530 
PROFESSOR CHAELIS ON THE PROBLEM OF THREE BODIES. 
Hence, since ^=--2ew^ sin p—5eV sin 2 j 9, the following equation will be obtained 
to terms of the second order with respect to e and e ' : — 
2 J - 2 -^ 1 « sm (p+sj) 
+ 2 . 
n-\-s{n 
■^) e'sinsj 
A‘) (2P+*?) 
+ 2 . 
n' + s[n 
2n^ / „ . , , dA.s\ , . , , 
^jee' sin;? cos {p+sq). 
8. Similarly we have to express as a function of t. Since ^ represents the 
partial differential coefficient of R with respect to r, 
dr 
dr ' dt 
dr dt 
the values of s being 0, 1,2, 3, &c 
But 
dr ^ f^R^ dr , ^ 
= coss{0-6), 
„ </As dks , , d’^Xs , , . c?^A., a'V® 
R,=A,+^ au-\-^ a!u -yrjTr'aduu! 
da’ 
dc? ’ 2 ' dada’ 
da'^ 
Hence 
Also 
dRs dr du/dXg d'^Xg _ t?^A 
dr dt ^dt\da' 
da^ 
dado! 
a'u'^ . 
du 
■^=en sin p-\-e^n sin 2p nearly. 
Consequently to terms of the second order with respect to e and e', 
«?R dr dXs . 12 / dK a d'^Xs\ . ^ , , d^X^ . , 
Also to the first power of e and e', 
2 eos5(^ — ^') = 2 cos sq — 2 8\nsq.s{v—v') 
= 2 cos sq-{-2es (cos (p+^^) — cos {p—sq)^ 
— 2e's (cos (p'+A';) — cos {p' — sq)^ 
.s being equal to 0, 1, 2, 3, &c. Or, if5= + 0, +1, +2, &c. on the right-hand side of 
the equality, 
2 cos s{0 — ^') = cos sq-\-2es cos {p-\-sq) — 2e's cos {p' -\-sq). 
Hence, placing the terms corresponding tO5=+0 apart from the others, and obser- 
ving that sin {p-\-}>q) cosp' is equivalent to sin;? cos {p-\-sq) when ^'= + 1, +2, &c., it 
