PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
535 
which will give rise in the right-hand side of the equation at the end of art. II, to 
the terms 
2a^ens\np' 
Hence, taking these terms into account, and putting the equation under the form 
>-(V) + . . 
(9.) 
we shall have 
3 «?Ao 1 P^e' 
/ 6 aAo , 
\'^na da 
2n da^ ‘ 
COS {•or — ■55-') — •CT 
2a dAo 
*” J 2 2/ t\‘2 
“ — s (w — n')‘‘ 2 
n da 
/7a dAo a^ d‘^Ao\ 
' ' n — n 
2. 
-2L 
L , M-fN \ , 
-«'))+s(2« + »(»-«0)/ 
W=^(cos(?jr — sin (z? — zsr')) cos/?^- 
2adn d'^Ao 
^2_^/2 
cos p i~\~ 2 • 
It may be remarked, that in the terms containing the disturbing force we have put 
for p and p\ for p', which is plainly allowable, because the reasoning might be 
repeated with these values in the place of /> and jo'. Also, it appears that the ex- 
pression for which W is substituted contains a term multiplied by /. This term 
might be included in p^, but it is more convenient to retain it in its present position. 
I proceed to develope r in terms of f by means of the equation (9.). 
13. This equation must give a result of this form, 
H and h representing respectively the terms which contain, and those which do not 
contain, the disturbing force. Hence, omitting &c.. 
Consequently, putting g for the last term within the brackets of equation (9.), 
= cos (p^+e\/l -IP) + sin (g, -j- e\/ 1 - H^) nearly. 
Hence H=cos (g^-l-c\/l — H^) 
and A=sin -IT) . 
By the first of these equations H may be developed in a series proceeding according 
to the powers of e, which will be found to be identical in form with the analogous 
