50 
Proceedings of the Royal Society of Edinburgh. [Sess. 
we have respectively the homogeneous integral equations 
2i(x) f K(x, s)2i(s)ds . . . . . (16) 
Jy 
= X'-^ I K(',r, s)v(.s)ds ..... (16a) 
Jy 
with the same series of characteristic numbers, whose solutions satisfy (13) 
and (13a). The functional relations (15) and (15a) are thus intermediate 
to the formation of the ordinary integral equations (16) and (16a). 
§ 7. An Important Particular Case. 
In general it is of advantage to have the equation MAc) + ot'p = 0 of as 
simple and well-known a form as possible, in order that (15) and (15a) 
may be relations connecting the solutions of (13) with known functions, 
such as the elementary transcendentals, Bessel or Legendre functions. A 
particularly simple and interesting case is that in which (13) occurs in the 
form 
LJu) + n^u = 0. . . . . .(17) 
n being an integer, and (13(i!/) reduces to 
, 2 A 
. (I7a) 
Then if K(ir, s) be a solution, periodic in s, 
equation 
L.(K)U^.„ 
of the partial ditferential 
m 
in which n does not appear, equation (17) will be satisfied a definite 
integral of the form 
I 
Also, if k[x,s) is a solution, periodic in s, of the equation adjoint to (18), 
and if u{x) is a solution of (17), it may be possible to find a path of 
integration y' such that 
nx ==C I Jr(s, x) ?i(s)d8 ..... (20) 
Thus, taking Bessel’s equation 
.X^u" + XU - 1 - (x- - n-)u == 0 . . . . .( 21 ) 
we associate with it the partial differential equation 
.02R 0K 
02K 
d.7x dx 
( 22 ) 
A solution of this equation periodic in s and having a periodic derivate 
