IN THE MOTIONS OF THE EARTH AND VENUS. 
115 
of which, on the principle of ( 8 ), &c., we are to take only 
m .r'r. <p'f. cos (d + v — O' — 0) 
f' 2 — 2 r' r . cos (d — v) + r 2 — 2 / r .f ~. cos (d + v — 2 3)}^ 
Expanding the denominator by powers of f 2 , this becomes 
m . d r . §\f. cos (d + v — 8' — 5) m . 3 r' 2 r 2 . $'f 3 . cos (d + v — 8' — 9). cos (F + v — Q6) 
{r fs — 2 dr. cos (t/ — v) + r 2 }" 
+ 
{d 2 — 2r'r. cos (w' — u) + r 2 j. 1 
or 
. r' r . $7* cos (o' + v — 8 f — 8) 3 . r /2 r 2 . $'f 3 . cos (2 + 2 w — 6' — 3 3) 
ffi 2 — 2 r r . cos [v — v) + r 2 f 
+ 
{ r ' 2 — %r r . cos ( v' — p) + r 2 p 
Section 19. 
Selection of the coefficients of cos (13 — 8 ) in the development of the two last 
fractions. 
63. If we compare the first fraction with the fraction developed in Section 8 , 
we perceive that the following are the only differences between them. The 
signs of the coefficients are different: and in the coefficient of the new fraction 
(and in every term of its development) there is f instead of f with the corre¬ 
sponding change of argument. From this it is readily seen that the coefficient 
of cos (13 — 8 ) will be formed from that in Section 8 (Art. 24), by changing the 
sign and multiplying by jr ; the argument always being changed according to 
the rules of (9). The coefficient is therefore 
( 9 ) (10) (11) (12) 
- M . e ' 3 <p 7 - M . e' 2 e <p'f - M . e' e 2 <p'f - M . e 3 <p'f. 
64. If we compare the second fraction with the fraction developed in Sec¬ 
tion 10 , we see that there are the same differences as those mentioned above, 
with this additional one, that the multiplier is double of the multiplier of the 
fraction in Section 10. Thus the coefficient of cos (13 — 8 ) is found to be 
( 10 ) 
- 2 N . e >'/ 3 
(ii) 
2N .ecp'ffi 
The sum of the terms in these two sets, multiplied respectively by the cosines 
of their proper arguments, constitutes the whole term of R which we have to 
consider. 
