48 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the solutions of (9a) will be expressible in the form 
00 00 
sin” 
0 1 
where D^ = 0 when = and C^ = 0 when n = l. The solutions of (9b) 
will be obtained by writing 1 — n for 7i. 
§ 5. Systems of Order n. 
We may readily extend the foregoing results to the general homogeneous 
linear equation of order the main theorem being as follows : — 
The solutions of the homogeneous integral equation 
u(x) ~ X I ~K(x, s)u[s)ds 
.1 a. 
satisfy the differential equation 
+ au = ln (^) 1 ”i" ‘ = 0 
(10) 
and the n periodic boundary conditions 
Uq(?^) = u(^) — u(a) ---0 
Jji{u) = u^%l^)-u^\a) = 9 [ . . . . (10') 
^ = 1, 2, , . , — 1 j 
provided the nucleus K(cr, s) satisfies the partial differential equation 
L^(K)-L,(K) = 0, ..... (11) 
where is the differential operator adjoint to L^, and satisfies also the 
periodic conditions 
K(c^, fi) = K(iT, a), K(/:), s) = K(a, s) ^ 
K/»(.7;, ^) = K/»(x, a) K/\f3,s) = KJ%a,s) . . (11') 
^'=1, 2 . . . , n- l) 
If the index of compatibility of (10, 10') is 1, the characteristic numbers X 
in the expansion 
= (12) 
must all be unequal ; if the index is i, not more than i of the characteristic 
numbers may be equal. 
Under like conditions, the adjoint differential equation 
L^(v) + av = 0 (10a) 
is satisfied by the solutions of 
ns 
v{x) = X / K(s, x)v(s)ds. 
J a 
* Bateman, Trans. Gamb. Phil. Soc., xxi (1909), p. 187. 
