PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
537 
A I • T T 2a dAn ^ L 
Also, since 
2cU cos /),= ^ cos Pi— 2-„i!_j,S(„_„()sCOS (p,+S9). 
Consequently, by putting for H its known expression from the elliptic theory, sub- 
d'^A d'^A 
■a foi' a -jznzn and making use of the foregoing equalities, with 
stituting 
.2^°. 
da 
dod- dado!’ 
the values of V and W given in art. 12, the equation (10.) becomes as follows 
r 1 dA(. ^ . 
a = + (cosp -COS 3 ;?^) 
r 3e dA^ e d'^A^ , d / . dA, „ d"^A,\ / . , x] 
+ -^J(cos(z^-?^)+2w^sm(z^r-tir)j|cos/y, 
dA, , d^A, 
'Cd 
2n . dAs 
1 n — w'***' ' "da 
CO^ sq 
r 
+ 42 . 
2n , c?As 
’A.+'-aT 
n — n 
v?-—s\n — n’f “(n + s(a— a'))2 — j 
—s){n — n’)—3n 
n + s{n — n!) 
/ 2i 
\n — 
2n dA,\ 
(2s®— s)nAs+ ((s® — 2s){n—7^)—n''^ 
— (n'4-s(a— a'))^ 
a' + s(?z — n') 
3n . 
/ ^s~ 
— TV ® 
d^As 
2 dcd 
A dA. 
a® d'^As 1 
2 dd^ J 
COS(/>, + 6'9) 
In this equation s has the values +1, +2, +3, &c. in the terms containing cos 
and cos(p^+>^^)j and the values +0, — 1, +2, &c. in the term containing cos(p'4-'^y)- 
On comparing this expression for the radius- vector with that obtained by Laplace*, 
terms will be found in the latter identical with all the above, with the exception of 
those that contain e' cosp^; and there are other terms to which none of the above cor- 
respond. These are only differences in form, arising from difference in the processes 
of integration. It is chiefly important to remark, that in the foregoing expression 
for r there is no term containing ent as a factor. The signification of that which 
contains e'nt will be presently considered. 
15. Having obtained the development of the radius -vector, it is easy to infer that 
of the longitude (^) from the equation (6.), viz. 
dA_ 
dt' 
A i 
r^J d^ 
J dVi 
dt in terms of t, which has been already found. Putting 
r^+^r for r, and taking Ir to represent the terms multiplied by the disturbing force. 
* Mecanique Cel. part 1. liv. ii. No. 50. 
