IN THE MOTIONS OF THE EARTH AND VENUS. 
85 
Terms of the first order, 
+ e'. cos (2 + 1 — 2 0) + e . cos (1 + 2 — 2 0) 
Terms of the second order, 
+ e 2 . cos (3 + 1 - 2 0) + e' e . cos (2 + 2 - 2 0) + f e 2 . cos (1 + 3 - 2 0) 
Terms of the third order, 
+ ~ e' 3 . cos (4 + 1 — 2 0) + ^ e' 2 e . cos (3 + 2—20) 
+ -g- e' e 2 . cos (2 + 3 — 20)+-^- e 3 . cos (1+4 — 20) 
On multiplying this by f 2 it will readily be seen that f 2 in the coefficient is 
always accompanied by — 2 0 in the argument, and that there is a necessary 
connexion between them. We may therefore omit 2 0; and thus we have for 
the development of f 2 . cos (d + v — 2 0) 
Term of the second order, 
f 2 . cos (1 + 1). 
Terms of the third order, 
+ e’f 2 . cos (2 + 1) + e f 2 . cos (1 +2). 
Terms of the fourth order, 
+ e' 2 f 2 . cos (3 + 1) + e' ef 2 . cos (2 + 2) + |- e 2 f 2 . cos (1 +3) 
Terms of the fifth order, 
+ ~ e' 3 / 2 . cos (4+1 ) -{■ — e' 2 ef 2 . cos (3 + 2) + e' e 2 f 2 . cos (2 + 3) 
+ -j e 3 f 2 . cos (1 + 4) 
