64
V 71
Probable error of observation.-On the assumption that the probable error of a single complet observation is the same for every pair of stars, a comparison of the results in the above Table furnishe means for determining its mean value from all the pairs, thus. Let v
be the seven differences for any pair between the several results for latitude from that pair (of which there has been n observations) and the mean result from that pair; and let such differences or residuals b similarly taken out. for all of the pairs which have been observed more than once; and let I be the sum of the squares of all these residuals, N the total number of determinations, M the total numbers pairs observed more than once, and Q (6745) the factor for reduction of mean to probable errors. Then & the probable error of observation, is found by the formula,
V
Q
=455
N-M V
N-M
.... (6)
Table III, below, exhibits the steps of the process for determining V
TO FIND THE PROBABLE ERROR OF OBSERVATION.
TABLE III.
No. of pair
Stars of the pair; Seconds of latitude
B. A. C. Nos.
Means
V
2,9.
of
the
w
0'.58
Here, V =
15,9288, N
42, M
16; whence, by (6), e — This value of the probable error of a single determination not only illustrates the high excellence 0.50) in the method, but also exhibits a very satisfactory agreement with the adopted value (e
United States' Coast Survey Department, where the method has been in use for some 35 years.
Determination of the Final Latitude.--The probable error of observation having been thus cal- culated, the observations can now be combined by weights in the usual way, and the final latitude and its probable error ascertained, as follows, it being assumed that the North Polar Distances in the Nine- Year Catalogue are correct.
n
Let n be the number of determinations of latitude from a pair of stars, the mean resulting latitude om those determinations, we the weight of this meau (=), and wel the product of the weight into the latitude; and let these quantities be taken out for the whole of the pairs observed. Then, if W be the sum of the weights, and X the sum of the values of wl, for all the pairs, the final latitude, L., is found by the formula,
X
W
..(7)
ون
Then, to find the probable error of L.; let v for each pair be the difference between L, and the value of for that pair; the product of t for that pair into the square of the residual e V the
sum of the values of wv for all the pairs observed; M the number of pairs; and e' the probable error of L; then,
#
9.55
0.68
4624
9.54
0.69
.4761
1
745-786
10.49
10.23
0,25
.0625
11.18
0.95
.9025
10.41
The following Table shows the steps of these computations,
0.18
.0824
e-.455
V (M-1) I
.(8).
n
being
1.12
13.59
0.54
.2916
2
872-901
12.26
13.05
0.79
TABLE IV.
.6241
18.31
0.26
.0676
FINAL LATITUDE AND PROBABLE ERROR.
3
602-518
13.97
13.39
0.65
.4225
No. of
No. of obs,
n
wl
4
999-1052
む
Wys
14.56
14.05
0,50
.2500
pair
===n
1.12
14.20
0.15
.0225
"
10.66
0.28
.0784
5
1087-1138
9.27
1
5
4.46
10.23
45.64
1.68
12.59
10.38
1.11
1.2321
11.22
2
2.68
13.05
34.97
1.14
3.48
0.84
7056
0.89
18.97
12.43
2.06
3,78
6
574-628
12.27
0.70
10.87
11.57
.4900
2.68
14.05
37.65
2.14
12.27
0,70
.4900
5
2.68
10.88
27.82
1.53
6.27
12,56
6
1.79
11.57
20.71
0.34
0.21
0.03
7
.0009
1045-1052
11.82
12.53
77
2.68
12,53
33.58
0.62
1.03
0.71
.5041
13.22
0.69
4761
8
1.79
12.19
21.82
0.28
0.14
9
1.79
13.29
23.79
1.38
3.41
8
11.97
1326-1346
0.22
12.19
.0484
10
0.89
11,70
10.41
0,21
0.04
12.41
0.22
.0484
11
2.68
10.29
27.58
1.62
7.03
9
1391-1408
13.37
0.08
13.29
.0064
12
2.68
9.84
26.37
2.07
11.48
13.22
0.07
13
1.79
13.16
23.56
1.25
2.80
.0049
10
14
1.79
11.16
19.98
0.75
1,01
1408-1487
11.70
15
1.79
12.12
23.48
1.21
2.62
10.57
0.28
16
2.68
13.36
85.80
1.45
5,63
粥
.11
.0784
1726-1778
9.74
10.29
0.55
2025
17
1.79
9.86
17.65
2.05
7.52
10.56
0.27
.0729
18
0.89
11.70
10.41
0.21
0.04
19
1.79
12.49
22.36
0.58
0.60
9.21
0.63
12
.3969
20
0.89
13.47
11.99
1.56
2.17
1837-1876
9.17
9.84
0.67
.4489
21
0.89
10.73
9.55
1.18
1.24
11.14
1.30
1,6900
22
0.89
13.14
11.69
1.23
1.35
13
510- 574
13.75
0.59
23
0.89
9.89
8.80
2.02
3.65
.3481
12.57
13.16
0.59
.3481
24
0.89
13.56
12.07
1.65
2.42
25
0.89
9.25
8.82
2.66
6.30
14
1377-1444
11.12
0.04
.0016
11.20
11.16
26
0.89
13.19
11.74
1.28
1.46
0.04
.0016
27
0.89
11.82
10.52
0.09
0,01
15
1420-1444
18.16
0.03
13.09
13.12
.0009
28
0.89
14.46
12.87
2.55
5.79
0.03
.0009
W 48,22
I
574.06
F= 106.32
18.24
0.12
.0144
16
1602-1687
14.90
13.36
1.54
2.3716
11.93
1.43
2.0449
17
1958-2001
9.83
0.03
9.86
.0009
*Here W Whence, by (7) and (8),
48.22, X= 574".06, 106".32, and M-28.
9.89
0.03
.0009
18
1408-1432
Leg
117.91 0.19
11.70
19
1883-1900
12.72
Therefore,
0.28
.0.529
12.26
12.49
0.23
.0529
21
24
2 7 2 * *
20
1990-2082
13.47
755-786
10.73
LATITUDE OF CENTRE OF OBSERVING PIER
(0.19 19.2 feet).
22° 18′ 11.91 ± 0.19
22
867-872
13.14
23
2133-2163
9.89
2398-2442
13.56
25
2398-2469
9.25
26
2486-2555
13.19
27
2617-2659
11.82
28
2700-2714
14.46
15.9288
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