THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 59 
The expression for (JZ) can be put in a form in which both the angles are mean 
anomalies. ‘Thus, resuming the expression for (/7), 
(Ca Fi cosie Gr cos f’— ch Gr cos f’—1.sin « (yr .cos f' 
r 
7 a\2 sin f’ a\2 sin f’ 6 a\2 sin f’ 
+U.coseé C) F su — a (“) a= £ +th’'.sine. @) a 
7 cos ¢ r cos * cos g’’ 
in which 
a 
hk = —.k.cos (i—K) 
h' = E- . ' k TI TK ee | vcos V 
Vv = —. cos p. cos 9’. k.cos (I— Kj) = gu. 
p C 5 1 v sin V 
(=i COS) Pr. k.sin (11 —K) = fu. aie 
a - 
U eee “yaa , k 9 (11 Pabek K,) ee | p cos Ie 
= ae CUS Gp : ,- Sin 1) = 3u.—=, 
a 
: a’\2 a\? sin f” 
we find the expressions for (<) cos f’, (5) as as follows. We put as before 
¢ 
@) ws" = Dar COs gy aie Y's cos 29’ + Y's GOs 3g + ete. 
'\2 sin f’ 5 ‘ : F BPS cu eae, 
(*) am = 8 sin g +0: sin 29’ + 0’, sin 8g’ + ete. 
c 0 7 , O 
If we differentiate (, cos f relative to q’ we have 
A 
d(G.cos 7’) = cos fl dr’ eis Tid sin f’ Cia sin /” 
dg’ a ala al” fala cos v? 
; dr’ ae’ sin f’ df’ @® 
since —— cae —_— 7 + COS 
dg cos g dg ip 
and hence 
a? G cos f’) a! 
ees SS cose 
