PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
529 
-\-v ' and for the sake of brevity put q for 
Then m=— ecos/?+^— -^-cos 2yo, u'=—e'cosp', 
0—6'=q-\-v — y'=^+2esin jo— 2e' sin j»'. 
It is not necessary for our present purpose to employ expressions containing higher 
powers of e and e'. 
7. It is next required to obtain an expression for dt in terms of t. Since each 
of the factors Ro, R„ R2, &c. is a function of r, r' and constants, it follows that 
Also, if As represent the value of R^when a is substituted for rand a' for r', we have nearly 
„ . , dA^ , dA^ , , 
Rs= A.+-^ a M 
* ' da da' 
A , ,dA , 
=As— ae^cos»— a e -^cos » . 
da ^ da' ^ 
Again, sin .s(^— ^')=^ sin r') 
— 5 sin^^ cos5(r — y')-l-5cos sq sin s{v—v'). 
Hence, by substituting the foregoing value of v—d, and retaining only the first power 
of e and e', it will be found that 
^ sin ^')=^ sin^9+e^^(sin (j9+'^5')H"Sin(p— 
— eV(sin (/>' +55') + sin ( jo' — sq)). 
By supposing 5 to have all negative as well as positive integer values, this equation 
may be more briefly written thus : 
S 
s sin s{6—&) = 2 siB i’^+e^^sin {p-\-sq) — eVsin {p' •\-sq). 
Now observing that .ssinA-^cosjo=.ssin (jo+a^'), because a— + 1, +2, +3, &c., the 
following result will be obtained by multiplying the foregoing values of R, and 
s sin s{d—d): 
wi, • sAs . . / as dAs\ . , 
R,s sin a(^— ^')=— sin sq-\- je sm {p+sq) 
Consequently, 
— sin (y +A^). 
2^'-^ J's.R.a sin . s(^— 
2s^ Ao — as 
dK 
da 
*? + 2 - — +s{n-ri) « (/'+*'?) 
25®As + <i's-^^ 
(?'+*?)■ 
