PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
531 
will be found that 
rfR dr dA^ . , , dA^ d^An\ . ^ i i d'^A^ . , 
2 * = sinp+ae‘n (^2 sm 2p-2aeaen sinp cosp 
I 'V dAg • / 1 \ 2 dAg • 
+ 2 . aen sin {p-\-sq) — Z. aems sin sq 
1 / 2 /I 1 \ dAg a^e^n d^Ag\ . 
+ ^.[ae‘n(l+s)-j^ ^ -jy j sin (2p+iy) 
^ / , dAg , , d'^Ag\ . /IV 
-Z.[2aeens -^-\-aa!een-^,J sinp cos {p'+sq), 
the values of s being now +1, +2, &c. 
9. We are now prepared to express the right-hand side of the equation (7.) in 
terms of t. The results obtained in arts. 7 and 8 give, 
^ dH Cd^ , dr 
*•*= 
„ dAp. . „ / dA(. d^Af\ . ^ , , , d^Ap. . , 
— 2aen sin/? — ae n f 2 — a \ sm 2p-\-2aea en ^^sin/? cos/? 
V • / 1 N f A , dAg'\ V 2 . , \ f ^ , dAg a% d'^A,'\ 
-2.esm{p+sj)|^A.+««-5j-j-2.e-srn(2p+i»9)-|„-:isA.-l-««-*-^ 
— 2.e^(sin (2/?+5^)— sin^^). 
'2s^Ag~as 
dAg'^ 
da 
\ dAg'] 
n + s{n — n’) \ 
f 2//,^ / 
+ 2.ee' sinp cos (p'+sj) .|^rf:^(^(2s’A,+ 
For the sake of brevity of expression the following substitutions will he made : — 
, dAg s dAg , 
_ 2n^ . , dA 
Li = fAg+an 
n — n * ' 
M: 
5n^ 
da 
dAg ahi d'^A 
2 da^ 
N: 
n 
n + s{n—n’) 
2rd- 
;^('2S’A.-«S^*) + 
dA 
P =- 
nn' 
■ans 
dAg 
da 
dAg 
' o A , t dAg\ , dAg , , d^Ag 
=S)(2^A,+as-^j + 2a«i-^ + aan^ 
n' + s{n 
After multiplying the right-hand side of the foregoing equation by dt, integrating, 
and substituting in the equation (7.), the following will be the result : — 
df-Tp^— r -t-'— 
^ dAp. , c/'dAp. a d'^Ar\ ^ 2aa!edn d^^Ap. , i . i • - 
2ae-^cosp+ae^{-^-^ -^{ncospcosji+n^mpsmp) 
y Le cos {p -I- sq) „ Me^cos {2p+sq) j _ 2 / cos {2p + sq) cos sq 
~r • nMsfn — n!) "• ’ 2n-\- s{n — rJ) * * 1 9.n.-i-s(n — 7)!\ stir). — nl 
- 2 . 
n-\-s[n—rv 
'Ped 
\2re-|-s(?? — 7i') s{n—n) 
'' ^(^n' + s{n-n')y {^ COS/? COS (/?'-l-^^) + (w'+5(?2-?2')) siu/? sin (/?'-+-«/)). 
4 A 
MDCCCLVI. 
