A. 
DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 149 
of which the first two correspond to a period nearly the same but slightly greater 
2 
than that of E, and the last two correspond to a period — times that of E. When 
m 
E is Jupiter, this last is - 2 /- times the period of Jupiter, or nearly 150 years; when E 
is Mercury, this period is approximately 200 years. As m is small we have approxi¬ 
mately 
p = 3 — -y-//n 2 , q = -y.pn 2 . 
To neglect m 2 would be to neglect the ratio 27E/S ; but we may remark in passing 
that if we put q = 0,p = 3, the equations give 
together with 
> k / + 2 < f> = C, a constant, 
j," + 1 +8Acose $ = 2C 
1 + 2X cos 0 
of which the integration can be completed in finite terms. For it may be verified 
that the equation 
(1 + 2X cos 0) <E>" + (1 + 8X cos 0) d> = 0 
possesses the two integrals 
sin 0 (l + 2X cos 0), 
cos 0-— 2X (1T sin 2 0) — 4X 2 cos 0 + 8X 3 cos 20+ 12X 2 sin 0 (l + 2X cos 0) \[s, 
where 
dO 
1 + 2X cos 0 * 
§ 8. We consider briefly, first of all, what would be the application of the method 
of infinite determinants to the equations (I.)", which we may now write, with x, y for 
d>, A, in the forms 
(1 + 2X cos 0) (x"~ 2y') = px, 
(1 + 2X cos 0) (y" + 2x') = qy. 
We should substitute 
x 
oo 
— oo 
and equate to zero the coefficients of the various powers of e lB . The substitution 
gives, if f = e ie , 
[l +X (£+ £ -1 )] 2 [A„ (/c+n)" + 2i (/c+n) B n ] f'WpSA,^” = 0, 
[l +X (£+f -1 )] 2 [B„ (k+h) 2 —2i(ic+n) A„]fi ! + gAB,J" = 0, 
and, denoting k + n by K n , we obtain for the unknown coefficients A„, B, s the equations 
^ (A„_ 1 /c 2 n _ 1 + 2«B n _ 1 K„_ 1 ) + A„ (/+“+p) + 27B n /c„+X (A. n+1 K- J n+l + 2iB n+1 K n+1 ) = 0, 
X (—2^A„_ 1 K: n _ 1 + B„_ 1 K''„_ 1 ) — 2iA„fc n + B„ {ic n 2 + q) + \ ( _ 2iA n+1 /c, 1+1 +B, 1+1 /c 2 „ +1 ) = 0 
