
Example 26. Given thirty comparisons of a Wallham watch and a Bond sidereal
chronometer made at intervals of one minute; to find the rate per minute of the watch
referred to the chronometer, a precise value of the watch time corresponding to the first
chronometer reading, and the average uncertainty of a single comparison.
The interval between any two chronometer readings minus the difference between the
corresponding watch readings is the loss of the watch as compared with the chronometer
during the interval. The quotient of the loss by the interval in minutes is a value of the
relative rate per minute. Thus, if
/c = interval between two chronometer times,
7w = interval between two watch times,
R = relative rate of watch per minute,
then
The solution of the first part of the problem may therefore be accomplished by grouping
the comparisons in pairs and applying equation (a). The mean of the resulting values of K
will then be the final result. The selection of the comparisons for the formation of the pairs
requires careful attention if the maximum of precision is to be secured. To obtain a criter-
ion for the most advantageous arrangement, consider the resultant error of observation in R
when derived from equation (a). Denoting the influence of the errors in the observed watch
times upon the interval 7W by e we find for the error of /?
Since c is independent of the length of the interval separating the comparisons, it follows
from (b) that the precision of R increases with the length of this interval.
It is desirable for the sake of symmetry in the reduction that the separate values of R
should be of the same degree of precision; and it is important to arrange the calculation so that
any irregularity in the relative rate will be revealed. The reduction will then give not only
the quantitative value of the final result, but at the same time will throw light upon the reli-
ability of the instruments employed.
We are thus led to the following grouping of the comparisons: i and 16, 2 and 17, 3 and
18 ....... 15 and 30; or, in general, the ath comparison is paired with the(is + »)th. The
fourth column of the table gives the values of 7W corresponding to this choice. The first of
these is derived by subtracting the first T« from the sixteenth; the second, by subtracting the
second 7"w from the seventeenth, and soon. The 15 values of 7« substituted into equation
(a), together with the constant value 7C = 15™, would give 15 separate values for R. The first
of these would depend upon data secured during the first 15 minutes of the observing period;
the last, upon those obtained during the last 15 minutes; while the intermediate values of /,'
would correspond to various intermediate 15-minute intervals. Any irregularity in the rate
will therefore reveal itself in the form of a progressive change in the separate values of /?.
But, since 7C is assumed to be constant throughout, equation (a) shows that constancy of 7W
will be quite as satisfactory a test of the reliability of the timepieces as constancy in R. It is
not necessary, therefore, to calculate the separate values of the relative rate; and for the der-
ivation of the final result we adopt the simpler procedure of forming the mean of the values
of 7W, which we then substitute into (a) with /c = I5m. We thus find mean 7W = I4m 57*65,
whence the mean relative rate of the watch referred to the chronometer is 05157 per minute of
chronometer time.
56
PRACTICAL ASTRONOMT
WATCH AND CHRONOMETER COMPARISON
No.
0B
TV,
/w
(n-i)R
7'w
V
I
oh 45"' 0:0
I0h 25m 25;9
Hm 5756
O'OO
25:90
— !II
2
46 o.o
26 25.8
.6
o. 16
25.96
—•'7
3
47 o.o
27 25.7
• 4
0.31
26.01
— .22
4
48 O.O
28 25-5
•5
0.47
25-97
—.18
5
49 o.o
29 25.1
.8
0.63
25-73
+.06
6
50 o.o
30 25.0
•7
0.78
25.78
+ .01
7
51 o.o
31 24.8
.8
0.94
25-74
+ •05
8
52 o.o
32 24.7
•7
I.IO
25.80
— .OI
9
53 o-o
33 24.4
• 7
1.26
25.66
+•13
10
54 o.o
34 24.2
.8
1.41
25.61
+.18
ii
55 o-o
35 24- i
•7
i-57
25.67
+ .12
12
56 o.o
36 239
•7
i-73
25-63
+.16
'3
57 o.o
37 23.8
•7
1.88
25.68
+.11
H
58 o.o 38 23.7
•7
2.04
2.5-74
+ .05
15
.59 °-°
39 23-7
•4
2. 2O
25-90
— .11
16
I 0 O.O
40 23.5
15) 9-8
2.36
25.86
—.07
17
I O.O
4i 23.4
H 57-65
2-51
25-9I
—.12
18
2 O.O
42 23.1
15 o.oo
2.67
2.5-77
+.O2
'9
3 o.o
43 23.0
15) 2-35
2.83
25-83
—.04
20
4 o.o
44 22.9
./? = o!i57
2.98
25.88
—.09
21
5 o-o
45 22.7
3-H
25.84
—•05
22
6 o.o
46 22.6
3-30
25.90
— .11
23
7 o.o
47 22.4
3-45
25-85
—.06
24
800
48 22.1
3-6:
25-71
+.08
25
9 o.o
49 22. o
3-77
25-77
+ .02
26
10 o.o
50 21.8
3-92
25-72
+ .07
27
II O.O
51 21.6
4.08
25.68
+ .11
28
12 O.O
52 21.5
4.24
2.5-74
+ -05
29
13 o.o
53 21.4
4.40
25.80
—.01
30
14 o.o
54 21. i
4-55
25-65
+ .I4
Precise 1
Comp. /
o 45 o.oo
10 25 25.79
30)23.69
+ 1-36
—1-35
M0 =
25-79
Rem. = — o.oi +0.01
30) 2.71
Average Residual = ± 0509
An examination of the individual values of 7W for the given problem affords no certain
evidence of a variability of the relative rate.
As for the second requirement of the problem, it is evident that were the observations per-
fectly made, with a watch whose relative rate was zero, the seconds and tenths of seconds of all
the watch readings would have been the same. Had they been made with the same errors of
observation as actually occurred, but with a watch of zero relative rate, they would have differed
among themselves only by the errors of observation. The mean of all the seconds readings
CLOCK COMPARISONS 57
would then have given a precise value of the watch time corresponding to the first chronom-
eter reading. The given problem may be reduced to this case by correcting each watch
reading by the effect of the rate during the interval separating it from the first observation.
To accomplish this we have only to add to the readings, in order, the quantities o/?, \R, iff,
.... 29^?; or, In general, to the «th reading, (» — i)/f. The values of these corrections are
in column five of the table, and the watch times, corrected for rate, in column six. These
results are given to two places of decimals in order to keep the errors ot calculation small as
compared with the errors of observation. The mean of the values of TV, ioh 25m 25579, is the
required precise watch reading corresponding to the first chronometer reading, o'1 45m o'oo.
To obtain a notion of the uncertainty of a single comparison, consider the corrected watch
readings, TV- If the true value of R has been used in applying the corrections for rate, and
if the true value of the first watch reading were known, the actual error of this and of each of
the remaining readings could at once be found by forming the difference between the true
value and each of the corrected watch times. The average of the errors would then indicate
the precision of the comparisons. But the true values of R and of the first comparison are
not known and cannot be found. We must therefore proceed as best we may; and, accord-
ingly, we use for the true relative rate the value calculated above, and for the true value of
the first watch reading, the mean of all the corrected readings. The differences between each
corrected watch time and the mean of them all are called residuals. The residuals will differ
but little from the corresponding errors, for the calculated value of R and the mean TV will
differ but little from the quantities they are taken to represent. Although the average of the
residuals will not exactly equal the average of the errors, it may be accepted, nevertheless, as
a measure of the precision of the observations; for, barring a constant systematic error, it is
evident that the more accurate the observations, i.e. the smaller their variations among them-
selves, the less will be the average residual.
Denoting the residuals by r, and the mean of the corrected watch times by A/0, we have
. » = A/0 — 7"w (0
The values of i< formed in accordance with (c) are in the last column of the table.
A valuable control may be applied at this point. It is easily shown that if the exact value
of Af0 be used for the formation of the residuals, their algebraic sum must be zero. (Num.
Comp. p. 17.) If, however, an approximation for M0 is used, the algebraic sum of the resid-
uals will equal the negative value of the remainder in the division which gives as quotient
the value used as a mean.
In the present case the algebraic sum of the residuals is -fo.oi; the remainder is — o.or,
which checks the formation of the mean and ihe residuals. The average residual, without
regard to algebraic sign, is ± 0509. This we may accept as the average uncertainty of a single
comparison.
The principles illustrated in the preceding reduction find frequent application in the
treatment of the data of observation. The example is typical and the methods followed in the
discussion should receive careful attention. In particular, the grouping of the observations
for the determination of the mean value of R should be examined; and the student should
investigate for himself the precision of the result when such combinations of the comparisons
as I and 2, 2 and 3, .... 29 and 30; i and 2, 3 and 4, .... 79 and 30; i and 30, 2 and 28, ....
15 and 16; etc, are employed in place of that actually used.
Example 27. To determine the average uncertainty of a single comparison of two time-
pieces by the method of coincident beats.
Ten successive coincidences of the beats of a Bond sidereal chronometer with those of a
Gregg & Rupp mean time clock are taken as the basis of the investigation. The method used
for the reduction is similar to that employed in Ex. 26. The comparisons are in the second
and third columns of the table. Since the chronometer beats halt-seconds and the clock sec-
onds, the interval between the successive coincidences is that required for the clock to lose
58
PRACTICAL ASTRONOMY
0:5 as compared with the chronometer. Denote the true value of this interval by I. To ex-
hibit the influence of" the errors of observation we find what the clock readings would have
been had they all been made at the same instant as the first. This is done by subtracting
from the readings, in order, o7, if, 2/, . . . . g7. The numerical values of the corrections are
in column five, and the reduced clock readings themselves, in column six. The value to be
used for 7 is one-fifth of the average of the intervals between the «th and the («-j-5)th clock
readings. The individual values of these intervals are in column four. Their mean is
I4'n55!2, whence 7= 2m59!O4. The variations in the values of 7" represent the influence of the
errors of observation. The average residual for the reduced clock readings is ± 2*94, which
may be accepted as the average uncertainty of the time of a coincidence. Since the clock
loses i" in 358*, the corresponding average uncertainty of a comparison is ± o'ooS.
COMPARISON BY COINCIDENT BEATS.
No.
0
T
'
T'
r
i
iyh 35m 57?o
2" 6- 1*0
i4nl 4550
Om O!O
2h 5m 6o!o
- 3!3
2
38 52.5
8 55-o
62.0
2 59-o
56.0
+ 0.7
3
4i .53-0
'i 5.5-0
60.0
.5 58-1
569
— O.2
4
44 52-5
'4 54-o
54-o
8 57-i
56.9
— 0.2
5
47 49-0
17 50.0
55-°
ii 56.2
53-8
+ 2.9
6
.5° 44-5
20 45 o
5)276.0
'4 55-2
49-8
+ 6.9
7
53 57-°
23 57-Q
5)'4 55-2
'7 54-2
62 8
— 6.1
8
56 55-5
26 55.0
7^= 2 59.04
20 53-3
61.7
- 5-0
9
59 49 'O
29 48.0
23 52-3
.55-7
+ I.O
10
18 2 46.5
32 45-o
26 51.4
536
+ 3-'
10)567.2 + 14.6
Ma= 56-7 -H.8
Clock loses i1 in 358". Rem=-f-o.2 - 0.2
Average uncertainty of a single 10) 29.4
comparison =r ± 2«94/35S = ± o'ooS. Average Residual = ± 2*94
35. The care of timepieces — All timepieces should be wound at regular in-
tervals. They should be protected from moisture, electrical and magnetic in-
fluences, and extremes of temperature, especially the direct rays of the sun. They
yield the best results when at rest, absolutely untouched, except as winding
may be necessary. Portable instruments must not be subjected to violent shocks,
jolts, or oscillatory motions. Chronometers are particularly sensitive to such dis-
turbances, especially oscillations. Timepieces of this sort are usually hung in gim-
bals, mounted in a substantial wooden case. When at rest, or when subjected to
the long periodic motions of a ship, they should hang free in the gimbals in order
that the mechanism may remain constantly horizontal in position. When trans-
ported from place to place on land, the gimbals should be locked. Otherwise the
unavoidable jarring may produce oscillations sufficient to change appreciably the
error and the rate. If the journey is such that shocks can not be avoided, it is
safer to stop the instrument and insert thin wedges of cork between the balance
wheel and the supporting frame, using just sufficient force to hold them in place.
In this way the delicate pinions of the balance may be guarded from injury. The
IHtlilZON AND VERXIEli 59
chronometer, so far as possible, should be kept in a fixed position with respect
to the points of the compass.
THE ARTIFICIAL HORIZON
36. Description and use. — The artificial horizon consists of a shallow dish
filled with mercury. The force of gravity brings the surface to a horizontal posi-
tion, and the high reflective power of the metal makes it possible to see the various
celestial bodies reflected in the surface. Any given object and its image will be
situated on the same vertical circle, and the angular distance of the image below the
surface will be equal to that of the object above. The angular distance between
the object and its image is therefore twice its apparent altitude. Strictly speaking,
this is true only when the eye of the observer is at the surface of the mercury,
but for distant objects the error is insensible.
The measurement of the distance between the object and its image therefore
affords a means of determining the altitude of a celestial body, and in this con-
nection the artificial horizon is a valuable accessory to the sextant. It can also
be used to advantage with the engineer's transit for the elimination of certain
instrumental errors.
The artificial horizon is usually provided with a glass roof to protect the
surface of the mercury from disturbances by air currents. It is important that
the plates of glass should be carefully selected in order that the light rays travers-
ing them may not be deflected from their course. The effect of any non-parallelism
of the surfaces may be eliminated by making an equal number of settings with
the roof in the direct and reversed position, reversal being accomplished by turn-
ing the roof end for end.
THE VERNIER
37. Description and theory. — The vernier is a short graduated plate attached
to scales for the purpose of reducing the uncertainty of measurement. It takes
its name from its inventor, Pierre Vernier, who in 1631 described its construction
and use. In its usual form the graduations are such that the total number of
vernier divisions, which we may denote by n, is equal to « — i divisions of the
scale, the graduation nearest the zero of the scale marking the zero of the vernier.
The vernier slides along the scale, the arrangement being such that the angle,
or length, to be measured corresponds to the distance between the zeros of the
scale and of the vernier. When the zero of the vernier stands opposite a gradua-
tion of the scale, the desired reading is given directly by the scale. Usually this
will not occur, and the vernier is then used to measure the fractional part of the
scale division included between the last preceding scale graduation and the zero
of the vernier.
The difference between the values of a scale and a vernier division is called
the least reading = / of the vernier. If
d = value of one division of the scale,
d' = value of one division of the vernier,
60 PRACTICAL ASTRONOMY
then, for the method of graduation described above,
(n — i ) d = nd'
whence
'-*-*-; (68)
The least reading of the vernier is therefore i/tith of the value of a scale division.
Now, for an arbitrary setting of the vernier, consider the intervals between
the various vernier graduations and the nearest preceding graduations of the scale,
beginning with the zero of the vernier and proceeding in order in the direction
of increasing readings. The first interval is the one whose magnitude is to be
determined by the vernier. Denote its value by i'. Since a vernier division is
less than a scale division by the least reading, I, it follows that the interval between
the second pair of graduations will be v — /; that between the third v — 2/; and so
on, each successive interval decreasing by /. By proceeding far enough we shall
find a pair for which the interval differs from zero by an amount equal to, or
less than 1/2, a quantity so small that the graduations will nearly, if not quite,
coincide. Suppose this pair to be n' divisions from the zero of the vernier. The
value of the corresponding interval will be v — «'/=£, and we therefore find
p = «V+*. (69)
In practice we disregard s and use
v = n'l. (70)
To determine the value of v, therefore, we count the number of vernier
divisions from the zero of the vernier to the vernier graduation which most nearly
coincides with a graduation of the scale. The product of this number into the
least reading is the value of v. The final result is the sum of v and the reading
corresponding to the last scale graduation preceding the zero of the vernier.
In practice the actual counting of the number of divisions between the zero
of the vernier and the coincident pair is avoided by making use of the numbers
stamped on the vernier. These give directly the values of n'l corresponding to
certain equidistant divisions of the vernier. Usually one or two divisions precede
the zero and follow the last numbered graduation of the vernier. These do not
form a part of the n divisions of the vernier, and are therefore to be disregarded
in the determination of /. They are added to assist in the selection of the coin-
cident pair when coincidence occurs near the end of the vernier.
38. Uncertainty of the result. — The error of a reading made with a per-
fectly constructed vernier is £, whose maximum absolute value is 1/2. The uncer-
tainty of the result is therefore 1/2.
The gain in precision resulting from the use of the vernier may be found
by comparing the uncertainty of its readings with that arising when the scale
alone is used. The latter may be fixed at O.osd, as experience shows that this
UNC&RTAINTT OP VEKXIER HEADINGS 61
is approximately the uncertainty of a careful eye estimate of the magnitude of r.
The inverse ratio of the two uncertainties may be taken as a measure of the
increase in precision, whence we find that the result given by the vernier is approx-
imately ;//io times as precise as that derived from an estimate of the fractional
parts of a scale division. It appears, therefore, that a vernier is of no advantage
unless the number of its divisions is in excess of ten.
The use of a magnifying lens usually shows that none of the vernier grad-
uations exactly coincides with a graduation of the scale. With a carefully grad-
uated instrument, it is possible, by estimating the magnitude of e, to push the
precision somewhat beyond the limit given above. To do so it is only necessary
to compare e with the interval between the next following pair of graduations,
or with that of the pair immediately preceding, according as £ is positive or
negative. The sum of the two intervals to be compared is /. It is therefore
possible to estimate s in fractional parts of the least reading.
The condition that » divisions of the vernier equal ;i — I divisions of the
scale must be rigorously fulfilled if reliable results are to be obtained. The matter
should be tested for different parts of the scale by bringing the zero of the vernier
into coincidence with a scale graduation, and then examining whether the (»+r)st
vernier graduation stands exactly opposite graduation of the scale. Information
may thus be obtained as to the accuracy with which the graduation of the instru-
ment has been performed.
The vernier should lie, preferably, in the same plane as the scale, and, in
all positions, should fit snugly against the latter. In many instruments, however,
it rests on top, the plate being beveled to a knife edge where it touches the scale.
With this arrangement the greatest care must be exercised in reading to keep
the line of sight perpendicular to the scale. Otherwise an error due to parallax
will affect the result.
THE ENGINEER'S TRANSIT
39. Historical. — The combination of a horizontal circle with a vertical arc
for the measurement of azimuth and altitude is known to have been used by the
Persian astronomers at Meraga in the thirteenth century, and it is possible that
a similar contrivance was employed by the Arabs at an even earlier date. The
principle involved did not appear in western Europe, however, until the latter
half of the sixteenth century. There it found its first extensive application in the
instruments of Tycho Brahe, who constructed a number of "azimuth-quadrants"
for his famous observatory on the island of Hveen. The vertical arcs of Tycho's
instruments were movable about the axis of the horizontal circle, and were pro-
vided with index arms fitted with sights for making the pointings. The adjust-
ment for level was accomplished by means of a plumb line, the spirit level not yet
having been invented. Magnification of the object was impossible, as a quarter
of a century was still to elapse before the construction of the first telescope. The
instruments were large and necessarily fixed in position ; and, indeed, there was
no need for moving them from place to place as they were intended solely for
astronomical observations. Though primitive in design, they were constructed
62 PRACTICAL ASTRONOMY
with the greatest care, and were capable of determining angular distances with an
uncertainty of only i' or 2'. They are of interest not only on account of the re-
markable series of results they yielded in the hands of Tycho, but also because
they embody the essential principle of the modern altazimuth, the universal instru-
ment, the theodolite, the engineer's transit, and a variety of other instruments.
None of these modern instruments is the invention of any single person,
but rather a combination of inventions by various individuals at different times. The
telescope, first constructed during the early years of the seventeenth century, was
adapted to sighting purposes through the introduction of the reticle by Gascoigne,
Auzout, and Picard. Slow motions were introduced by Hevelius. The vernier was
invented in 1631, and the spirit level, by Thevenot, in 1660. All these were com-
bined with the principle of the early azimuth-quadrant to form the altazimuth,
which appears first to have been made in a portable form by John Sisson, an
Englishman, about the middle of the eighteenth century. At the beginning of the
nineteenth century the design and construction were greatly improved by Reich-
enbach, who also added the movable horizontal circle, thus making it possible
to measure angles by the method of repetitions. The universal instrument was
then practically complete, and the transition to the engineer's transit required
only the addition of the compass and such minor modification as would meet
the requirements of precision and portability fixed by modern engineering
practice.
For a detailed description of the engineer's transit, the student is referred
to any standard work on surveying. Certain attachments, notably the compass
and the telescope level, are not required for the determination of latitude, time,
and azimuth. On the other hand, it is desirable that the instrument used in the
solution of these problems should possess features not always present in the mod-
ern instrument. In particular, the vertical circle should be complete, and should
be provided with two verniers situated 180° apart. A diagonal prism for the
observation of objects near the zenith, and shade glasses for use in solar obser-
vations are a convenience, though not an absolute necessity.
40. Influence of imperfections of construction and adjustment. — It is assumed
that the student is familiar with the methods by which the engineer's transit may
be adjusted, and that observations will not be undertaken until the various adjust-
ments have been made with all possible care. But since an instrument is never
perfect, it becomes of importance to determine the influence of the residual errors
in construction and adjustment, and to establish precepts for the arrangement of
the observing program such that this influence may be reduced to a minimum.
In the instrument fulfilling the ideal of construction and adjustment, the fol-
lowing conditions, among others, are satisfied :
1. The rotation axes of the horizontal circle and the alidade coincide.
2. The planes of the circles are perpendicular to the corresponding axes
of rotation.
3. The centers of the circles lie in the corresponding axes of rotation, and
the lines joining the zeros of the verniers pass through the axes.
/.V.SV/iY.M/A'.V X1. 1 A BRRO/tS
63
4. The vertical axis of rotation is truly vertical when the plate bubbles are
centered.
5. The horizontal rotation axis is perpendicular to the vertical axis.
6. The line of sight, i.e. the line through the optical center of the objective
and the middle intersection of the threads, is perpendicular to the hori-
zontal axis.
7. The vertical circle reads zero when the line of sight is horizontal.
It is the task of the instrument maker to see that the first three of these con-
ditions are satisfied. The observer, on the other hand, is responsible for the re-
mainder.
No. I is of importance only in the measurement of horizontal angles by the
method of repetitions. The error arising in such measures from non-coincidence
of the vertical axes may be eliminated by the arrangement of the observing pro-
gram described in Section 47.
No. 2. It can be shown that the error due to lack of perpendicularity of the
circles to the axes is of the order of the square of the deviation. In well con-
structed instruments it is therefore insensible.
No. 3. If the third condition is not satisfied the readings will be affected by
an error called eccentricity.
0
Fig. 7.
In Fig. 7 let C be the center of the graduated circle OV^^ ; a, the point
where the rotation axis intersects the plane of the circle; O. the zero of the
graduations; and Ft and V '•> the< zeros of the verniers. The distance aC=e
is the eccentricity of the circle. The perpendicular distance of a from the
line joining Vl and V '„ is the eccentricity of the verniers. The reading of F,
is the angle OCF,, and of F,, OCV „. Denote these by R, and /?,, respect-
ively. The angles through which the instrument must, be rotated in order
that the zeros of the verniers may move from O to the positions indicated, are
/l=A1 and OaV,,=A.,, respectively. Al and A., are therefore to be regarded
64 PRACTICAL ASTRONOMY
as the angles which determine the positions of the verniers with respect to 0
for the pointing in question. The relations connecting At and A2 with the vernier
readings, Rl and Rs, are
A,=Rt + £. — £„
/*, = *, + £„ — £„ (72)
where £„, £u and £2 are tne corrections for eccentricity for the points O, V ^
and Vz. The mean of (71) and (72) is
y> (A, + A,) = % (Rt + R.) + E. + % (E, - E,). (73)
For any other pointing of the telescope, we have the analogous equation
A.') = #(*/+*.') +•£. + # (£.' — £,'). (74)
It is easily shown that £„—£, and £,'—£/ are of the order of tt"/r2, where
e' is the eccentricity of the verniers and r the radius of the circle. The last terms
°f (73) ar>d (74) are entirely insensible in a well constructed instrument. The
difference of (73) and (74) is therefore
XW+A;) - ^(At+At) = #(/?,•+*.•) - #(*.+*,). (75)
The left member of (75) is the angular distance through which the instru-
ment is rotated in passing from the first position to the second, and the equation
shows that this angle is equal to the difference in the means of the vernier read-
ings for the final and initial positions. The eccentricity is therefore eliminated
by combining the means of the readings of both verniers.
It can be shown that the eccentricity will also be eliminated by combining
the means of any number of verniers, greater than two, uniformly distributed
about the circle. In practice it is sufficient to use the degrees indicated by the
first vernier with the means of the minutes and seconds of the two readings.
Nos. 4 — 7. Horizontal Angles: In the measurement of horizontal angles
an error of adjustment in No. 7 has no influence. To investigate the effect of
residual errors in Nos. 4 — 6, let
*=inclination of the vertical axis to the true vertical,
90° — . y=inclination of the horizontal axis to the vertical axis,
^^inclination of the horizontal axis to the horizon plane,
90° -f- r=inclination of the line of sight to the horizontal axis.
The quantities b and c are the errors in level and collimation, respectively.
Then, in Fig. 8, which represents a projection of the celestial sphere on the plane
of the horizon, let Z be the zenith, Z' the intersection with the celestial sphere
of the vertical axis produced, O an object whose zenith distance is £0, and A
the intersection of the horizontal axis produced with the celestial sphere when
O is seen at the intersection of the threads. The sides of the triangles ZAZ'
INSTRUMENTAL ERRORS
65
and ZAO have the values indicated in the figure. Finally, let k, K and / be the
directions of ZA, ZO, and Z'A, respectively, referred to ZP.
Applying equations (13) and (15) to triangle ZAZ\ we find
sin b = sin/cos / -f cos/sin /cos/,
cos b sin £ = cos/ sin /.
(76)
(77)
In a carefully adjusted instrument i, j, and b are very small, and we may neglect
their squares as insensible. Equations (76) and (77) thus reduce to
(78)
(79)
Equation (13) applied to triangle ZAO gives
— sin c = sin £ cos za + cos £ sin za cos (AT
(80)
Since c and 90° — K ' + k are also very small, equation (80) may be written
— c = b cos 20 -f (90° — K + k) sin za
or
K — 90° — k = b cot za + c cosec z0.
(81)
Were there no errors of adjustment, the direction of A referred to P
would be AT — 90°. The direction given by the instrument, determined by the
angle through which it must be rotated to bring A from coincidence with ZP
to its actual position, is /. Since the verniers maintain a fixed position with re-
spect to A, the difference K — 90° — / represents the effect of the residual errors
5
66 PRACTICAL ASTRONOMY
on the horizontal circle readings. But by (79) l=k, sensibly, whence it follows
that the amount of the error is given by (81). If, therefore, R be the actual
horizontal circle reading, and R{,, the value for a perfectly adjusted instrument.
we have
R, = R + 6cotsQ + ccosecs0, C. R. (82)
in which the value of b is given by equation (78). Assuming that equation
(82) refers to that position of the instrument for which the vertical circle is on
the right as the observer stands facing the eyepiece (C. R.), we find by a
precisely similar investigation for circle left (C. L.),
b,=j— z'cos/, (83)
a, C. L. (84)
where Rt is the circle reading less 180°, and £,, the inclination of the horizontal
axis to the plane of the horizon for C. L. The mean of equations (82) and
(84) is
— *,) cot*., (85)
or, substituting the values of b and b, from (78) and (83)
Ro= %(R+Rt) + zcos/cot20. (86)
It therefore appears that the mean of the readings of the horizontal circle
taken C. R. and C. L. for settings on any object is free from the influence of
j, c, and the component of i in the direction of the line of sight, viz., » sin /.. More-
over, for objects near the horizon the effect of i cos /, the component of i par-
allel to the horizontal axis, is small, for it appears in (86) multiplied by cot "„.
If the instrument be provided with a striding level, the values of b and bl
may be determined by observation. Their substitution into (85) will then give
the horizontal circle reading completely freed from i, j, and c-
The readings may also be freed from the influence of b by combining the
results of a setting on O with those obtained by pointing on the image of 0 seen
reflected in a dish of mercury, both observations being made in the same position
of the instrument, either C. R. or C. L. The reflected image, O', will be on the
vertical circle through O, and as far below the horizon as 0 is above. Since the
horizontal axis is not truly horizontal, it will be necessary to rotate the instru-
ment slightly about the vertical axis in turning from 0 down to 0'. A will thus
move a small amount to a new position A'.
To investigate the effect of the errors for a pointing on O' we must therefore
consider the triangle A'ZO' in place of AZO in Fig. 8. The sides of A'ZO' are
ZA' = ZA = go°—b, A'0' = A0 = go°+c, and ZO' = 180°— s0. The angle at
•^is K — k' where fe' is the direction of ZA' referred to ZP. We then find, simi-
larly to equation (80),
INSTRUMENTAL ERRORS • <>7
— sin c — — sin bcos 50 -)- cos b sin *0cos (K — £'),
whence
K — 90° — k' — — ^ cot sa -\- c cosec sa,
and, finally, if /?' he the horizontal circle reading for the setting on O'
A>0 — R'—b cot z0 + c cosec s0. C. R. (87)
Equations (82) and (87) both refer to C. R. Their mean is
A'o = y2 ( /? -|- A") + c cosec za, C. R. (88)
By the same method we find from the reflected observation. C. L.,
A'0 = AJ, ' + l\ cot £„ — c cosec s0, C. L. (89)
in which /?,' is the circle reading less 180° for C. L. This equation combined
with (84) gives
.R, = 14 (A1, + A5,') - f cosec sa, C. L. (90)
Equations (88) and (90) show that the mean of the horizontal circle read-
ings for direct and reflected observations of an object in the same position of
the instrument is free from the influence of any adjustment error in level.
Finally the combination of (88) and (90) gives
A'0 =K(/?+ K' + A, + AY), (90
in other words the mean of the readings, direct and reflected, for both positions
of the instrument, is free .not only from b, but from the collimation error as well.
Vertical Circle Readings : To investigate the influence of i. j and c upon
the readings of the vertical circle, consider again Fig. 8. The true zenith dis-
tance of 0 is Z0 = z»; that given by the vertical circle readings is equal to the
angle Z'AO. From the triangle ZAO we find
cos sa = — sin /;sin c + cos b cos c cos (ZAO).
The squares and products of the errors of adjustment are ordinarily quite insen-
sible, whence we find with all necessary precision.
.?„= Angle ZAO,
Denoting the instrumental zenith distance Z'AO by s, we find £„ — £=angle-
ZAZ', and from triangle /.AZ'
cos 6s\n(ca — £) = sin /'sin /,
68 PRACTICAL ASTRONOMY'
or, since b, c0 — 2, and i are very small,
z0 = z + /sin /, C. R. (92)
A similar investigation gives for the reversed position of the instrument
£„ = £, + ?' sin/, C. L. (93)
in which s, is the instrumental zenith distance for C. L.
The angles s and £, are not read directly from the circles. The ordinary
engineer's transit reads altitudes, but if there is any deviation from the condi-
tion expressed in No. 7, the readings will not be the true altitudes, for they will
include the effect of the index error. If r and rt be the vertical circle readings
for C. R. and C. L., respectively, and / the reading when the line of sight is hor-
izontal, we have
z = 90° — r + /, C. R. (94)
^=90°-r,-/, C. L. (95)
Substituting (94) and (95) into (92) and (93)
,;o = 90° — r + / + /sin /, C. R. (96)
z0 = go° — 1\ — /-fz'sin/, C. L. (97)
The mean of (96) and (97) is
s0 = 90°— iX(r + r,)-Hsin/. (98)
For an instrument whose vertical circle is graduated continuously from o°
to 360° it is easily shown that the equation corresponding to (98) is
£„ = % (v, — •',) + / si n /, (99)
in which vt and v3 are the circle readings, the subscripts being assigned so that
»!— »1by 90°+^, and Z' a position Zt'. The triangle
ZA^O leads to an equation differing from (84) only in that bt is replaced by b.
The mean of the new equation and (82) is simply
R.= %(R + R,), (105)
where R and R1 are the horizontal circle readings; the latter having been reduced
by 180°. The result is therefore free from both b and c.
Again, from triangle ZA^Z^, we find for circle left analogously to (97),
z0 = 90°— r^— I— z'sin/, C. L. (106)
in which the vertical circle reading r1 is not the same as the rl of (97), for
(106) presupposes that the instrument is relevelled after reversal, while (97)
assumes that no change is made in the position of the vertical axis during the
observations. The mean of (96) and (106) is
^0 = 90— lA(r-\- r,), (107)
which is free from b, c, and /. For a circle graduated continuously we have
similarly,
2,= y2(vl—-i',) (1 08)
where as before the readings are to be taken in such an order that their dif-
ference is less than 180°.
It is assumed throughout that the pointings are always made by bringing
the object accurately to the intersection of the threads. It is important that this
be done, even though the threads be respectively horizontal and vertical; for
observing at one side of the field is equivalent to introducing an abnormal value
INSTRUMENTAL ERRORS— THE LEVEL 71
of the collimation, while pointings above or below the horizontal thread corre-
spond to a modification of the index error of the vertical circle.
41. Summary of the preceding section. — The preceding results may be
summarized as follows:
No. I. Non-coincidence of vertical axes enters only when the horizontal
circle is used by the method of repetitions. Error eliminated by proper arrange-
ment of observing program. See Section 47.
No. 2. Non-perpendicularity of circles to axes usually has no sensible
influence on circle readings.
No. 3. Eccentricity of circles and verniers eliminated by forming means
of readings of both verniers. See equation (75).
Nos. 4 — 7. Horizontal circle readings: Component of deviation of vertical
axis from vertical in direction of line of sight, non-perpendicularity of axes, and
collimation eliminated by forming mean of readings taken C. R. and C. L. Com-
ponent of deviation from vertical which is parallel to horizontal axis appears mul-
tiplied by cots0. See equation (86). Correction for the latter may be made by
observations with the striding level. See equation (85). All errors in Nos. 4 — 6
eliminated by forming mean of readings direct and reflected, for both C. R. and
C. L. See Equation (91). The error in No. / — index error of vertical circle —
does not enter. Vertical circle readings: All errors in Nos. 4 — 7 excepting com-
ponent of deviation of vertical axis from vertical in direction of line of sight
insensible or eliminated from mean of readings C. R. and C. L. See equation
(98) or (99). All errors in Nos. 4 — 7 insensible or eliminated from mean of
readings, direct and reflected, in same position of instrument. See equations
(103) and (104). Desirable to observe both C. R. and C. L., however, to reduce
graduation error of vertical circle.
All errors under Nos. 4 — 7 eliminated from mean of readings C. R. and
C. L. for both horizontal and vertical angles provided plate bubbles have same
position in tubes for both positions of the instrument. See equations (105)
and (107) or (108).
42- The level. — The adjustment of the engineer's transit with respect to
the vertical is usually made by means of the plate bubbles, any residual error
being eliminated by some one of the methods of Section 40. In some cases,
however, it is desirable to remove the effect of this error by measuring the
inclination of the horizontal axis to the horizon and applying a suitable cor-
rection to the circle readings. This method of procedure requires a knowl-
edge of the theory of the striding level.
The striding level is more sensitive than the plate bubbles, its tube is
longer, and the scale includes a larger number of divisions. It is made in two
forms, one with the zero of the scale at the middle of the tube, the other with
the zero at the end. Theoretically the two forms are equivalent. The adjust-
ment of the level tube within its mounting should be such that the bubble
stands at the middle of the tube when the base line is horizontal. The scale
reading of the middle of the bubble for this position is called the horizontal
reading. Owing to residual errors of adjustment, the horizontal reading will
7J PRACTICAL ASTRO NOMT
not usually be zero, even for the form in which the zero of the scale is at the
middle of the tube. Its value must be determined and applied as a correc-
tion to the scale readings, or else its influence must be eliminated. The latter
is easily accomplished by combining readings made in the direct and the
reversed position, reversal being made by turning the level end for end.
-Let d= the angular value of one division of the level scale.
h = the horizontal reading.
Further, for any inclination of the base line, let m' and m" be the readings of
the middle of the bubble, and b' and b" the corresponding observed inclina-
tions, for the level direct and reversed, respectively. Finally, assume that all
readings increasing toward the right are positive, and all toward the left, neg-
ative. We then find, whatever the position of the zero of the scale,
b'=(m'—h)d, (109)
b" = (m" — K)d. (i 10)
Since h has opposite signs for the two positions of the level, the mean of (109)
and (no) is
b=y2(m' + m")d, (in)
in which the mean of the observed inclinations has been written equal to b.
Denoting by r', /', and r", /", the readings of the ends of the bubble for two
positions, and writing
D=y4d, (112)
we find from (in)
("3)
This result depends only upon the readings of the ends of the bubble and
the value of one division of the scale, and is therefore free from the horizontal
reading. The convention regarding the algebraic sign is such that when b
calculated from (113) is positive, the right end of the level is high.
Since b' and b" are two observed values of the same quantity, we find from
the difference of (109) and (no)
h=^(r' +/' — /'—/"), (114)
which may be used for the calculation of h when a complete observation has
been made.
43. Precepts for the use of the striding level. — The level is a sen-
sitive instrument, and great care must be exercised in its manipulation if pre-
cise results are to be obtained. The inclinations to be measured should be
small and the horizontal reading should correspond as closely as possible with
the scale reading of the middle of the tube. The points of contact of the level
with the pivots upon which it rests must be carefully freed from dust particles.
THE LEVEL 73
The length of the bubble, which is adjustable in the more sensitive forms,
should be about one-third the length of the tube, and ample time should be
allowed for the bubble to come to rest before reading. The instrument should
be protected from changes in temperature, and, to this end, it should be
shielded from the rays of the sun, and from the heat of the reading lamp and
the person of the observer. The right end of the bubble should always be
read first, careful attention being given to the algebraic sign, and the time of
reversal for each observation should be noted. Mistakes in reading may be
avoided by noting that r' — /', the length of the bubble, must equal r" — /".
The following, in which 6' represents the sum of the four readings, is a
convenient form for the record:
Time
r' r
r" r"
r' +/" "I
r>' 4- /' \ -S, 6 = Sd.
S is most easily found by forming first the diagonal sums of the four
readings written as above, for both r' and I" and r" and /' will be opposite in
sign and approximately equal in absolute magnitude.
Example 28. The following illustrates the record and reduction of level observations".
The first observation was made with a level whose zero point is in the middle of the tube; the
second, with one whose zero is at the end. The values of D are 8'.'i6 and 0*032, respectively.
0=6h 15'" T=9b I2m
+ 14.1 - 9.7 +31-0 + 16.4
+ io.\ —13.8 —20.3 —35.0
+ °;
44. Determination of the value of one division of a level. — The ob-
server should be familiar with the sensitiveness of all the levels of his instru-
ment, even though he depends entirely upon a simple centering of the bubble
for the adjustment. If the striding level is to be used, a knowledge of the an-
gular value of one division of its scale is an essential.
The investigation of levels is most easily carried out with the aid of a
level trier, which is an instrument consisting essentially of a rigid base carry-
ing a movable arm whose inclination to the horizon may be varied by a known
amount by means of a graduated micrometer screw. The entire transit may
be mounted on the arm, or the various levels may be attached separately for
the investigation. The determination of the change in the inclination of the
arm of the level trier necessary to move the bubble over a given number of
divisions gives at once the angular value of one division of the scale.
74
PRACTICAL ASTRONOMY
Example 29. The following shows part of the reduction of observations made with a
level trier for the determination of the value of one division of a level. The bubble was run
from the left to the right end of the tube and back again, for both level direct and reversed,
by moving the micrometer head through four divisions at a time. The ends of the bubble
were read for each setting of the micrometer. Column two of the table gives the micrometer
settings; and columns three and four, the corresponding means of the end readings of the
bubble for level direct. The fifth column contains the means of the quantities in the two pre-
ceding columns; and column six, the differences between the »th and the (6-f«)th readings
in column five. The principle used in combining the observations is the same as that em-
ployed in Examples 26 and 27. The length of the arm and the pitch of the micrometer screw
are such that a rotation of the micrometer head through one division changes the inclination
by i". Each of the displacements of the bubble in column six therefore corresponds to a
change in inclination of 24". The quotients formed by dividing the displacements into 24"
are the values of one division of the level for different portions of the tube. A similar reduc-
tion of the readings taken with the level in the reversed position gave for d the values in column
eight. The means for the two series are in the last column. A glance at the results in this
column is sufficient to show that the curvature of the level tube is variable.
ONE DIVISION OF A LEVEL — LEVEL TRIER
No.
Microm.
Reading
Reading Middle of Bubble
Dis-
place-
ment
,/
L to R
R to L
Mean
Direct
Rev'sed
Mean
i
164
'2-35
12-95
12.65
'4-45
i '.'66
''•'73
i '.'70
2
160
15-5°
15-7.5
15.62
13-33
i. 80
1.76
i 78
3
'56
17.70
18.00
I7-85
13-03
1.84
1.84
1.84
4
152
19.80
20.50
20.15
12-73
1.89
i .90
1.90
5
148
22.45
23.00
22.72
12.30
i-95
i .90
1.92
6
144
24-55
25.20
24.88
12.22
i .96
1.92
1.94
7
140
26.95
27-25
27.10
8
136
28.90
29.00
28.95
9
132
30.70
31-05
30.88
10
128
32.60
33-15
32.88
it
124
34-95
35- «o
35-02
12
120
37-05
37-15
37-io
The investigation may also be carried out by a method first proposed by
Comstock in which the circles of the transit are used to change the inclination '
of the level tube by a known amount. If the instrument is levelled, the levels
themselves being in adjustment, the bubbles will remain centered when the
alidade is rotated about the vertical axis. If now the vertical axis be deflected
from the true vertical by a small angle z, the bubbles will not remain centered
as the instrument is rotated. For any given level, however, there will be two
readings of the horizontal circle, differing by 180°, for which the bubble will
stand at the middle of the tube; and by rotating slightly about the vertical
axis it can be brought to any desired position in the tube. The change in the
inclination of the level tube corresponding to any given displacement of the
bubble can be expressed in terms of i and tfte observed change in the reading
ONE DIVISION OF THE LEVEL
75
of the horizontal circle, whence the angular value of one division of the level
may be determined as before.
To express d as a function of i and the horizontal circle readings, let HC
and HC in Fig. 9 represent portions of the horizontal circles for the normal
and the deflected positions of the vertical axis; L, any position of the level,
which is supposed to be attached with its axis perpendicular to the radius
through L and parallel to the plane of the circle; and b, the corresponding
inclination. In the spherical right triangle HLC the angle //is equal to i, the
Fig. 9
deflection of the axis from the vertical, while that at L is 90° — b. Now, if ra
and r be the horizontal circle readings corresponding to the inclinations zero
and b respectively, we find
Arc//£=90° — (r — r,),
whence from the triangle HLC,
tan b = tan i sin (r — rc).
(US)
The angle b is very small and, for i equal two or three degrees, r — ra will
never exceed one degree. We may therefore use the approximate relation
= (r — r0) tan
(116)
with an error not exceeding o"oi.
For any other inclination, £,, we have the analogous equation
bi — (rl — ra) tan i,
which, combined with (112) gives
bi — b = (rt — r) tan i. (117)
The angle rl — r is the change in the horizontal circle reading correspon-
ding to the change in inclination £, — b. The latter, however, may be written
equal to sd, where s is the displacement of the bubble in scale divisions, and d
the angular value of one division.
We thus have finally as the expression for d
r, — r
tan /.
(118)
PRACTICAL ASTRONOMT
The angle i should be two or three degrees for the investigation of the
ordinary transit levels. For very sensitive levels it should be less. If the
instrument be provided with a telescope level, the deflection of the vertical
axis may be accomplished as follows: Level the instrument and center the
telescope bubble. Then change the vertical circle reading by the angle i and,
by means of the levelling screws, bring the telescope bubble back to the
middle of its tube, taking care at the same time that the transverse plate
bubble is also centered after the deflection. This precaution is necessary in
order that the deflection may have no component perpendicular to the plane
of the vertical circle. In the absence of a telescope level, level the instrument,
sight on a distant object, change the vertical circle reading by i, and bring the
object back to the intersection of the threads by means of the levelling screws.
The observations may be made either by displacing the bubble through a
definite number of divisions and noting the corresponding change in the hori-
zontal circle readings, or by changing the circle readings by a definite amount,
say 10', and observing the variations in the position of the bubble. For short
tubes with only a few graduations the former method is more convenient, while
the latter is to be preferred for the long finely graduated tubes of sensitive
levels.
The bubble should be run from one end of the tube to the other and then
back again, in both positions of the instrument. Such a series of readings
constitutes a set.
The instrument must be as rigidly mounted as possible, preferably on a
masonry pier. It is desirable to check the constancy of i by deflecting through
this angle toward the vertical at the end of a set and noting whether the
instrument is then levelled.
Example 30. Observations were made by the deflected axis method for the determina-
tion of the value of one division of the striding level of a Berger transit. The deflection was
3°. The graduations of the tube are in two groups of three each, the groups being separated
ONE DIVISION OF A LEVEL — DEFLECTED Axis
Level
Divisions
Hor. Circle Reading
Mean
r^ — r
Position
L to R
R to L
i and 4
3 and 6
34i° SS.'o
342 S.o
341° 56/0
342 8.0
55- '5
8.0
12-5
Direct
Direct
3 and 6
i and 4
162 6.5
161 53.0
162 6.3
161 53.0
6.4
13-4
Reversed
Reversed
' = 3°
, — r= 12/95
.« = 2
Mean 12.95
tan i ----- 8.719
colog .« 9.699
log d — 9.530
, — v, < 180°. If the
observations are direct, one v will represent the mean of all the circle readings
C.R.; the other, the mean of all C.L. If the artificial horizon has been used,
one v will represent the mean of all the direct readings; and the other, the
mean of all the reflected readings.
The observed altitude, or zenith distance, thus derived must be corrected
for refraction and parallax in accordance with Sections 8 and 9.
Example 31. The following is the record of partial sets of observations made with a
Buff & Buff engineer's transit at the Laws Observatory, on 1908, Oct. 2, Friday P. M., for the
determination of the altitudes of Polaris and Alcyone. The measures were all direct. The
timepiece used was an Elgin watch.
An inspection of the readings for Polaris shows that the measures are consistent. The
relatively large difference in the readings C. R. and C. L. reveals the existence of an index
error of 2' or 3'.
80 PRACTICAL ASTRONOMY
POLARIS ALCYONE
Vertical Circle Vertical Circle
Watch Ver. A Ver. B Clrcle Watch Ver. A Ver. B Clrcle Rate
19" 39° 26' 39° 26' R 9>>3sm2i!4 20° 40' 20° 40'
33 25 26 26 R 37 1.2 20 59 20 59 L
36 50 33 33 L 4i i-4 21 38 21 38 R ^ _
36 ii 34 34 L 43 1.4 22 i 22 I R
8" 35-1 1- 39° 29'. 8 91139"' 6H 2i°i9-5
For Alcyone the close agreement of the values for the rate of change in altitude per
minute of time given in the last column is evidence of the consistency of the measures.
The quantities in the fifth line are the means. The angles are the apparent altitudes
corresponding to the watch times immediately preceding. To obtain the true altitudes a
correction for refraction, which may be obtained from Table J, page 20, must be applied.
Example 32. The following observations were made with a Berger engineer's transit
on 1908, October 15, Thursday P. M., for the determination of the altitude of the sun. The
measures were all direct and were made by projecting the image of the sun on a card. The
transits were observed over the middle horizontal thread, the telescope being shifted after
each transit. The timepiece was the Fauth sidereal clock of the Laws Observatory.
Fauth
Clk.
Vertical
Ver. A
Circle
Ver.B
Limb
Circle
Rate
l6h 37m
5'»
24° r
24° i'
F
R
40
46.8
23 33
23 33
F
R
9/6
43
34-3
22 35
22 35
P
R
45
19.1
22 19
22 19
P
R
9.2
47
. 7-5
22 26
22 27
F
L
48
47-3
22 I I
22 II
F
L
9.0
50
11.7
21 25
21 25
P
L
51
29.9
21 \i
21 12
P
L
10. 0
Means i6h 45™ 3855 22° 2y'.8
46. The measurement of horizontal angles. — It is assumed that
the two objects whose difference of azimuth is to be determined are a terres-
trial mark and a celestial body, either the sun or a star. The directions given
in Section 45 for making settings in the measurement of vertical angles apply
here with only slight and obvious modifications. The conditions determining
the arrangement of the observing program are similar to those enumerated in
the present section. Although the details may vary with circumstances, the
following will serve to indicate the essentials. The first arrangement is in-
tended for use when only an approximate result is required, while the second
and third are designed for more precise determinations. The first two include
only direct observations, while the last is arranged for measures in which the
artificial horizon is employed. In direct observations care should be taken to
keep the bubbles centered throughout, but when the artificial horizon is used,
the levelling screws must not be touched between any direct observation and
its corresponding reflected setting. For settings on the mark the zenith dis-
tance will usually be so nearly equal to 90° that the error due to the deviation
of the vertical axis from the true vertical will be quite insensible, even
though no special effort be made to eliminate its influence.
HORIZONTAL ANGLES
81
DIRECT OBSERVATIONS
1 setting on mark \
2 settings on star i
2 settings on star 1
I setting on mark /
DIRECT OBSERVATIONS
i setting on mark C.R.
I setting on mark C.I..
3 settings on star C.I,.
3 settings on star C.R.
i setting on mark C.R.
i setting on mark C.L.
DIRECT AND REFLECTED OBSERVATIONS
setting on mark C.R.
setting on mark C.L.
setting on star, direct 1 _,
setting on star, reflected /
setting on star, reflected \ „ R
setting on star, direct J
setting on mark C.R.
setting on mark C.L.
Both verniers of the horizontal circle should be read for each setting, and
for those made on the star, the time should be noted in addition.
The required difference of azimuth will be the difference between the
means of the readings on the mark and on the star. Its value will correspond
to the mean of the times. If more precision is desired than can be obtained
from a single set, several sets may be observed, each of which should be re-
duced separately. To reduce the influence of graduation error, the horizontal
circle should be shifted between the sets. If the number of sets is n, the
amount of the shift between the successive sets should be 36o°/«.
Example 33. The following is the record of a simultaneous determination of the altitude
of Polaris and the difference in azimuth of Polaris and a mark.
ALTITUDE OF POLARIS AND AZIMUTH OF MARK No. 2
1908, Oct. 13, Thursday P. M.
Station No. 2
Buff & Buff Engineer's Transit No. 5606
Jrw = — 38*7 at 7h 59™ P.M., and —31:4 at 9" 54™ P.M.
Observer Sh.
Recorder W.
Object
Mark
Watch
Hor. Circle
Ver. A Ver. B
147° 23/5 327° 23/0
Vertical Circle
Ver. A Ver. B
Circle
R
Polaris
9b 35m
23'
322
in
.0
142
IO.O
39° 5i'
39° 5i'
R
Polaris
40
35
8
o
8,
0
Si
5'
R
Polaris
44
33
142
6
•5
3"
6
•5
59
59
L
Polaris
48
8
4
•5
5
.0
59
58
L
Mark
327
-M
•5
H7
23
.0
L
Means
9h42mIO, (Star 322'
\Mark 147 23.
Difference of Azimuth 5 — M = 174 44.06
7.'3'\
23- 2.5 /
39° 54-'9 = Appt. Alt.
47. The method of repetitions. — The precision of the measurement of
the azimuth difference, D, of two objects, A and B, may be increased materi-
ally by making a series of alternate settings on A and B such that the rotation
from A to B is always made with the upper motion of the instrument, and that
from B to A with the lower motion. Assuming that the graduations of the
horizontal circle increase in the direction AB, each turning from A to B will
6
82 PRACTICAL ASTRONOMY
increase the reading by the angle D, while that from B back to A will produce
no change since during this rotation the vernier remains clamped to the circle.
If the turning from A to B is repeated n times, the difference between the
circle readings for the final setting on B and the initial setting on A will be
nD; and if the initial and final readings be R, and R,, repectively, we shall
have
D = ^=A. (I20)
The method of repetitions derives its advantage from the fact that the
circle is not read for the intermediate settings on A and B. Not only is the
observer thus spared considerable labor, but, what is of more importance, the
errors which necessarily would affect the readings do not enter into the result.
Consequently, that part of the resultant error of observation arising from the
intermediate settings is due solely to the imperfect setting of the cross threads
on the object. For instruments such as the engineer's transit, in which the
uncertainty accompanying the reading of the angle is large as compared with
that of the pointing on the object, the precision of the result given by (120)
will be considerably greater than that of the mean of n separate measurements
of the angle D, each of which requires two readings of the circle. But for
instruments in which the accuracy of the readings is comparable with that of
the pointings, as is the case with the modern theodolite provided with read-
ing microscopes, the method of repetitions is not to be recommended.
Although there is even here a theoretical advantage, it is offset by the fact
that the peculiar observing program required for the application of the method
presupposes the stability of the instrument for a relatively long interval, and
hence affords an unusual opportunity for small variations in position to affect
the precision of the measures. Moreover, experience has shown that there
are small systematic errors dependent upon the direction of measurement, i.e.
upon whether the initial setting is made on A or on B; and, although these
may be eliminated in part by combining series measured in opposite direc-
tions, it is not certain that the compensation is of the completeness requisite
for observations of the highest precision. With the engineer's transit, how-
ever, the method of repetitions may be used with advantage.
Since rotation takes place on both the upper and the lower motions, any
non-parallelism of the vertical axes will affect the readings; and the observing
program must be arranged to eliminate this along with the other instru-
mental errors. For any given setting the deviation of the axis from parallel-
ism,/, unites with the inclination of the lower axes to the true vertical, i', and
determines the value of z, the inclination of the upper axis to the vertical, for
the setting in question. For different settings i will be different, for a
rotation of the instrument on the lower motion causes the upper axis to
describe a cone whose apex angle is 2p and whose axis is inclined to the true
vertical by i'. But no matter what the magnitude of i may be, within certain
limits easily including all values arising in practice, it may be eliminated by
forming the mean of direct and reflected readings made in the same position
METHOD OF REPETITIONS 83
of the instrument, provided that i is the same in direction and magnitude for
both settings. This follows from the discussion on pages 66 and 67 whose
result is expressed by equation (88). Hence, if after a series of n repetitions
observed C.R. direct, n further repetitions be made C.R. reflected, such that
the vernier readings for the corresponding settings in the two series are
approximately the same, the instrumental errors i' and /will be eliminated.
Equation (88) shows that j, the deviation of the upper vertical axis from per-
pendicularity with the horizontal axis, will also be eliminated. To remove
the influence of the collimation, c, the entire process must be repeated C.L.;
and to neutralize the systematic error dependent upon the direction of meas-
urement, the direct and reflected series should be measured in opposite direc-
tions. We thus have the following observing program, in which A' and £'
denote the reflected images of A and B, respectively:
Level on the lower motion.
(Set on A and read the horizontal circle.
Direct -j Turn from A to B on the upper motion » times.
••Read the horizontal circle for last setting on B.
^ C.R.
(• Set on B' and read the horizontal circle.
Reflected <. Turn from B' to A' on the upper motion » times.
I Read the horizontal circle for last setting on A'.
Repeat for C.L.
The circle reading for the first setting on B' must be the same, approximately at least,
as that for the last setting on />'.
The mean of the values of D calculated from the four series is the required azimuth
difference of A and B.
Uusually one of the objects, say A, will be near the horizon, in which case
reflected settings on A' will be impossible. A must then be substituted for
A' in the above program. The error due to i will not be eliminated from these
settings; but, owing to the presence of the factor cot20, it may be disregarded.
When the artificial horizon is not used the program must be modified.
Were i' zero, i would constantly be equal to/, although the direction of the
deflection would change with a rotation of the instrument on the lower
motion. If a series of n repetitions C.R. be made under these circumstances,
equation (82) shows that each setting will be affected by an error of the form
/cot «0 -f- p cos /cot 20 -f- c cosec 20.
The first and last terms of this expression will have the same values for all
pointings on the same object. Equations (82), (84), and (86) show that they may
be eliminated by combining with a similar series made C.L. The values of
the second term will be different for each setting owing to the change in I, but
their sum will be zero if the values of / are uniformly distributed throughout
360°, or any multiple of 360°. In order that this may be the case, approxi-
mately at least, it is only necessary that n be the integer most nearly equaling
k 36o°/Z>, where the k is any integer, in practice usually I or 2.
It is also easily seen that, if after any arbitrary number of settings the
instrument be reversed about the loiver motion and the series repeated in the
84 PRACTICAL ASTRONOMT
reverse order, the sum of the errors involving/ will be zero, provided that the
circle readings for corresponding settings C.R. and C.L. are the same, or ap-
proximately so. The reversal of the instrument on the lower motion changes
the direction of the deflection/ by 180°. The values of / for corresponding
settings C.R. and C.L. will therefore differ by 180°, and the errors will be oppo-
site in sign and will cancel when the mean of the two series is formed. The
reversal also eliminates the influence of j and c as indicated in the preceding
paragraph.
The above assumes that the deflection of the lower axis, z', is zero. If
this is not the case, each setting will be affected by an additional error of the
form i' cos/' cot 20, in which /' is constant so long as i' remains unchanged
in direction. If i' be the result of a non-adjustment of the plate bubbles, the
error which it produces may be eliminated from the mean of two series, one
C.R. and one C.L., by relevelling after reversal. (See page 70.) This will
change the direction off by 180°. Consequently, the values of /' for C.R.
and C.L. will differ by 180°, and the errors for the two positions will neutralize
each other when the mean is formed.
The consideration of these results leads to the following arrangement of
the observing program.
Level on the lower motion.
Set on A and read horizontal circle. ^
Turn from A to B on upper motion « times. \ C.R.
Read horizontal circle for last setting on B.
Reverse on lower motion and relevel.
Set on B and read horizontal circle. \
Turn from B to A on upper motion « times. I C.L.
Read horizontal circle for last setting on A. >
The circle reading for the first setting on B, C.L. should be the same, approximately at
least, as that for the last setting on B, C.R.
The mean of the values of D calculated from the two series is the required azimuth
difference of A and B.
With this arrangement the instrumental errors i',p,j, and c will be com-
pletely eliminated, whether the settings are distributed through 360° or not,
provided only that the instrumental errors remain constant during the obser-
vations. Practically, it is desirable that the value of n should be such that nD
equals 360°, or a multiple of 360°, at least approximately; but when D is small
this may unduly prolong the observations. The maximum number, of repeti-
tions which can be made advantageously depends upon the stability of the
instrument and must be determined by experience.
If the instrument is provided with a striding level, the influence of i\ p,
and / may be taken into account by measuring the inclination of the horizontal
axis for each setting and applying a correction to Rl and R3 of the form £cotz0,
in which b denotes the sum of all the observed inclinations for settings on
A and B respectively.
When one of the objects, say B, is a star, the time of each setting on B
must be noted. The calculated value of D will then correspond sensibly to
the mean of the times, provided the observing program be not too long.
THE SEXTANT
85
Example 34. On 1909, April 9, the following observations of the difference in azimuth
of Polaris and a mark were made by the method of repetitions with a Buff & Buff engineer's
transit. The recorded times are those of a Fauth sidereal clock whose error was +6m36".
After four repetitions C.R., the instrument was reversed on the lower motion, relevelled, and
the series repeated in the reverse order. Since the azimuth difference is approximately 174°,
720° must be added to the readings on the star before combining them with those on the
mark. The results for the two halves are derived separately, although the means for the set
are also given.
Hor.
Circle
Object
Of
Ver. A
Ver. B Circle Means
Mark
179° 59-5
359° 59'. 5 R
179°
59'
30"
Polaris
9h 27"
'3°'
353 32
R
Polaris
30
48
R
Polaris
33
36
R
Polaris
35
49
"54 13-5
334 13-5 R
874
13
3°
4) 7
43
4)694
H
o
9 3'
56
0 = 9h
3S">32' S — Jk
r=»73
33
3°
Polaris
9 39
12
154 13-0
334 13-5 L
874
13
'5
Polaris
42
28
L
Polaris
44
9
L
Polaris
46
20
L
Mark
179 51.0
359 5i-o L
179
Si
o
4) 12
9
4)694
22
'5
9 43
2
0 = 9"
49m 38* S — Ik
r=i73
35
34
Final Means
0 = 9
44 5 5— A
r=i73
34
32
THE SEXTANT
48. Historical and descriptive. — The instruments typified by the engi-
neer's transit may be used for the measurement of horizontal or vertical angles
only. Simultaneously with the development of the altazimuth principle there
was gradually evolved a contrivance adapted for the measurement of angles
lying in any plane. Beginning with the astrolabe of the ancients, the applica-
tion of various ideas gave in succession the Jacob's staff, or cross-staff, which
dates apparently from the middle of the fourteenth century, the back-staff, or
Davis quadrant; the sextant of Tycho, which was also used by the Arabs in .
the tenth century; the octants of Hooke and Pouchy, in which a mirror was
used for the first time; and, finally, the reflecting octant whose principle was
due to Newton, although the construction was first carried out by John Hadley
about 1731. The instrument of Hadley. has been improved in design, but no
essential modification has been made in its principle. In its modern form it
is known as the reflecting sextant, or more generally, simply as the sextant.
With the exception of the astrolabe and the large fixed sextants of Tycho,
the various forms mentioned are characterized by the fact that they may be
held in the hand during observations, small oscillations and variations in the
position of the instrument offering no serious difficulty in the execution of the
measures. These instruments have therefore played an important part in the
86 PRACTICAL ASTRONOMY
practice of navigation, and to-day the sextant is the only instrument which
can advantageously be employed in the observations necessary for the deter-
mination of a ship's position. In addition, its compactness and lightness, and
the precision of the results that may be obtained with it render it one of the
most convenient and valuable instruments at our command.
The modern sextant consists of a light, flat, metal frame supporting a
graduated arc, usually 70° in length; a movable index arm; two small mirrors
perpendicular to the plane of the arc; and a small telescope. The index arm
is pivoted at the center of the arc and has rigidly attached to it one of the
mirrors, the index glass, whose reflecting surface contains the rotation axis
of the arm and the attached mirror. The position of the index glass corre-
sponding to any setting may be read from the graduated arc by means of a
vernier. The second mirror, the horizon glass, is firmly attached to the
frame of the sextant in a manner such that when the vernier reads zero the
two mirrors are parallel. Only that half of the horizon glass adjacent to
the frame is silvered. The telescope, whose line of sight is parallel to the
frame, is directed toward the horizon glass, and with it a distant object may
be seen through the unsilvered portion. When the frame is brought into
coincidence with the plane determined by the object, the eye of the observer,
and any other object, a reflected image of the second object may be seen in
the field of the telescope, simultaneously with the first, by giving the index
arm a certain definite position depending upon the angular distance separat-
ing the objects. If the position of the arm is such that the rays of the second
object reflected by the index glass to the horizon glass, and then from the
silvered portion of the latter, enter the telescope parallel to the rays that pass
from the first object through the unsilvered portion of the horizon glass, the
two images will be seen in coincidence. This being the case, the relative incli-
nation of the mirrors as shown below, will be one-half the angular distance
separating the objects; and, since the construction is such that the inclination
may be read from the graduated arc, it becomes possible to find the angular
distance between the objects. The use of the instrument is simplified by
graduating the arc so that the vernier reading is twice the inclination of the
mirrors, and hence, directly, the angular distance of the objects. With the
usual form of the instrument the maximum angle that can be measured is
therefore about 140°. The two mirrors and the telescope are provided with
adjusting screws, which may be used to bring them accurately into the posi-
tions presupposed by the theory of the instrument. In addition, the tele-
scope may be moved perpendicularly back and forth with respect to the frame
thus permitting an equalization of the intensity of the direct and reflected
images by varying the ratio of the reflected and transmitted light that enters
the telescope. Adjustable shade glasses adapt the instrument for observa-
tions on the sun.
49. The principle of the sextant— In Fig. 10 let 0V represent the
graduated arc; / and H, the index glass and the horizon glass, respectively;
and IV, the index arm, pivoted at the center of the arc and provided with a
THE SEXTANT 87
vernier at V. When V coincides with 0, the mirrors are parallel. The posi-
tion indicated in the figure is such that the two objects 5, and S, are seen in
coincidence, for the rays from 5, pass through the unsilvered portion of H
and enter the telescope in the direction HE, while those from S, falling on /
are reflected to H and thence in the direction HE. The two beams therefore
enter the telescope parallel.
It is to be shown that the inclination of / to H is one-half the angular
distance A separating the objects. / + A,
whence
a = b+ Y2A.
But in the triangle IHN
a = b + M.
Therefore,
But M, being the angle between the normals to the mirrors, measures their
inclination, and is equal to the angle subtended by the arc 0V, whence
A = 20y. (121)
But since the arc is graduated so that the reading is twice the angle subtended
by OF the angular distance between the two objects is given directly by the
scale.
50. Conditions fulfilled by the instrument. — The following conditions,
among others, must be fulfilled by the perfectly adjusted sextant.
88 PRACTICAL ASTRONOMT
1. The index glass must be perpendicular to the plane of the arc.
2. The horizon glass must be perpendicular to the plane of the arc.
3. The axis of the telescope must be parallel to the plane of the arc.
4. The vernier must read zero when the mirrors are parallel.
5. The center of rotation of the index arm must coincide with the center
of the graduated arc.
Since the positions of the mirrors and the telescope are liable to derange-
ment, methods must be available for adjusting the instrument as perfectly as
possible. This is the more important inasmuch as it is impossible to eliminate
from the measures the influence of any residual errors in the adjustments.
Although elimination is impossible, it should be remarked that the errors
arising in connection with Nos. 4 and 5, at least, may be determined by the
methods given in Sections 52 and 53, and applied as corrections to the read-
ings obtained with the instrument. Conditions Nos. 1-4 are within the control
of the observer. No. 5 must be satisfied as perfectly as possible by the
manufacturer.
51. Adjustments of the sextant. — No. i. Index glass. To test the
perpendicularity of the index glass, place the sextant in a horizontal position,
unscrew the telescope and stand it on the arc just in front of the surface of
the index glass produced. If then the eye be placed close to the mirror, the
observer will see the reflected image of the upright telescope alongside the
telescope itself. By carefully moving the index arm, the telescope and its
image may be brought nearly into coincidence. If the two are parallel, the
index glass is in adjustment. The telescope should be rotated about its axis
in order to be sure that it is perpendicular to the plane of the arc. If the
adjustment is imperfect, correction must be made by the screws at the base of
the mirror. Some instruments are not provided with the necessary screws,
and in such cases the adjustment had best be entrusted to an instrument
maker.
The test can also be made by looking into the index glass as before, and
noting whether the arc and its reflected image lie in the same place. If not,
the position of the mirror must be changed until such is the case.
No. 2. The horizon glass. The adjustment of the horizon glass may be
tested by directing the telescope toward a distant, sharply defined object,
preferably a star, and bringing the index arm near the zero of the scale. Two
images of the object will then be seen in the field of view — one formed by the
rays transmitted by the horizon glass, the other, by those reflected into the
telescope by the mirrors. The reflected image should pass through the direct
image as the index arm is moved back and forth by the slow motion. If it
does not, the horizon glass is not perpendicular to the plane of the arc, and
must be adjusted until the direct and reflected images of the same object
can be made accurately coincident.
No. 3. The telescope. The parallelism of the telescope to the frame may
be tested by bringing the images of two objects about 120° apart into coin-
cidence at the edge of the field nearest the frame. Then, without changing
INDEX CORRECTION 89
the reading, shift the images to the opposite side of the field. If they remain
in coincidence, the telescope is in adjustment. If not, its position must be
varied by means of the adjusting screws of the supporting collar until the test
is satisfactory.
No. 4. Index adjustment. If the fourth condition is not fulfilled, an index
error will be introduced into the angles read from the scale. To test the
adjustment, bring the direct and reflected images of the same distant object
into the coincidence as in the adjustment of the horizon glass. The corres-
ponding scale reading is called the zero reading = R0. If R0 is zero, the
adjustment is correct. If not, set the index at o°, and bring the images into
coincidence by means of the proper adjusting screws attached to the horizon
glass. It is better, however, to disregard this adjustment and correct the readings
by the amount of the index error.
It can be shown that the errors affecting the readings as a result of an
imperfect adjustment of the index glass, the horizon glass, and the telescope
are of the order of the squares of the residual errors of adjustment. If care be
exercised in making the adjustments, the resulting errors will be negligible
as compared with the uncertainty in the readings arising from other sources.
52. Determination of the index correction. — Make a series of zero
readings on a distant, sharply defined object, a star if possible. If the zero of
the vernier falls to the right of the zero of the scale, do not use negative
readings, but consider the last degree graduation preceding the zero of the
scale as 359°, and read in the direction of increasing graduations. The zero
reading is what the instrument actually reads when it should read zero. The
index correction, /, is the quantity which must be added algebraically to the
scale readings to obtain the true reading. We therefore have
7=0° — ^, (122)
7=360° — Ra. (123)
The latter expression is to be used for the determination of 7 when the
zero of the vernier falls to the right of that of the scale for coincidence of the
direct and reflected images of the same object.
When observations are to be made on the sun, the index correction should
be determined from measures on this object. Since it is impossible, on account
of their size, to bring the solar images accurately into coincidence, we deter-
mine the zero reading as follows: Make the two images externally tangent,
the reflected being above the direct, and read the vernier. Let R, represent
the mean of a series of such readings. Then make an equal number of settings
for tangency with the reflected image below. Call the mean of the corres-
ponding readings R,. The mean of R, and R, will then be the value of the
zero reading, and we shall have
7=o° -#(*,+*.), (124)
7 =360°-^ (/?, + *,). (125)
90 PRACTICAL ASTRONOMY
The readings thus obtained will also give the value of 5, the sun's semi-
diameter. Since the center of the reflected image moves over a distance of
four semi-diameters in shifting from the first position to the second, we have
Owing to the brilliancy of the solar image, its diameter appears larger
than it really is— a phenomenon known as irradiation. Should the value of
S be required for the reduction of observations on the sun (see Section 55),,
the value calculated .from equation (126) should be used rather than that
derived from the Ephemeris, in order that the influence of irradiation may be
eliminated.
53. Determination of eccentricity corrections. — Any defect in the
fifth condition introduces an eccentricity error into the readings. Since, with
the usual form of the instrument there is but a single vernier, this cannot be
eliminated. Each sextant must be investigated specially for the determination
of the eccentricity errors affecting the readings for different parts of the scale.
These may be found by measuring a series of known angles of different mag-
nitudes. The mean result for each angle, A, gives by (71) an equation of.
the form
A=R + f+E. — E, (127)
where R is the sextant reading for coincidence of the two objects whose
angular distance is A; I, the index correction; and E0 and E, the eccentricity
corrections for those graduations of the scale which coincide with vernier
graduations for the readings R0 and R, respectively. The readings of the
coinciding graduations when the vernier reads Ra and R may be denoted by
R0' and R', respectively. £„ — E is the correction which must be applied to
the sextant reading, freed from index correction, in order to obtain the true
value of the angle. Denoting its value by e, (127) may be written
e=A — (R + f). (128)
Having determined e from (128) for a considerable number of angles dis-
tributed as uniformly as possible over the scale, the results may be plotted as
ordinates with the corresponding values of R' as abscissas. From the plot a
table may be constructed giving the values of s for equidistant values of R',
from which the value of s for any other reading, R, can then be derived. Care
should be taken always to enter the table with the R' corresponding to the
given R as argument. It should be noted that the usefulness of the table
depends upon / remaining sensibly constant, for if the index correction
changes by any considerable amount, Ra' may change sufficiently to render
the tabular values of e no longer applicable.
The chief difficulty in investigating the eccentricity of a sextant consists
in securing a suitable series of known angles. A simple method is to measure
with a good theodolite the angles between a series of distant objects, nearly
MEASUREMENT OF ALTITUDES 91
in the horizon, care being taken to tilt the instrument so that in turning from
one object to the next no rotation about the horizontal axis is necessary.
54. Precepts for the use of the sextant. — The following points should
carefully be noted in using the sextant: Focus the telescope accurately.
The image of a star should be a sharply defined point; that of the sun must
show the limb clearly defined and free from all blurring. For solar observa-
tions, use, whenever possible, shade glasses attached to the eyepiece rather
than those in front of the mirrors; and reduce the intensity of the images as
much as is consistent with clear definition. If the use of the mirror shade
glasses cannot be avoided, select those which will make the direct and
reflected images of the same color, and reverse them through 180° at the
middle of the observing program to eliminate the effect of any non-parallelism
of their surfaces. If a roof is used to protect the surface of the mercury from
wind, it also should be reversed at the middle of the program. In all cases
make the direct and reflected images of the same intensity by regulating the
distance of the telescope from the frame. Make the adjustments in the order
in which they are given above, and always test them before beginning obser-
vations. The index correction should be determined both before and after
each series of settings. Make all coincidences and contacts in the center of
the field. Finally, the instrument should be handled with great care, for a
slight shock may disturb 'the adjustment of the mirrors and change the value
of the index correction.
55. The measurement of altitudes.— Although the sextant may be
used for the measurement of angles lying in any plane, it finds its widest
application in practical astronomy in the determination of the altitude of a
celestial body.
At sea the observations are made by bringing the reflected image of the
body into contact with the image of the distant horizon seen directly through
the unsilvered portion of the horizon glass. To obtain the true reading the
plane of the arc must be vertical. Practically, the matter is accomplished by
rotating the instrument back and forth slightly about the axis of the tele-
scope, which causes the reflected image to oscillate along a circular arc in the
field. The index is to be set so that the arc is tangent to the image of the
horizon. The corresponding reading corrected for index correction, dip of
horizon, and refraction is the required altitude. The correction for dip is
necessary, since, owing to the elevation of the observer, the visible horizon
lies below the astronomical horizon. The square root of the altitude of the
observer above the level of the sea, expressed in feet, will be the numerical
value of the correction in minutes of arc. The observations are not suscept-
ible of high precision, and the correction for eccentricity may be disregarded
as relatively unimportant.
For observations on land the artificial horizon must be used. The meas-
urement of the angular distance between the object and its mercury image
gives the value of the double altitude of the object. Some practice is
required in order to be able to bring the object and its mercury image into
92 PRACTICAL ASTRONOMY
coincidence quickly and accurately. In case the object is a star, care must be
taken that the images coinciding are really those of the object and its reflec-
tion in the mercury. The following is the simplest method of procedure:
Stand in a position such that the mercury image is clearly visible in the
center of the horizon, and direct the telescope toward the object. By bring-
ing the index near zero the reflected image will appear in the field. The
telescope is then turned slowly downward toward the mercury, the index
being moved forward along the arc at the same time at a rate such that the
reflected image of the object remains constantly in the field. If the plane of
the sextant is kept vertical, and if the observer is careful to stand so that the
mercury reflection can be seen, its image seen directly through the unsilvered
portion of the horizon glass will come into the field when the telescope has
been sufficiently lowered. Both images should then be visible. The varying
altitude of the object will cause them to change their relative positions. The
index is set so that the images are approaching and clamped. When they
become coincident the time is noted and the vernier read. The instant of
coincidence is best determined by giving the instrument a slight oscillatory
motion about the axis of the telescope and noting the time when the reflected
image in its motion back and forth across the field passes through the direct
image.
To obtain an accurate value of the altitude, a series of such settings
should be taken in quick succession, the time and the vernier reading being
noted for each. It is not necessary to use the method described above for
bringing the images into the field for any of the settings but the first; for if,
after reading, the index be left clamped and the telescope be directed toward
the mercury image, the plane of the arc being held vertical, the reflected
image will also be in the field. If it is not at once seen, a slight rotation
about the axis of the telescope will bring it into view, unless too long an
interval has elapsed.
Measures for altitude may also be made by setting the zero of the vernier
accurately on one of the scale divisions so that the images are near each other
and approaching a coincidence. The time of coincidence and the vernier
reading are noted. The index is then moved 20' so that the images will again
be approaching coincidence. The time and the reading are noted as before
and the process is repeated until a sufficient number of measures has been
secured.
The consistency of the measures should always be tested, as in the case
of the engineer's transit (see page 79) by calculating the rate of change of the
readings per minute of time. If however, the observations have been made
by noting the times of coincidence for equidistant readings of the vernier, the
constancy of the time intervals between the successive settings will be a
sufficient test.
If R denote the mean of the vernier readings, the apparent double altitude
of the object will be given by
(129)
ME A S UREMENT OF AL TITUDES 93
in which /is the index correction, and e the correction for eccentricity. The
true altitude corresponding to the mean of the observed times is found from
where the refraction, r, may be derived from Table I, page 20, or if more
accurate results are required, by equation (3), page 18. If the zenith distance
is desired instead of the altitude, we calculate z' from
«' = 90°— h\ (130)
and z from
z — z' + r. (131)
For measures on the sun coincidences are not observed, but, instead, the
instants when the images are externally tangent. To eliminate the influence
of semidiameter, the same number of contacts should be observed for both
images approaching and images receding. If for any reason this cannot be
done, a correction for semidiameter must be applied. Let
«a = number of settings for images approaching,
nr = number of settings for images receding,
n = total number of settings,
5= the semidiameter of the sun calculated by equation (126).
We shall then have for solar observations
„/,' jp_i_wa — nf c i / L , f Upper sign, altitude decreasing. "1
= Rd —^~S+I+e' JLowersign, altitude increasing }
in which h' is the apparent altitude of the sun's center corresponding to the
mean of the observed times; and the term involving S, the correction for
semidiameter. The true altitude and zenith distance are then given by
A = A' — r+p, (133)
2 = 2' + r— /, (134)
The solar parallax,/, may be obtained from columns four and eight of Table
I, page 20. For approximate results r — / may be taken from the fifth
and tenth columns of this same table.
Example 85. On 1909, April 10, the following sextant observations of the altitude of
the sun were made at the Laws Observatory near the time of meridian transit. The error of
the timepiece was J0F = + 6ra 37'. The observations will be reduced later for the determina-
tion of latitude.
94
PRACTICAL ASTRONOMT
Readings on Sun
for
Index
Correction 0F
Reading
Limb
R
i
X,
O*1 59"
1 22'
117°
18'
50"
Lower
0°
31'
30"
359°
28'
o"
i i
28
118
23
10
Upper
3i
40
28
20
2
29
118
25
5°
Upper
32
0
28
O
4
9
117
24
0
Lower
32
10
28
1O
5
19
117
24
3°
Lower
o
31
TO
359
28
g
6
53
118
28
40
Upper
j
8
27
118
29
o
Upper
ro Reading =
= 359
59
59
12
8
117
24
10
Lower
Index Corr. = -f- I
Rl-R2= i 3 42
Semidiameter = 15 56
CHAPTER V
THE DETERMINATION OF LATITUDE
56. Methods. — On page 34 it was shown that if the zenith distance or
altitude of a star of known right ascension and declination be measured at a
known time, the latitude of the place of observation can be determined by
means of equation (31). The preceding chapter indicates the methods that
may be employed for the measurement of the zenith distance. It is the pur-
pose of the present chapter to determine the most advantageous method of
using the fundamental equation and to develop the formulas necessary for
the practical solution of the problem.
To establish a criterion for the use of equation (31), it is to be noted that
the resultant error of observation in — d,
or
— 8) — 2 cos y cos d sin' ^ /. (148)
Let z be the observed value of the coordinate, #„ the meridian value, and Zthe
reduction to the meridian. We then have
z+Z=za. (149)
Substituting into (148) we find
cos (za — Z) =cos sa — 2 cos ip cos 8 sin* Yz t. (150)
To express Z explicitly we may replace the left member of (150) by its expan-
sion by Taylor's theorem. Since Z is small the convergence will be rapid.
Introducing at the same time
A = cos
9.8908
f
38°
57'
cos — A^) 9.8034
The calculated ^ is larger
— IV) are negative. Since
cos (tf — N} is positive, y> — ^V is in the fourth quadrant.
5. ALTITUDE OF POLARIS
65. Theory. — The peculiar location of Polaris with respect to the pole
makes it possible to simplify the fundamental latitude equation for use in
connection with this object. Since the latitude is by definition equal to the
altitude of the north celestial pole, the problem may be solved by finding an
expression for the difference in altitude of the pole and Polaris. The polar
distance of Polaris is about i° n', consequently, the required difference will
always be a small angle. To this fact is due the possibility of a simplification
of equation (31). (See Num. Comp. pp. 14 and 16.)
Replacing z and d in (31) by the altitude, ft, and the north polar distance,
IT, respectively, we find
sin h = cos n sin
°
y> = 35°
? = 40°
p = 45°
= h — i: cos
K.
(169)
in which 0 is the sidereal time of observation, and a and 3 the apparent right
ascension and declination, taken from the Ephemeris, pp. 312-323. Inter-
polate /ffrom Table VIII with /, or 24h — /, and an approximate value of "39« = i054'45"
log^l 0.1744' log dy 1.1849
sin/ 8.5334 9
6
o.oooo
0.1092
18
7
9.9981
0.1072
17
8
9.9963
0.1055
16
9
9.9948
0.1039
IS
10
9.9936
0.1028
'4
ii
9.9928
O.IO2O
13
12
9.9926
O.IOlS
12
Further, logG may be tabulated, as in Table IX, for a mean value of the lati-
tude with t as argument. Since ,
from table IX with t as argument. Then calculate A from
A = i8o°— TrGsecysint. (184)
If JT be expressed in minutes of arc, the last term of (184) will also be
given in minutes of arc.
Example 43. Determine the azimuth of the mark from the data given in Ex. 34, p. 85.
The latitude of the place of observation is 38° 56' 52".
Equations (180) are used for the calculation, the results for the two positions of the
instrument being reduced separately. The azimuth of the mark is found by subtracting the
difference 5 — M, taken from p. 85, from the calculated azimuth of Polaris. The difference
of the two values of M is not to be taken as an indication of the precision of the result, as
these quantities are affected by instrumental errors whose influence is not eliminated until the
mean is formed.
AZIMUTH FROM ZENITH DISTANCE
113
if 3°
sec/p
tan?r
log//
ih 25™ 19"
i° 10' 46"
» 56 5i
0.10918
8.31362
9.90756
8.42280
8.22118
C. R.
C. L.
0
9h 38". 32«
9>> 49"' 38'
t
8 13 13
8 24 19
t
123° 18' 15"
126° 4' 45"
cos/
9.73964"
9- 77005n
h cos t = A
7. 96082 n
7-99'J3n
B = log G
9.99605
9-99577
sin/
9.92209
9 -9°75Z
tun A
8. 34094 „
8.326090
A
178° 44' 38"
178° 47' 10"
S—M
'73 33 30
«73 35 34
M
5 ii 8
5 'i 36
Mean
5°'"
22"
3- AZIMUTH FROM AN OBSERVED ZENITH DISTANCE
73. Theory. — Equation (26) rewritten in the form
— cos A =
sin 3 — cos.sr sin a>
-- •—
sin 2 cosy
A < 180° when the object is west of the meridian
(185)
expresses the azimuth as a function of 8, £, and
O.II5I
* = 7T
I 10.9
sin I
9-7737.1
c = 90 — (f,
51 3-1
lOg^N
1-7394
s
51 10. i
AS
+54-9
s — a
i 3-9
A
180° 54:9
s — 6
49 59-2
S — M
174 44.1
s — c
o 7.0
M
6 10.8
sin (5 — c)
7 . 30882
sin (s — a)
8 . 26920
cosec (5 — £)
0.11583
cosec i
o. 10847
cot %A
7.90116
A
180° 54.8
Ck.
76. Influence of an error in the time. — An uncertainty in the clock
correction, or any error in noting the time of the measurement of -S — M, will
introduce an error into the final result, for the calculated azimuth of the object
will not correspond to the observed azimuth difference. The magnitude of
this error for any given given error in 6 depends upon the position of the star.
Its value may be calculated from the differential relation between A and 0,
dA = cos (? cos ^ cosec zdO, (187)
which is derived from (175).
Similarly, when the azimuth of the object is calculated from measures
of its zenith distance there will be an uncertainty in the result due to the error
affecting 2. The relation in this case is, by (176),
INFLUENCE OF ERROR IN TIME 115
dA = cos q sec ,
S— Af=8i°24'.7. Find the azimuth of the mark, calculating the azimuth of the sun both by
method i and method 3.
The computation of the solar azimuth by (177) and (178) is in the first column; that for
(186), in the second. In the latter instance the time is required only with such precision as
as may be necessary for the interpolation of declination from the Ephemeris for the instant
of observation.
C.S.T. 3» 59" 26:5 /*' 33° I9'.6
Col. M.S.T. 3 50 8.2 r— p 1.3
E i 24.7 a = z 56 41.7
t = Co\. A.S.T. 3 52 32.9 « = ,r 76 8.9
/ 58° 8'.2 c = oo-y, 51 3.,
«5 +13 Si-' * 91 56.9
tan ,J 9-39196 sin(i — c) 9.81604
cost 9-7"55 sin(s — a) 9.76132
tan N 9.66941 cosec s 0.00025
-W 25° 2'.2 cosec (.« — *) 0.56498
—JV) 0.61902
tan^l 0.78268
A 80° 38'.!
CHAPTER VII
THE DETERMINATION OF TIME
77. Methods. — The determination of time means, practically, finding
the error of a timepiece. To accomplish this the true time 6 or T\s calculated
from observations on a star or the sun and compared with the clock time at
which the observations were made. The required error is given by
J0=0 — 0't (193)
or
jT=r—r, (194)
according as the timepiece is sidereal or mean solar, 6' and T' being the clock
values of the time of observation.
The fundamental equation for the determination of time is
0 = «+7. (195)
Applied to any celestial object this equation gives the sidereal time, from
which the mean solar or apparent solar time may be derived by the transform-
ation processes of Chapter III. For the sun, however, the hour angle t is
directly the apparent solar time, and, in case of observations on this object, the
mean solar time may be found from (42) written in the form
T=t + E. (196)
When the timepiece is solar the use of (196) is simpler than that of (195).
Since a and E may be regarded as known, the problem is reduced to the
determination of the hour angle of the object for the instant of observation.
As indicated on page 34 this may be accomplished by measuring the zenith
distance of the object at a place of known latitude and using equation (38)
or (39)-
The problem can also be solved by determining the clock time 00' of the
instant for which the hour angle of the object is zero. For this case the
fundamental equation reduces to
6 = a, (197)
and
J6 = a — da'. (198)
In outlining the methods that may be employed for the determination of
00' it will be assumed that the object is a star and that the timepiece used is
sidereal. The modifications necessary for the removal of these limitations will
be considered in connection with the discussion of the details presented in the
following sections.
116
METHODS 117
To determine #„' we may note the time 6, when a star has a certain zenith
distance, or altitude, east of the meridian, and, again, the time 0, when it has
the same zenith distance west of the meridian. Since the celestial sphere
rotates uniformly, we shall have
0: = x(o, + o.). (199)
The method is known as that of equal altitudes.
The clock time of meridian transit, 00', may also be determined by noting
the instant of passage of an object across the vertical thread of a transit
instrument mounted so that the line of sight of the telescope lies in the plane
of the meridian. This is the meridian method of time determination.
Finally, 00' may be found by observing the transit of an object across the
vertical thread of an instrument nearly in the plane of the meridian. The
application of a small correction to the observed time depending upon the
displacement of the instrument from the meridian gives the clock time for
which f = o. In practice the deviation of the instrument is such that the line
of sight lies in the plane of the vertical circle passing through Polaris at a
definite instant. The process is accordingly known as the Polaris vertical
circle method of time determination. It is of special interest on account of
the fact that it is readily adapted to a simultaneous determination of time and
azimuth.
There are other methods of determining the true time, but those outlined
afford a sufficient variety to meet the conditions arising in practice. We there-
fore proceed to a detailed consideration of
1. The zenith distance method.
2. The method of equal zenith distances or altitudes.
3. The meridian method.
4. The Polaris vertical circle method.
I. THE ZENITH DISTANCE METHOD
78. Theory. — The formulae necessary for the calculation of / from S, j.6'7
2 30 a 3 42 3.2
h 21 17 o 0 22 17 19.9
'5 +23 49 23 C.S.T. 9 40 51.4
cosec/ — tan 3cott)dd, (203)
in which / is one-half the interval between the two observations expressed in
solar units, , respectively. Further, the line
of sight will not be exactly perpendicular to the horizontal axis, but will form
with it an angle 90° +£. The quantities a, d, and c are known as the azimuth,
level, and collimation constants, respectively. In general, therefore, the star
will not be on the meridian at the instant of its transit across the vertical
thread, but will have a small hour angle t whose value will depend upon the
magnitude of the instrumental constants a, b, and c and the position of the
star. To obtain the clock time of meridian transit we must subtract t from the
clock time of observation, 6', whence
THE MERIDIAN METHOD 121
e,' = B' — t, (209)
and by (208)
J6 = a — 0'-\-t. (210)
The values of a, b, and c can always be found. Consequently J0 can be
determined by (210) when / has been expressed as a function of the instru-
mental constants. For this purpose we make use of equations (82), (89), and
(33). The last two terms of (82) express the influence of the level and colli-
mation constants, b and c, upon the reading of the horizontal circle of the engi-
neer's transit for C. R., or, what amounts to the same thing, the amount by
which the azimuth difference of the point A and the object 0, when on the
vertical thread, exceeds 90°. The last two terms of (89) express the corres-
ponding quantity for C. L. These results may be applied directly to the
meridian transit to determine the azimuth of the star at the instant of its
transit across the vertical thread. For, denoting this azimuth by As, and
assuming that a, the azimuth of the point A referred to the east point, is
measured positive toward the south, we have at once
A, = a + b cot z ± c cosec z, (211)
in which the upper sign refers to C. R., and the lower to C. L. In the present
case, however, the positions of the instrument are less ambiguously expressed
by circle west (C. W.) and circle east (C. E.), respectively. We may now use
(33) to determine the hour angle of the star when its azimuth is equal to^s.
Replacing A in (33) by As and writing As and / instead of their sines, which
we may do since both are very small angles, we find
Ats\nz (212)
whence by (211)
/cos d = asin z -f-£cos2± c. (213)
Equations (211) and (212) become indeterminate for .3 = 0, on account of the
presence of A, but the conditions of the problem show that there can be no
such discontinuity in the expression which gives / as a function of a, i>, and c.
Equation (213) is therefore valid for.s = o, and becomes inapplicable only for
stars very near the pole. Since the star is near the meridian at the instant of
observation, z in (213) may be replaced by the meridian zenith distance .
Writing
A = s\n(
. For any given latitude C itself
may be tabulated with d as argument. The third column of Table X contains
such a series of values for the latitude of the Laws Observatory, viz., 38° 57'.
The vertical circle method is easily adapted to a simultaneous determi-
nation of time and azimuth. If the horizontal circle be read at the instant of
setting on Polaris, and if in addition, readings be taken on a mark, the azimuth
of the mark will be given at once; for the azimuth of Polaris is calculated in
the course of the reduction of the observations for time, and the horizontal
circle readings give the azimuth difference of the star and the mark. Since aa
is measured from the north point positive toward the east, the azimuth of the
mark measured in the conventional manner will be
Am = M—S + a0—i&o° (232)
in which S and M are the means of the horizontal circle readings on the star
and the mark, respectively; and am the mean of the calculated azimuths of
Polaris.
The vertical circle method of time determination, like that of the meridian
method, is not dependent upon the reading of graduated circles, and in conse-
quence, yields results of a relatively high degree of precision. It possesses
the further advantage that no preliminary adjustment in the plane of the
meridian is necessary. It is especially valuable for use with unstable instru-
ments, for the constancy of the quantities a, b, and c is assumed for only a
very short interval, much less than in the meridian method. It is necessary
that the azimuth and level constants remain unchanged only during the
interval separating the setting on Polaris and the transit of the time star
immediately following, and this need not exceed two or three minutes. More-
over, each set of two time stars is complete in itself and gives a complete
determination of the error of the timepiece.
The instrument used should be carefully constructed, however, for any
irregularity in the form of the pivots is likely to produce serious errors in the
results.
THE VERTICAL CIRCLE METHOD 129
85. Procedure. — The instrument is carefully levelled, and three or four
minutes before the transit of a southern star across the vertical circle through
Polaris, the telescope is turned to the north, and Polaris itself is brought to
the intersection of the vertical and horizontal threads. The instrument is
clamped in azimuth and the sidereal time of setting, 00, is noted. The tele-
scope is then rotated about the horizontal axis until its position is such that
the southern or time star will pass through the field of view. The transit
of the time star is observed, and the entire process is then repeated for a
second time star, with the instrument in the reversed position. The data thus
obtained constitute a set and permit a determination of the error of the time-
piece.
If a simultaneous determination of time and azimuth is required^ the
program for a set will be
Set on the mark and read the H. C.
Set on Polaris, note the time, and read the H. C. \ C. W.
Observe the transit of the time star.
Set on Polaris, note the time, and read the H. C. )
Observe the transit of the time star. V C. E.
Set on the mark and read the 11. C.
in which C. W. and C. E. are to be interpreted as meaning that if the instru-
ment be turned from the mark to the north by rotating about the vertical
axis, the vertical circle will then be west or east, respectively. The plate levels
should be carefully watched, and if there is any evidence of creeping, the in-
strument should be relevelled.
The observing list with the settings for the time stars should be prepared
in advance. It is also desirable, in order to save time in observing and to avoid
errors in the identification of the stars,' to calculate in advance the approxim-
ate times of transit. Disregarding the errors in level and collimation we have
from (225)
d' = a + a0A-JO (233)
in which JO represents an approximate value of the clock correction. To de-
rive a value for the term aaA we combine equation (226) with the value of A
from (214), and write
We thus find
aaA=P(tand — tan =4™7 sin (f). — ih3Om). (23?)
^ r r \o w> / • \ J .//
Since aaA need be known only very roughly, we may use a constant value
for #„, choosing for this purpose the sidereal time corresponding approxi-
mately to the middle of the observing program.
9
130 PRACTICAL ASTRONOMY
/•having been calculated ^01^(235)' we find the value of a0A for each time
star from (234) by introducing the corresponding value of d. One or two
places of decimals are ample for the calculation.
The observations having been secured, the first step in the reduction is
the determination of an approximation for the clock correction of sufficient
accuracy for the calculation of the azimuth of Polaris. Neglecting errors in
level and collimation we have from (225)
M. = a—d'+aaA, (236)
which applied to the time star transiting nearest the zenith will give the
required approximation. For the term a0A we may introduce the value calcu-
lated by (234) in preparing the observing list. Collecting results we have the
following notation and formulae:
«0, TT, and a, 8 are the coordinates of Polaris and the time star,
respectively;
00 and d', the sidereal clock times, respectively, of their observation;
S and M, the readings of the horizontal circle for settings on Polaris
and the mark, respectively;
Am, the azimuth of the mark rqeasured from the south, positive
toward the west;
Ad, the error of the timepiece, and J00, an approximation for this
quantity.
/„= 00+^00 — «„, «0 =
A = sin ( (237)
C —
C e ~i L- w
Log G or log G sec tp is to be taken from Table IX, which is reprinted here
for convenience, with the argument ta; E or C\ from Table X with the argu-
ment d. The subscripts w and ^ refer to observations made circle west and
circle east, respectively. Finally, calculate
Am = Y-z \M, — (S— a.). + Mm — (S— OJ — 180°, (238)
where the subscripts attached to M refer to settings made with the instrument
in such a position that if turned toward the north by rotation about the verti-
cal axis, the circle would then be west or east, respectively, according to the
subscript.
For the determination of the error of the clock, a0 should be expressed in
seconds of time; for the determination of the azimuth, in minutes of arc. The
values of A are needed to four places of decimals, and when once obtained,
should be preserved, since, for a given latitude, they may be used unchanged
THE VERTICAL CIRCLE METHOD
131
for several months. If the coHimation is known to be small and the declina-
tions of the two time stars do not differ too greatly, it will be sufficient to
take the mean of the values of Jd' for C. W. and C. E. as the error of the
timepiece.
TABLE IX, 1910.0
TABLE X
<0
log G
log G sec
4 and 5, Sec. 26,
60,
last,
64,
eq-(72),
7o,
J7,
73,
75, prec. eq. (u7),
80,
81, 4, Ex. 33,
85,
96, prec. eq.
98,
98,
ERRATA
Many nebulae show continuous spectra, indicating that they may not
be wholly gaseous in constitution.
for cos 0 co (p read cos z cos