DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
137 
§4. In the equation considered by Hill (‘Coll. Works,’ I., p. 268) the ratio 
4&iA/( n 2 — l) is about (2785) \ and there is a term slightly greater than 4&A (2785) _r 
arising in the terms in A r+1 in the series for k, in which 4&A is about O'5704 ; and 
the series fails absolutely in cases in which n is an integer. Then the assumption 
that k is a power series in A 2 , and the terms in X which are independent of A, must 
be modified, for reasons above given. The series when n is an integer has been 
considered by Tisserand, ‘Bull. Astr.,’ IX., 1892; modifying his procedure, so as to 
include the case when n is near to 1 as well as that in which n — 1, we may write, in 
accordance with the suggestion of such examples as that above quoted, 
w 2 = 1 + 4A h x + 4A 2 h 2 +..., 
and then, denoting e 2lrt + e 2lTt by w r , consider the equation 
+ [l +4A (h 1 + k 1 w 1 ) +4A 2 ( h 2 + k 2 w 2 ) +...]x = 0. 
By the changes 
t — 2 it, f = e T , 
dx 
-ix = e - i(1 + 2 ?D[U—V£], TT = e -i(i +3s )t[ U + y^, 
(JLl 
the differential equation may be replaced by the pair 
where 
€L_ ? U = -^(tr-Vf), = -*(U£-'-V), 
«>, = r+r r , 
<p — A (/?] + kiWi) + A 2 (A '2 + k 2 w 2 ) + .... 
Assuming here 
q — \q l + \ 2 q 2 J t- ..., 
IT = 1 + AW) + A 2 w 2 + ..., "V — B (l + \v 1 + \ 2 v 3 +...), 
in which B is a constant, and u r , v, are polynomials in £ and £ _1 , we find, equating 
coefficients of like powers of A, 
du r tt 
— i —tgyw r _ 1 + (72 w r-2+ ••• + QV — H r , 
(XT 
. dl' TT 
— -j— + guV-i + q 2 v r _ 2 + • •. + q r = X r , 
in which 
H r = (/q + k{wk) (w r _ x -( , Bv r _ 1 ) + (h 2 + k 2 w 2 ) (u r _ a —£ Bv r _ a ) + .,. + {h r +k r w r ) (1 -£B), 
K r = f^B- l H r . 
VOL. CCXVI.-A. U 
