AnnualReport-1881

Here, V = 15.9288, N = 42, M = 16; whence, by (6), e0 = 0′′. 53 (or 0′′. 50) in

This value of the probable error of a single determination not only illustrates the high excellence of the method, but also exhibits a very satisfactory agreement with the adopted value (e.g., the 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 calculated, 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.

Let n be the number of determinations of latitude from a pair of stars, l the mean resulting latitude from those determinations, w the weight of this mean (= n/2), and wl 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, L0, is found by the formula,

X. L0 = W ...(7)

Then, to find the probable error of L0; let v for each pair be the difference between L0 and the value of l for that pair; wv2 the product of w for that pair into the square of the residual v; V the sum of the values of wv2 for all the pairs observed; M the number of pairs; and e' the probable error of L0; then,

V e'2 = 0.455 (M - 1) W ...(8).

n = 22 being = 4e2 = 1.12.

The following Table shows the steps of these computations,

TABLE IV.
FINAL LATITUDE AND PROBABLE ERROR.

No. of pair n No. of obs. W w l wl v wv2 1 5 4.46 10.23 45.64 1.69 12.59 2 3 2.68 13.05 34.97 1.14 3.48 3 1 0.89 13.97 12.43 2.06 3.78 2 1.79 14.05 25.15 2.14 8.20 4 3 2.68 10.38 27.82 1.53 6.27 2 1.79 11.57 20.71 0.34 0.21 5 2 1.79 12.53 22.43 0.62 0.69 6 2 1.79 12.19 21.82 0.28 0.14 7 2 1.79 13.29 23.79 1.38 3.41 8 1 0.89 11.70 10.41 0.21 0.04 9 2 1.79 10.29 18.42 1.62 4.70 10 3 2.68 9.84 26.37 2.07 11.48 11 2 1.79 13.16 23.56 1.25 2.80 12 2 1.79 11.16 19.98 0.75 1.01 13 2 1.79 13.12 23.48 1.21 2.62 14 3 2.68 13.36 35.80 1.45 5.63 15 2 1.79 9.86 17.65 2.05 7.52 16 1 0.89 11.70 10.41 0.21 0.04 17 2 1.79 12.49 22.36 0.58 0.60 18 1 0.89 13.47 11.99 1.56 2.17 19 1 0.89 10.73 9.55 1.18 1.24 20 1 0.89 13.14 11.69 1.23 1.35 21 1 0.89 9.89 8.80 2.02 3.63 22 1 0.89 13.56 12.07 1.65 2.42 23 1 0.89 9.25 8.23 2.66 6.30 24 1 0.89 13.19 11.74 1.28 1.46 25 1 0.89 11.82 10.52 0.09 0.01 26 1 0.89 14.46 12.87 2.55 5.79 W = 48.22 X = 574.06 V = 106.32

Here W = 48.22, X = 574′′.06, V = 106′′.32, and M = 28.

Whence, by (7) and (8),

L0 = 11′′.91 ± 0′′.19

Therefore,

LATITUDE OF CENTRE OF OBSERVING PIER = 22° 18' 11′′.91 ± 0′′.19 (0′′.19 = 19.2 feet).