IN THE MOTIONS OF THE EARTH AND VENUS. 
89 
Terms of the fifth order, 
+ 2 e '/ 4 . cos (3 + 2) + 2 e / 4 . cos (2 + 3). 
27. This is now to be multiplied by cos ( kvhv), the expansion of which 
has been performed in (16). Effecting this operation, we have for the deve¬ 
lopment of cos (kv 1 — kv) ./ 4 . cos (2 v' + 2 v — 4 0), 
Term of the fourth order, 
yr/ 4 . cos (k + 2 — k — 2 ) 
Terms of the fifth order, 
f ^-k + l) e '/ 4 . cos (k -f 3 — k — 2) + (— k + l) e / 4 . cos (£ + 2 — k— 3) 
Section 10. 
Selection of the coefficients of cos (13 — 8 ) in the development of 
^ /y&> 
— m . — .- - ./ 4 . cos (2«/+ 2 v — 4 $). 
{V s — 2 r 1 r. cos (u f — y) + 
3 
28. We must suppose the expression of (27) to be multiplied by ~^m, and by 
(/*) / . (fr) 
the expression for — r ? (which will be formed from that of (18), puttingC ? 
(Jc)\ 
for Ci j. Then giving to k different values, we must select the terms in the 
product whose argument is (13 — 8 ). It is easily seen that 10 and 11 are the 
only admissible values of k. Thus we get these coefficients; 
k— 10 . 
mx{-|( 0 , 0 )++ 1 , 0 ) ) + .</•*.(N < 10 ) .e'/‘) 
*= 11 . 
r 07 3 3 (11) 
m X | -g- (0,0) + yg (0,1) j C f . ef . 
N 
MDCCCXXXII. 
( 11 ) 
• . (N* .c/ 4 ) 
