IN THE MOTIONS OF THE EARTH AND VENUS. 
79 
Section 4. 
,(*) 
Expansion of — T r , to the fifth order. 
m 
17. We suppose — ^^2 _, £ r r r _ cos y _ p ) + , the first term in the expres- 
sion of ( 14 ), to be expanded in the form 
\ m ri } — mTj ) . cos {v' — v) — . cos (2 v' -2r) - &c. 
(k) 
— mY x .cos(kv' — kv) — &cc. 
2 
(0) (1) 
where Y x , r x , &c. are functions of r and r only. 
Let 
\/ {a B — c 2 a 1 a cos (d — v) + a 3 } 
= y Cx '* H- C x \ cos(v'— v)-\~c[ \ cos(2z;’ — 2i>) + &c. + Cx \cos(A;y'— ^y)+&c. 
(k) (k) 
then I\ is the same function of r and r that C i is of a and a. Consequently, 
if r' = a' (1 +q'), r = a (1 +9) : and if for convenience we use the notation 
(0 
(m, n ) C 4 
to express that which is commonly written 
j m 
a ' m . a n 
-\-n Q(k) 
d a lm . d a n 
(k) 
we shall have for — I\ the following expression : 
.(l,0)cf. ? ’-(2,0)cf.^ - (3; 0)cf.^ -(4,0)c' l, .g -(5,0)0™ 
,(*> 
>'' 2 -. ' ' ' 2 
•(0,l)cf. ?- (l,l)cf. (2,l)cf . (3,l)cf . ^ - (4,l)c‘‘ 
cf-^-(2,2)cf.^-(3,2)C l 
(k) (lc) 
v)c; .9-(i,i)c; 
-( 0 , 2 )cf.^-(l, 2 )cf.^| 
,(*) 
'5 
(0,3) C f.^ _(l,3)cf.^-(2,3)C^ 
•(0,4)0;.?- — (1,4) C* 
