DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. . 
135 
where </> r is an integral polynomial of order r in £ and £ -1 , the quantity e ie being 
denoted by £ Then we have 
2A>"„ + 2i {a + k 2 A 2 + ...) 2Ay» 
+ [ — 2<x/c 2 A“ — (2 cta 4 + /cy) A 1 + ... + 2 A/q cos 0 + ...] []l + 2A n <pn\ — 0. 
The terms in A give 
<p" 1 + 2ia<p' 1 + k 1 (£+f -1 ) = 0, 
which, if we denote (<x + r) 2 — a 2 or r(2o- + r) by u r , so that the result of substituting 
for (f> in <p" + 2ia(f>' is — u r £ r , leads to 
The terms in A 2 give 
(f>i — kA— + 
\u. 
0^2 + 2ia<p r 2 + k x (f+ f " _1 ) (pi + k 2 (<U + £ 2 ) — 2cr/c 2 — 0, 
which, if we write 
leads to 
and 
02 — A 2 £"+A_ 2 £ 
A 2 = lfi !+ U), A_ 2 = — R+AA) 
w 2 \ Itj/ W_s\ W_i/ 
2cr\U l U_J <r(4<x~ —l) 
By the terms in A 3 , A 4 , we similarly find the coefficients in 
and also 
*4 
03 — A 3 £ 3 +A_ 3 £ ,j + B 1 ^ + B_ 1 ^ 1 , 
A = A 4 £ 4 + A_ 4 £; 4 + B 2 £ 2 +B_ 2 £- 2 , 
6 O 0 - 4 —35o- 2 + 2 74 3 , „ 1 /2 
4cr 3 (a- 2 -l)(4«x 2 -l ) 3 1 + 2a(a- 2 -l)(4a- 2 -l) 1 2 4a-(a- 2 -l) 2 ’ 
If we change the notation, writing 0 = 2 t, 2or = n, so that the differential equation 
becomes 
dfa 
dt 
j + [ , h 2 + 8AA; 1 cos 2t+8\ 2 k 2 cos 4 1+ ...~\x = 0 
and 
we have 
k = \n 
£ — e 2lt , x = e 2wt X, 
2k 2 \ 2 , „ 15A 1 —35w 2 + 8 7 , 4i 
1 A -i A . \ . . /Ci + 
12 k 2 k 2 
2k 2 
n(ri 2 — l) ' l w? {n 2 — 4) {n 2 — l) 3 1 n {n 2 — 4) (n 2 — l) n{n 2 — 4) 
It is clear that k is essentially real so long as this series converges. 
