182 
PROF. H. F. BAKER ON CERTAIN LINEAR 
This gives 
q i — (ai + a 2 + a 3 )‘ — (yi + y3 + y 3 )' 
= X 2 h 2 -X 3 h (h 2 + %k 2 ) + \* ^h i +^h% 2 -^-k 1 i -k 2 2 +U 2 k 2 ) 
as far as terms in X 4 . This result is for the equation 
d 2 x 
wherein n — 2 and 
V' 
dt 2 
+ (n 2 + \k) x = 0, 
w — Xh,+ Xk x (£ 1 + £) + X“k a (£ 2 +f“)+... 
o 
and agrees with the result previously found (§6) when in this last we replace 
h, k x , k 2 respectively by 2 h, 2k u 2 k 2 , as is necessary, taking account of the difference 
of notation for ^ in the two cases. By an independent investigation for the case 
when 
'A _ 
xA 1 +xA 1 (r i +f)+x% 2 +x 2 A 2 (r 2 +r)+x%3+^ 3 (r 3 +^)+... 
we have found (above, p. 142), 
cf = h 2 X 2 -} h {h 2 + %k 1 2 -2h 2 ) X 3 
+ X 4 {(W+W-h 2 ) 2 + h 1 i + ^h 1 % a -2h 1 % + 2hA~(h-^h a ) a } + ..., 
which, replacing h by h 1 + h. 2 X + h 3 X 2 , arises from the preceding result. 
§ 21. Now consider the equations 
(X, Y) = u (X, Y), 
where 
A 
(It 
u = / -<p, t n <p' 
\ —<p 
and n is not 1 or 2, but is an integer if u is a periodic matrix. 
With 
<p — Xh + Xk y (£ -1 + £ ) + X'A> + +••• 
we have, retaining only to X 2 , 
a, = — 
c, = 
(jj dr = -xhr + Xk x (f 1 - £) + f 2 —D, 
£-*+<&■ = - [x^-”+x^ 1 (r»- i +r” +i )+x 2 A 2 (r”- 2 +r” +2 )jdr 
2 n 
= l -u(f-D+ui^ + LA _ + -£A! _ „ 
n \n + l n — l n 2 —lj \n-\-2 n — 2 n~ — 4 
