DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
153 
If herein we assume 
<f>2 — A 2 i'“ + A_2^”' + H, = B 2 £“ + B_ 2 £ “+K, 
and equate terms in £ 2 , <A 2 , f°, we obtain 
A 2 (o- 2 2 +j>) + 2Pfir 2 B 3 =_p(A 1 -l), A_ 2 (o- 2 _ 2 +£>) +2P»<r_ a B_ a =p( A_ 3 -l), 
— A 2 .2Q %<T 2 -\- (o"2 2 + c[) B 2 = (/ (Bj — l), — A_2- 2Q?-CT_2 _ t (cr 2 _ 2 + q) B_ 2 ~~ q (B_J —l), 
and 
(cr 2 +p)(H-K)+2^(fP + (r) =_p(A 1 +A_ 1 -2) J 
-(a 2 + g)(H — K)+2 /c 3 (-«:Q + o-) = g(B 1 + B_ 1 -2), 
wherein the coefficients of H —K and k 2 have for determinant 
(o- 2 +_p) (—?Q + cr) + (o- 2 + g) (?P + cr), 
which is 
a (l — m 2 y 
and is not zero. That TI, K should not be determinable separately is obvious 
a priori; to regard H as zero would be equivalent to dividing X, Y by a power 
series in A 2 with constant coefficients. We notice that the successive coefficients 
A], A_j, ..., B 2 , B_ _ 2 are all real. The value found for k 2 is 
7 —6o- 2 
cr (l —2cr 2 ) (l —4<r 2 )' 
A similar procedure can be continued. The differential equations for <p 3 , \[s 3 can be 
solved by forms 
A = A 3 ^+A_ 3 r 3 +H 1 f+H_ i rb V-3 = B 3 r + B_ 3 r 3 +K 1 ^+K_ i rb 
the differential equations for <p if \fr i by forms 
= A 4 r+A_ 4 r 4 + M 3 r+MJ- 2 +M, 
^ = B 4 r+B_ 4 rwN 2 r+N_ 2 r 2 +N, 
and then the terms in A will involve the unknown quantities 
(A +p) (M — N) + 2k 4 (iP + cr), 
-(<r 2 + g)(M-N) +2K i {-iQ + a), 
from which K i is found. And it serves as verification of the computation to see that a.- 4 
involves H, K only in the combination H —K, as it must in order to be determined 
without ambiguity. 
VOL. CCXVI.-A. 
Y 
