DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
175 
where A' r+1 is derived from A r+1 by change of the sign of r, and similarly C ' r+1 from 
C r+ , 
Thus, when in Q ( u ) we pick out the coefficient of t, as it occurs explicitly, 
independently of its occurrence in £ and in this coefficient put - — 0, we shall obtain 
a series of the form 
a i, —yA + / ot 2 , —y 2 \ + ..., 
7u — “i/ \y 2 , — a 2 / 
where the first of these comes from Q u and involves terms in X and higher powers, 
the second comes from QuQu and involves terms in X 2 and higher powers, and so on. 
And the equation for q will be of the form 
namely, 
a 1 + a 2 +q, — yi—y 2 — ... 
- 0, 
yi+y*+"-> - a i~ a 2~ •••— q 
q 2 — (oq + a 2 + ... — (y! + y 2 + ... )\ 
Further, if the part of Q u which is independent of explicit powers of -, consisting 
of elements which are polynomials in f, £ _1 and periodic with period ri, be denoted 
by Pj, and similarly the periodic part of Q uQa be denoted by P 2 , &c., then the 
periodic matrix P above spoken of will be 
P = 1+P 1 + P 2 +.... 
Proceeding to the computation, retain first only to terms in X. Then 
CL X = — j <p dr = —Xhi — Xk x {^—C, 1 ), 
Jo 
c, = - U-^dr = - fV^XA + XMt + r 1 )]^, 
Jo Jo 
= \h (?"-!) + ^ I (iAA_r). 
Hence 
oq = — \tl, yj = —Xlc, 
and q is given by 
q‘ = \‘(h°-k°). 
In the case when the differential equation is that considered by Hill, this gives at 
once a very near approximation, as he remarks, being equivalent to his formula 
c 2 = i + kv-i) 2 -^ 2 }* 
(Hill’s ‘ Collect. Works,’ I., p. 260). 
