544 
ON THE VARIATIONS OF THE POSITION OF 
[484 
C’A = b + db, 
= 6 + ( — tan & sm <f>' + tan cf) cos d> sin 6), 
= d ~sà® (p '- :pcos 
Z.C = C + dC — <3> — cos 6 tan <f> + cos 6' tan cf)', = $ — q + q. 
Suppose v, v' are the longitudes of the planets in their two orbits respectively; 
that is 
whence 
v = OA + Am = 0 + 0 + Am, 
v = QB + Bm' = 0 + 6' + Bm', 
Cm = C'A + Am, = v — ® — (p' — p cos <E>), 
C'm' = C'B + Bm, = ?/ — ©' + s - r ~ c p (P ~ P cos < ï > )> 
Z C = <4> - q + q ; 
say these values are v — © + x, v' — 0' + x, d> + y. Then if H is the angular distance 
mm' of the two planets, 
cos H = cos (y — © + x) cos (v — ©' + x') + sin (v — © + x) (sin v' — ©' + cc') cos (4> + y), 
= cos (v — ©) COS (V' — ©') + sin (v — ©) sin (v' — ©') COS 
+ X [— sin (V — ©) COS (V' — ©') + COS (V — ©) sin (v' — ©') cos <J>] 
+ X [— COS (V — ©) sin (v — ©') + sin (V — ©) cos (v‘’ — ©') cos $] 
+ y [— sin (v — ©) sin (V' — ©') sin <J>], 
= cos H + V suppose. 
The disturbing function for the planet m disturbed by m' is 
fl = m' 
r cos 
r 2 _}_ /2 _ 2rr' cos H 
(il = — R, if R is the disturbing function of the Mécanique Céleste) ; and the term 
hereof which involves V is 
d . cos H 
where after the differentiation cos H is replaced by cos H, 
= m 
(r 2 + r' 2 — 2rr' cos Hy r 
+ 5iV,