538 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
we shall have to the same approximation as before, 
h 
dt' 
=-"- 17 ; 
2Sr\ 1 
h 2 h , 1 , , rt?R , 
=y^—^{ 1 -i-3e cos (1 -{-2e cos pj | ; 
. d~z-\-\-^dt — 1 ( Ta + 
2h 
1 rt?R 
H-~2 J ^ e cos p^dt. 
The development of the term ^-^dt will be of the same form as in the elliptic theory. 
An equation obtained in art. 7 gives, 
. , 2s^As— 
1 ^ c?R 1 As , 1 ^ da / , X 
n-\-s[n — 7i’) 
dA., 
2s^As + a's-^, 
+ ^2. — rT~7 rr cos(p-{-SQ) '■ 
’ 2a" n +s[n — n) 1 ' 2 / 
JC 
2s2A,- 
as 
d^ 
da 
' (n + s{n — n')Y 
dA 
esin(p^+.sry) 
2s\As + a'5-^, 
+gg ^ ■ >..! si" (p:+»?)- 
2a‘^ ’(a' + s(a — a'))^ 
Since the relations of the constants h, a, n and e are expressed in our problem in the 
same manner as in the elliptic theory, we have h=nd^\/\-~e~. Hence we may put 
na^ for h in the terms involving the disturbing force. Consequently, omitting e% &c.. 
j^8recos/>,rf<=J^‘(-^ 
T 
■^2 
da 
da 2a — s^[n — n’) 
cos sq^e co^ pflt 
/\2 
6e </Ao 
3ne 
V a — lu da ' / I V 
2 "T~sin p , — 27 7 “ — rw 2 — s — 07 {P,-Tsq). 
arr da « (a + s(a — n)) — s^{n — dp ^ 
Also, to the same approximation, 
^ dt^e cos CO'S: sq cos p^dt 
A, 
f(!«j V9 "V 
— — 2-- 
7^sin(p,+.sg). 
In all these equations the values of s are +I5 +2, +3, &c. 
Putting now, for the sake of brevity, D for 2Ai— 2 a^J— and P, G and H for 
(XU/ (XU 
the coefficients of cos sq, cos cos {p\-^sq) respectively in the expression for 
and observing that 
J tdtcosp-- 
tsmp^ cospj 
^2 » 
