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). 
2 7 . This is now to be multiplied by cos {kv'~ hv), the expansion of which 
has been performed in (16). Effecting this operation, we have for the deve- 
lopment of cos (k v' — kv ) . / 4 . cos (2 v' + 2 v — 4 0), 
Term of the fourth order, 
\ffi . cos {k + 2 — A- — 2) 
Terms of the fifth order, 
( ^k + l) e'/ 4 . cos ( k + 3 — k — 2) + (— \ k + l) e/ 4 • cos (/fc + 2 — k — 3) 
Section 10. 
Selection of the coefficients of cos (13 — 8) in the development of 
— m . — . . / 4 . cos (2 v' + 2v — 40). 
{ r 12 — 2 dr. cos (d — v) + r 2 }^ 
3 
28. We must suppose the expression of (27) to be multiplied by and by 
(k) / . (t) 
the expression for — T a (which will be formed from that of (18), putting C 5 
(*)\ 
for Cr ). 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. 
r q 3 1 (io) (io) 
(0,0) + ^ (1,0) j C f . e'ffi (N . e'ffi) 
k= 11. 
r 27 3 1 (11) 
m X (0,0) + TB (0.1) } V • «/‘ 
MDCCCXXXII. 
N 
( 11 ) 
• • (N .effi) 
