IN THE MOTIONS OF THE EARTH AND VENUS. 
85 
Terms of the first order, 
+ d . cos (2 + 1 — 2 Q) + e . cos (1 + 2 — 2 6) 
Terms of the second order, 
+ e ‘ 1 . cos (3 + 1 - 2 6) + e' e . cos (2 + 2 - 2 6) + e 2 . cos (1 + 3 - 2 6) 
Terms of the third order, 
+ y e' 3 . cos (4 + 1 — 2 &) + y d 2 e . cos (3 + 2 — 2 6) 
— e’ e 2 . cos (2 + 3 - 2^) 4-je 3 . cos (1 + 4 — 2 6) 
On multiplying this by f 2 it will readily be seen that f 2 in the coefficient is 
always accompanied by — 2 6 in the argument, and that there is a necessary 
connexion between them. We may therefore omit 2 6 ; and thus we have for 
the development of f 2 . cos (f + v — 2 6) 
Term of the second order, 
f 2 . cos (1 + 1). 
Terms of the third order, 
+ d f 2 . cos (2 + 1) + ef 2 . cos (1 + 2). 
Terms of the fourth order, 
+ ^ d 2 f 2 .cos (3 + 1) + d ef 2 . cos (2 + 2) + |- e 2 f 2 . cos (1 + 3) 
Terms of the fifth order, 
+ y d 3 f 2 . cos (4 + 1) + ~ d 2 ef 2 . cos (3 + 2) + y d e 2 f 2 . cos (2 + 3) 
+ | e 3 / 2 . cos (1 + 4) 
