536 
PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES. 
series in the elliptic theory. In the other equation terms involving e"x disturbing- 
force are to be omitted, and cosjo^ may therefore be put for H. Hence 
/z = sin ^mp){eh c,oip,—g), 
or, to the same approximation as before, 
h= -—g sin p^{\-\-2e cos Pi). 
Thus, since -=1— e(H+/i), we finally obtain 
1 — eH + ^(U + e(V+2Ucos;?J + e'W). (10.) 
It will be seen that in this process sinpi has disappeared from the denominator. 
14. The expressions represented by U, V and W admit of simplifications which 
will render them more convenient for substitution in the equation (10.). In art. 9 
we have 
_ . dA.^ 
L — 7 A.4- an -r- 
71 — rv ® ' da 
N: 
2s^A,—as 
dAs\ dA, 
— as-^ 1 -\-ans 
da J 
da 
Hence 
also, 
' 7 i-\-s{n — n’) 
TVT 2A I / s^n—n’) ^ 
N= — ; — r — — 7 t( 2n A.-\-an{n—n)-:r~ I = — rA AL ; 
n + s{n — 7i)\ * ' ^ ' da j /i + s(n — vl) 
y. It A . f^As (An d’^As _ , . cP-n d'^Ag 
M = — 2- •^=L+— , A.— 
n— n! * 
71 — rJ * 2 dd^ 
Using these values, and putting s{n—n') under the form {s{7i—n')-\-n)—n, it will be 
readily found that 
2 / 2L , M + N 
I — n' \ 
(^7i + s{n—7i')')‘^—7 
2 V ra 
s(7i + s{7i — 7i')')~^s{^2n + s{7i — n'))i 
2iP . dAs\ (s^ — s)(?i— ra') — 3n 3?i^ a^a 
— n’ j’ 7i + s{7i — 7i') ' ?i — 7i' ^ 2 da^ J 
t?Ao 
Again, since A^ and-^ are homogeneous functions of a and a' of the dimensions — 1 
and —2 respectively, by a known theorem we have 
. dAg . dAg 
, d'^Ag dAg d^Ag 
^ dada' da ^ da^ 
By these equations the differential coefficients of A^ with respect to a' may be elimi- 
nated. Thus by substituting in the expression for P in art. 9, and reducing, the 
following- result will be obtained : — 
2n 
w'-l- «(?!—?/) " 
Hence, putting .9=1, 
^ • j ( 2.9^ — s)?iAg-]-(^{s'^ — 2s){n — n')— w /) a 
da 
d^Ag 
da^ 
ilA, 
Pi = 2nA, — 2/?a -gg — ad^n 
d^ 
da^ 
