46 
Proceedings of the Royal Society of Edinburgh. [Sess. 
satisfied by K(x, s), we can replace K(a?, s) by s), or by a nucleus 
of higher rank in which the condition is satisfied.] Then since lLp_i{x, s) 
satisfies (2), we have 
r \ . ( d ) 
P=b 2, 3, . . . J 
Hence by the classical theory of simultaneous equations, whether finite 
or infinite in number,* 
{ka5 — Lg = 0 . . . . . (7) 
provided \ is not equal to any other number in the X-sequence. Thus the 
single product (p^{x)(p^(s) satisfies (2) ; it may likewise be shown to satisfy 
the boundary conditions (2', 2 "). Consequently (p^(x) is a solution of the 
system (1, 1') for a particular value of a. 
It has thus been proved that provided the nucleus K(a?, s) of the 
integral equation ( 3 ) satisfies the partial differential system ( 2 , 2 ', 2 "), pro- 
vided the expansion of the nucleus in fundamental functions and its second 
derived series are uniformly convergent in the square (a, /3) — a condition 
which can be ensured by iteration, — and provided that to the characteristic 
number there corresponds only one fundamental function then 
<pr{x) is a solution of the differential system (1, V). It follows immediately 
that, under the same conditions, solutions of (1, 1') exist which are not 
orthogonal to the nucleus lL{x, s) chosen, thus justifying the assumption 
made in | 2. 
If we suppose two of the characteristic numbers equal, e.g. if X2 = X^, 
then we can conclude no more than that + ^2(^)9^2(®) satisfies 
( 2 , 2 ', 2 "). It does not follow from this that (})^(x) and satisfy ( 1 , I'); 
but if one of these satisfies the system, the other also satisfies it, and hence 
the linear combination -h C.^02(^c) satisfies the system. This may 
arise when the index of compatibility is 2, but must be definitely excluded 
in the more general case of index I. 
§ 4. An Example illustrating the Method. 
The foregoing theory includes all the known cases of the solution of 
linear differential equations with periodic coefficients by integral equations.f 
* Kowalewski, Einfuhrung in die Determinantentheorie, §§ 22, 156. 
t Whittaker : (1) Proceedings International Congress^ Cambridge^ 1912, i, p. 367 ; Modern 
Analysis, § 19 '21. 
(2) Proc. R.S.E., xxxv(19l4), pp. 70-77, 
(3) Proc. Lond. Math. Soc. (2), xiv (1915), pp. 260-268 ; Modern Analysis, 
§ 23-61. 
(4) Proc. Edin. Math. Soc., xxxiii (1915), pp. 14-23. 
Reference may also be made to A. Milne’s integral equation for the parabolic cylinder 
functions, Proc. Edin. Math. Soc., xxxii (1914), p. 8. 
