136 
PROF. H. F. BAKER ON CERTAIN LINEAR. 
As an immediate application take the equation in Brown’s ‘ Lunar Theory,’ p. 107, 
d 2 x 
dt 
where 
Put 
j + n 2 x {l + fm 2 —(3m 2 + J ^m 3 + —|^m 4 ) cos cos 4£} = 0, 
^ = (n—n') t + e—e', m = n’ln. 
m __ __h— m = _dh —^ n dt = (l +m x ) d £; 
1 —m n—n 1 +m l 
then the equation becomes 
d 2 x 
d, 
hh + ic {1 +2m 1 +fm 1 2 — 3V^i 4 + w h 3 (3 + ^ 9 -w 1 + i 3 1 m 1 2 ) cos 2^+%%q 4 cos 4£} = 0, 
which is of the form above, £ replacing t. We may then take 
A = , n 2 — 1+ 2m 1 + fm 1 2 — 3 %m x 4 , & x = 3 + 1 ^-m 1 + --f-m 2 , k 2 — 33. 
8 
Here m 1 is a small quantity and 
A 2 m, 
n 2 — 1 64 (2m x + ...) 12'8 
is of the order m x 3 , while similarly A 4 /(?i 2 —l)' ; is of the order m x 5 . Also 
n = (1 + m x ) {1 + f m x 2 (1 — 2m 1 + 3m 2 ) — 
= ( 1 + m,) ( 1 + f m 2 — | w x s + -y^-wii 4 ). 
Thus 
ij-, 4A 2 A 2 1 
2 1 n 2 (n 2 — 1)1 
(n 2 -l)j 
= \ (1 + m } ) (1 + f m x 2 - +Hibb 4 ), 
which is easily seen to agree with the result given by Brown, or by Adams, ‘ Coll. 
Works,’ I., p. 187, when we take account of the fact that 
2 ug — 2 Ik (n—n') t — 2A 
n—n 
n 
nt , 
so that, in terms of the quantity denoted by g, 
k = ^(l+mjg. 
This example is chiefly useful here as calling attention to the fact that n 2 , while 
not exactly equal to 1, is near to it, and consequently the factor \/(n 2 — l) is only 
small of the first order in m v The same weakness occurs in the terms in £ _1 , ..., in 
the solution. 
