1921-22.] Linear Differential Systems and Integral Equations. 47 
As a further example, let us consider the determination of K(.t, s) so that 
its fundamental functions may be periodic solutions of the equation 
d'^u 
^2 
a + A;2 cos2 x — 
n{n— 1) 
( 8 ) 
where 0 ^ < 1. This is a generalisation * of Mathieu’s equation 
' + (a + cos2 = 0, 
whose even periodic solutions were shown by Whittaker f to be the funda- 
mental functions of the integral equation 
f2.1T 
ulx)^X\ 
Jo 
Let us assume the corresponding nucleus for (8) to be of the form 
K(x, s) = ® ® siii^ X sin^ s ; 
substituting this in the partial differential equation 
82K 
dx'^ 
COs2 x—k"^ COs2 s — 
n(n \) _^ n{n 
sin^ X sin^ s J 
we find that r = n or 1 — n. 
Thus, supposing the conditions enumerated above to be satisfied, we can 
conclude that equation (8) is satisfied by the fundamental solutions of 
f2n 
u(x) = X e* ® ® sin” iT sin” s w(.s)(is . . . (9a) 
Jo 
and 
f2n 
2 i(x)=-X e* ® * sin^“” o; sin^“” s . . . (98) 
Jo 
The condition as to the convergence of the development presents no 
difficulty ; the condition as to the inequality of the characteristic numbers 
is satisfied for sufficiently small values of n. 
When n = 0, (9a) reduces to Whittaker’s integral equation for the even 
Mathieu functions cej^x ) ; and (96) reduces to 
f2ir 
u{x) •= A I e* cos a: cos s ^ n{s)ds, 
Jo 
an integral equation for the odd Mathieu functions se^(x). 
Since the expansion of e^cosxcoss form 
CO 
0 
and that of e^cosa:coss ^ g-^^ ^ -g form 
00 
* It is virtually the equation exhaustively treated by M. i^braham, J[Iath. Ann., Hi 
(1901), pp. 81-112. f hoc. cit. (1). 
