534 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
2Me^ / 2/i . . , \ 1 
— — 27 TTa 1 h sm sq SID »+cos sq cos p hr::- 
An-^—s^{n—n'Y\s[n—n') j r ^ i t Js\np 
2Ne^ cos(^ + s^) 
s{n — 'n!){2n + s{n — Y)^ sinjp 
, 2aa!ee'n d'^Ao cosp' 
*' n^ — 71'“^' da da'" sin^ 
„ Ped cos (p' + sq) 
~^^'n^ — (n' + s(n—n'}y' sinp 
Since s=^l, +2, &c., the following equalities are true: 
2*COS .^5 COS7?= 2.COS + 
sin S 5 ' sin JO cos(jo + S 5 ') 
Hence it will be found that the term in the above equation which involves cos (p-\-sq) 
is 
2 / 2Le^ (M + N)e" \cos(j(j + sg) 
2 . / 
—n V 6 
s(?i + s(w — 7*0) s{2n + s{n—n'))J sin|j 
Consequently the equation (8.) becomes 
/ 3 (^Ao 1 d^Ao\ r 1 /-ii 7 ^ 
a dAa 
(Ye (YAa 
2aa!dn d^An 
L 2 
■ 2 • 2 s 7 /\2 cos sq “i“ / • 2 • 
/r — s'^in — rt \ 2 ' n — n 
+ 2 - 
rr — — n' 
Pd 
2Le 
. , (M + N). 
s^n + s(ra — n')) s(2n + s(7* — 7*0) / \ P~T^2> 
'■ — {id + s{n — n')Y 
n 
cos {p'-\-sq) 
12. Before advancing to the next operation, our attention must be directed to the 
failure of the term containing the denominator 7Y—{rd-\-s{n — n')^^ in the case of a==1. 
As the denominator vanishes for this value of s, it is necessary to retrace our steps 
and consider that case separately. Referring to the equation at the beginning of 
art. 9, it will be seen that if .9=1 and Pi represent the consequent value of P, the last 
term becomes PjCe' sin/v cos (y+*y). Also, since 
p' -\-q='rdt-\-^ — tzr'+wi+s — idt — £'=jo+ro — cj', 
we have I'sin/* cos (/>'+y)«/^=J sin y? cos 
(2y>4-ra'— w')— ^sin(w — ot'). 
Thus the equation (8.) will contain the term 
\ CP.eddtp 1 , , t . 
2 .^j 'shi^ ( + ) - 2 sin (*^7- J , 
