IN THE MOTIONS OF THE EARTH AND VENUS. 
79 
Section 4. 
(k) 
Expansion of —T x , to the fifth order. 
m 
17 . We suppose — ^ ^ a r i r , cos ^ ^ + r a, , the first term in the expres¬ 
sion of (14), to be expanded in the form 
l (o) (i) (2) 
—I\ — mT^ .cos (v—v)—mT^ . cos (2 v' — 2v)— &c. 
(k) 
— mT, . cos (kv 1 — kv) — &c. 
where \ ri \ kc. are functions of r' and r only. 
Let 
*/ {a ! 2 — 2 a! a cos (1/ — v) + a 3 } 
= ci ) -f Cx cos(«/—v)+Ci \ cos(2v'—2t>) + &c.+ ci \cos(A;i/—A;i;)+&c. 
(A) . (A) 
then T, is the same function of r and r that C 4 is of a! and a. Consequently, 
if r' = a! (1 +q'), r = a(\-\-q): and if for convenience we use the notation 
(0 
(m, n) Ci 
to express that which is commonly written 
+ n c (k) 
a ! m . a n . 
d a lm . d a n 
,(*) 
we shall have for — I\ the following expression 
,(*) 
(A) 
(1,0)0, • 
(A) 
s'-(2,0) c;. 
q n 
' 2 
(A) 
(3,0) C 4 
■ 0 
(A) 
(4,o)c;. 
q'* 
24 
(A) 
~(5,o)c;. 
qf 5 
I 20 
(A) 
( 0 , 1 ) c;. 
(A) 
9-(i,i)c;. 
q'q~ 
(A) 
(2,1)Ci . 
q'*q 
2 
(A) 
(3,i)c;. 
q' 3 q 
6 ' 
(A) 
-(4,i)c;. 
q'*q 
24 
(A) 
~ (0,2) Ci . 
O 
(A) 
(l,2)Cl . 
f a 
9 T 
2 
(A) 
(2,2) Ci . 
q 1 - q 2 
4 
(A) 
-(3,2)c; . 
9 ,3 f 
12 
— 
(A) 
( 0 , 3 ) c; j . 
q 3 
6 ~ 
(A) 
(l,3)Ci . 
q' q i 
6 
(A) 
-(2,3)c; . 
q 12 q 3 
12 
— 
(A) 
(0,4) Ci . 
<L 
24 
(A) 
~(i,4)cy. 
9 f 9 * 
24 
(A) 
-(0,5)0, . 
9 5 
120 
