152 
PROF. H. F. BAKER ON CERTAIN LINEAR 
and, in general, 
H„ = P l<pn-W</> n -l + W 2 <t> n -2- ••• +(-w) n ]> 
K„ = Pg[^-#,_i + wVn->- ••• +(-w) n ], 
where H„, K n are the coefficients of X” respectively in 
In particular 
so that 
and, as previously, 
while, if we write 
X"-2Y' + 2A(X'-Y)-/c 2 X, 
Y" + 2X' + 2 A (Y' + X) — k 2 Y. 
H 0 = —2?itP — a 2 , K 0 - 2?<r — <x 2 P, 
o- 2 +p + 2fo-P = 0, 2 icr — (cr 2 + g)P, 
O' 4 — (T 2 + = 0, 
p - a2+ P n - 0-2 +9 
^ 0 * 3 o ' 5 
— 2‘lar Zlcr 
which are both pure imaginaries, we have PQ = 1. 
Next 
Hi = (j>'\ — + 2 icr {(p\ — P\^i) — 
K x = P + 2Q <j>\ + 2 la (A + Qfa ) — <tV i] > 
putting these respectively equal to p^—w), P q(\Js 1 — tv), we obtain two differential 
equations for <p x and Ai- If we assume 
0i — A-X+A_X _ \ A = BX+B_X -1 > 
and notice that 
(ry = w-r, (D" = -ap, 
we find, writing <r„ for <r + n, 
If 
these give 
Aj (<r*+p) + 2PA 1 B 1 = P, A_J (o-_i 2 +p) + 2PAr_ 1 B__ 1 = £>, 
-A x . 2QAj+' {o- 2 + q) B x = q, -A_ 1 2Qfir_ 1 + (<r_i+q) B_ x — q. 
Aj = —crd + im 2 , 
A]A X = (<r 2 + q)p + ~{<r 2 +p) q, 
AA = (<p 2 +_p)g+-(°' 2 +g).p, 
<T 
with similar equations for A_ ls B_ x . 
Proceeding similarly to equate terms in X 2 , we find 
(j)' 2 + 2i<T(j/ 2 — cr 2 02 —2P + ftOl/'g) —2 /Co (tP + ix) = P ( 02 — W(j) 1 + w 2 ), 
y]s" 2 + 2 i(r\Is , 2 — O- 2 10 2 + 2Q (0 / 2 + ?ct0 2 ) — 2/c 2 (— 'iQ + cr) = g (\ls 2 — W\Js 1 + tV 2 ). 
