PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
539 
the following equation is readily obtained : 
V j. 2 dAo . , \ Se dAo , e d'^Ao De' 
}-^orat= t+ 
J na da ' 
{ 3e «?Ao , e 
an^ da 
da^ 4a^n^ 
COS 
(ar— zy')|sin 
De' 
(/>:+?) 
the value ^=1 being excluded from the last term. 
Substituting the results of these integrations, the following will be found to be the 
value of 6 : 
. , / , 2 </Ao\ , 5e2 . ^ , e3/13 . „ . \ 
«=5+ («+j;-s-;<+T ^P‘+1 [J ®'" 3;>,-sin p, J 
, , 3e dAo e d^Ao , D e' / , , ,_\'l . 
dAi 
2a^n^ 
, 1 V / 2wF _A, \ . 
~^2a^ n') s(w— 
2 [3«+| ("-^C”-"'))] F-^ir -2 «g} sin (ft+i?) 
—2.- 
o-«' / 
ra' + s(: 
1 / 
(m— w') 
(2s®— s) As— as 
da 
-l-2wHjsin {'p',-\-sq), 
2(^«' + s(re— «')) 
where s has the values +0, —1, +2, +3, &c. in the last term, and the values +1, 
+2, +3, &c. in the other terms, and A_s has the same value as A,. 
16. The expressions for r and 6 obtained in arts. 14 and 15 may be put under 
forms more compact, and more convenient for drawing inferences, by making the 
following substitutions : 
1 dAo 
A=“-S5-*+T 
e «?®Ao De7 
E=e-i-, ^“-^sin (®-b') 
4a® </a® 
_ / 1 d'^Ap 
^ yaa da "^2a* c?a® ' 
Bd 
2a* c?a® ' 4aa®a 
cos( 
nr — ra 
o 
N=?^+“ "^5 so that je>^=N/+£— n 
^f=-2rfa ■^+8A» 
d 
e'g-. 
2a®a® 
+^)’ 
4 B 
MDCCCLVI 
