174 
PROF. H. F BAKER ON CERTAIN LINEAR 
t in the solution of the (X, Y) equations, and forming from this, after putting r — 0, 
a determinantal equation (§ 14). 
We are to calculate in turn Q u, QuQu, &c., and arrange the result according to 
powers of X. First we have 
Q u = /«!, b\ , 
where 
\ C 1> ^1/ 
Cti = — <p dr, />! = <j)£ dr, 
Jo 
c x = — i dr, d x — j cp dT ; 
Jo Jo 
thus, as (f> is unaltered by changing the sign of t, by can be obtained from c x by 
changing the sign of t, and similarly dy from a v This we denote by writing 
Then 
by = c'y, dy — Ct!y. 
uQu = / -<p, i<p\ /cq, c'A, 
— £ 0 
,c l5 ct lx 
and hence 
where 
■ — <pc'y + i<pa\ \ , 
-f l (pay + (pcy, — f“Vc'i + 0a , 1/ 
QwQw = /a 2 , c'A , 
v C 2 , a g/ 
c' 2 = 
= I (p{-ay + ^Cy)dr, 
Jo 
C 2 = I + a' 2 = 
Jo 
0(-c'i + ^a'i)dT, 
I 
r 
<f>(a\-£- 1 c'y)dT, 
so that a' 2 is obtained from ct 2 by changing the sign of r throughout, and similarly c' 2 
from c 2 . In general, in passing from a term of ii (u) involving r integrations to one 
involving (r+l) integrations, we shall have a law expressible by 
A r+1 — (p (—A r + £C r ) dr, C r+1 — £ V ( — A,. + (,(J r ) dT, 
and the new term, like that from which it is derived, will be of the form 
