102 
MR. GALLOWAY ON THE PROPER 
and A, the right ascension of the star and of the point Q. It is more convenient, 
however, to compute the side QS (which is required in the subsequent calculations) 
in the first place, and to make use of it in computing -dj , because the same logarithms 
which are required in proceeding after this manner serve also for computing the 
coefficients of the equations of condition. 
The ordinary trigonometrical formulse give 
cos QS=cos QP' cos SP'+sin QP' sin SP cos QP'S. 
Now since Q is assumed to be on the north side of the equator, and all the stars 
included in the present investigation are on the south side, we have 
cos QP'=cos (90°+D) = — sin D ; sin QP'=sin (90°+D) = + cos D. 
cos SP' = cos (90°— e$) = +sinci ; sinSP'=sin (90°— ci) = -j-cos£; QP'S=a— A. 
Denoting QS by %, and substituting these values in the above formula, we get 
cos x— — sin D sin^+cosD cos S cos (a— A). ...... (2.) 
Having found or QS, the angle %|/ is computed from the formula 
sin \p r = 
sin QP' sin QP'S cos D sin (« — A) 
(3.) 
sinQS sin% 
This sine belongs to two angles. In general there will be no difficulty with respect 
to the quadrant to which it belongs ; but in a case of ambiguity, which may occur 
when $ is near 90°, recourse may be had to the formula for cot $ given above, 
which, on substituting for QP', SP', and QSP' their expressions in terms of D, and 
(a — A), becomes 
-cot^'=+ sin^cot(a-A)+^-^ =I y (4.) 
In computing from the above formulse attention must be paid to the changes of 
sign of cos (a — A) and sin (a— A). The most convenient mode of proceeding, perhaps, 
is to take the stars in the order of right ascension, beginning at the declination circle, 
passing through Q, and adding 360° to all the values of a which are less than the 
assumed value of A, that is, to the right ascensions of all the stars excepting those 
which lie between the declination circle which passes through Q and that which 
passes through the first point of Aries. The values of (a— A) will thus be expressed 
in a series proceeding from 0° to 360°, and the sign to be prefixed to the cosine or 
sine becomes known from the value of the angle. 
The equations of condition are formed as follows : — 
Differentiating equation (4.) on the supposition that A andD are the variable quan- 
tities, we get 
dV 1 f . V . t. V / .4 7* . COS 5 7T . 
2 = -r— tt j sin o — tan D cos 5 cos (a— A) iaA+^— 7 - — a > „ ~ 2 r> dD ; 
sirr rp sin (a — A) I 1 v J sin (a — A) cos P 
now 
sin 4^' 
cosD 
sin (a— A) sinp^’ 
therefore 
7 ,. cosDf t-v-v. .-r>. v , . 4 , cos 8 sin (a — A) , . 
d-ijy 1 = 7-2— lcosDsin£+ sin D cos & cos (a— A) dD. . . (5.) 
sm x l 
sm x 
