IN THE MOTIONS OF THE EARTH AND VENUS. 
77 
+ H kd z . sin (3 + 0) — — k£ . s in (0 + 3).(C) 
12 1 
+ ^ k e'* . sin (4 + 0 ) - k *. sin ( 0 +4).(D) 
- 4 - * ^5. sin (5 + 0)-I29|£e5.sin(0 + 5).(E) 
The cosine is 
cos {k — k) . cos (A + B + C + D + E) 
— sin {k — k) . sin (A + B + C + D + E) 
or 
cos {k — k) . 11 — 
A 2 + 2AB + B 2 + 2AC + 2AD + 2BC , A 4 + 4A 3 B 
2 
+ 
24 
} 
. „ n f, ,D,n.n,r A 3 + 3 A 2 B + 3 A 2 C + 3 A B 2 , A 5 ) 
- sin {k — k) . | A + B + C + D + E-g-f- ^ > 
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 {k v' — kv): 
Principal term, 
cos {k — k) 
Terms of the first order, 
+ &e'. cos (£+ 1 — k) — k e . cos {k—k — 1) 
Terms of the second order, 
& 2 + kje' 2 . cos (k + 2 — k} — k 2 e’ e . cos ^+1— k— 1) 
+ ^4" k 2 — ^ k') e 2 . cos (k—k — 2^ 
Terms of the third order, 
(~jj ^+ •4^ 2 +^^) e,3 > cos(&+3 — &) + (—■ ^k?—^k 2S Jd 2 e.cos(k-\-2 — k— l) 
