DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
147 
further 
p = A+(HK)* = f 
, q = A— (HK)* = f 
1 - 1 - 
nv 
2\r 
so that 
Also 
(iy 
_p + g = 3, pq — iw 2 . 
w = (1 + 2X cos 0) (X-fYY), u = (l+ 2X cos 0) (X—fY), 
whereby the equations (I.) become 
(1 + 2X cos 0) (u" + 2ivl) = Aw + Hu, 
* 
(1 -f 2X cos 0) (v" — 2 iv' ) = Kw+ Av, _ 
m which v! = du/d6, &c., and then 
<t> = fGw + HY, Ah = K-w—HY, 
so that d>, 'h are both real, and 
<b + Ak = 2K*(l + 2X cos 0) (XqYY), <f> —Ah = 2H 1 (1+ 2X cos 0) (X— iY), 
and the equations (I.) become 
(1 + 2X cos 0) (<l> // — 2V) = p4 
(1 + 2X cos 0) ('V + 2V) = gY, 
in which, beside the eccentricity 2X, there are the two constants p, q, which are 
dependent upon the single number m. 
The equations (II.), by means of the changes 
IJ = (l + 2X cos 0) (x + iy), -Y = (l + 2X cos 0) (x—iy), 
(I •)' 
become 
(1 + 2X cos 0) (U" + 2 AY) —| (U + Y) 
3M 
2w 
(1— w 2 )v, 
3M 
(1 + 2X cos 0) (Y" —2fY') —f (U + Y) =^±(l-w)u. 
2/x 
(ii. y 
Consider now the equations (I.)". We know that the solutions are of the form 
<f> = Ce^F + G^F. + G^F.+G^F,, 
w = Ce i ' tfl G+C 1 e^ 0 G 1 +C 2 e^G 2 + C 3 e^G 3 , 
where C, Cj, C 2 , C 3 are arbitrary constants, F, F 1; ..., G 2 , G s are definite functions of 
period 2 tt, and k, k 1} k 2 , k s are definite constants. When X = 0, substituting in the 
equations 
<f> = e ia \ A = Ve ia \ 
x 2 
