542 
ON THE VARIATIONS OF THE POSITION OF 
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planes of reference and origins of longitude therein, these last being however absolute 
constants not affected by any variation of the elements; so that as regards the variation 
of the elements the disturbing functions are in fact given as explicit functions of the 
variable elements 9, 6', <£>, ff; and where </>, ff and therefore also p, q, p', q' are only 
of the order of the disturbing forces. 
I proceed to work out this idea, for the present considering the development of 
the Disturbing Function only as far as the first powers of p, q, &c. For comparison 
with the ordinary theory, observe that in this theory the disturbing function contains 
only the second powers of the p, q, &c., made use of therein; these are in fact of a 
form such as P+p, Q + q, ... where P, Q are absolute constants and p, q, ... are the 
p, q, ... of the present theory; the ordinary theory gives therefore in the disturbing 
function a series of terms involving (P+p) 2 , (P+p) (Q + q), ... which I now take 
account of only as far as the first powers of p, q, ... viz., they are in effect reduced 
to P 2 + 2Pp, PQ + Pq + Qp, &c. ... The present theory is thus not now developed to 
the extent of giving the p, q, ... of the ordinary theory in the more complete form 
as the solutions of a system of simultaneous linear differential equations, but only to 
the extent of obtaining for these p, q,... respectively the terms which are proportional 
to the time. 
I commence with the following subsidiary problem. Consider a spherical triangle 
ABC (sides a, b, c, angles A, B, C, as usual), and taking the side c as constant, but 
the angles A and B as variable, let it be required to find the variations of G, a, b 
in terms of variations dA, dB and the variable elements G, a, b themselves. Although 
the geometrical proof would be more simple, I give the analytical one, as it may be 
useful. 
We have 
and thence 
that is 
or finally 
Next 
or, differentiating, 
cos G = — cos A cos B + sin A sin B cos c, 
— sin GdC = (sin A cos B + cos A sin B cos c) dA 
+ (sin B cos A + sin A cos B cos c) dB 
sin B sin c , . sin A sin c 7 „ 
= —,—j— dA + — dB, 
tan b tan a 
sin G ir . sin B cos b 7 . sm A cos a 7 _ 
—— dG = .—— dA H : dB, 
sm c sm b sm a 
— dG = cos bdA + cos adB. 
sm A 
sm a = sm c —.—~ , 
sm G 
sm c 
cos a da = g - n ^ (sin G cos Ad A — cos G sin AdC)