IN THE MOTIONS OF THE EARTH AND VENUS. 
91 
differentiating the coefficients and retaining the same cosines. The coefficients 
(8) 1 (8) (8) t 
will be differentiated by changing (0,0) Ci into (1,0) Ci , (3,2) into 
1 (8) 3 (8) 
(4,2) Ci + (3,2) Ci , &c. Thus new terms will be introduced whose cal- 
culation is rather troublesome. It is desirable, then, to inquire whether it is 
probable that the term depending on will be comparable in magnitude to 
the other term which has the same argument. 
31. Now if we put A . cos {13 {rit + s') — 8 (nt -f- s) + B] or A cos (13 — 8), 
for one of the terms, we find 
" = _ 3 .13.".A.sin (13-8) 
whence 
. XT , , 3. 13. re' 3 a' . . ^ . 
» = N' + A . cos (13 - 8) 
( 13 re' — 8 n) fx, 1 
(where N' is constant and = mean value of ri) 
re' 2 a 1 
2 re' a n d A 
dt 
= +3.13 . -jj-A.t. sin (13- 8)+-^-. ^ cos (13- 8) 
whence 
s' =E' — 
S . 1 3 re' 2 a' 
3.1 3 n n a! 
(13 re' - 8re)/*' A * * ‘ C0S ( 13 8 ) + (13 re' - 8 re)>' A ' Sin ( 13 ~ 8 ) 
, 2 re' a' 2 d A . . 
+ (13rf-8n)^ •g7 51n ( 13 ~ 8 ) 
(where E' is constant and = mean value of s') 
and rit + s' (which, by (1), is the first term of v') becomes 
XT , , . -c, . f 3. 13 re' 2 a' , 2 re' a' 2 rfAl . /1n 
Ni + E +{(T3re'-8re)V A+ (13 re' -8 re),*' ‘ ^jsm(13-8). 
The ratio of the two coefficients of the inequality sin (13 ~ 8) is 
39 re' . , d A 
2 ' 13 re' — 8 re ' a da' 
or nearly 4800 X A : a ^ 
,d A 
