IN THE MOTIONS OF THE EARTH AND VENUS. 
77 
+ l|Ae'3.sin(3 + 0)-l|Ae 3 .sin(0 + 3) (C) 
+ be'* • sin (4 + 0) — sin (0 + 4) (D) 
+ ^A 4 ». 8 m(6+0)-^A^sm(0+5) (E) 
The cosine is 
cos ( k — k ) . cos (A — !B — f— C — D — J— E) 
— sin (>t — ^:) . sin (A + B + C + D + E) 
or 
,, , v f A°- + 2AB + B 3 + 2AC + 2AD + 2BC , A* + 4 A 3 B ") 
cos {k-k) . |1 2 r 24 / 
. /7 / \ ("a i t> , i tv i t' A 3 + 3 A 2 B + 3 A 2 C + 3 A B 2 , A 5 1 
— sin (k-k ) . | A + B + C + D + E g I 20 } 
omitting all products of an order above the fifth. 
16. In expanding the powers of A, B, &c., and in multiplying the expan- 
sions by cos {k — k) and sin {k — k), the rules of (8) must be strictly followed. 
Thus we find at length for the value of cos {kv 1 — kv ) : 
Principal term, 
cos (k — k) 
Terms of the first order, 
-f -ke' . cos (k -\- 1 — k) — ke . cos {k — k— 1) 
Terms of the second order, 
(~k 2 -\- -£■ k^e ' 2 . cos (k + ’Z — Jc^ — k 2 e'e. cos 1 — k— l) 
+ ^4- k 2 — ^ k) e 2 . cos (k — k — 2 ) 
Terms of the third order, 
(^k?+^k 2 + l ^k^e'3.cos(k+3 — J^ + (—^W—^k 2 ^d 2 e.cos(k+2 — Jc^-\^ 
