1921-22.] Linear Differential Systems and Integral Equations. 51 
with respect to s, and thus fulfilling the limit conditions, is ^{x, s) = e* 
Hence Bessel’s equation admits of solutions of the form 
1{X)= 
^ ^ Jo Sin 
The real and the imaginary parts may be separated, giving integrals of 
the form 
1 L cos 
j , . I cos , . .cos , 
= 4 sin sin”-’* 
• (23) 
such as occur in the classical theory of Bessel functions. 
The adjoint partial differential equation is 
Jd^K . 0K 
02K 
. (22a) 
0,^2 ' 0.r ' 0.s2 
and has solutions and These lead to the two 
X X 
equations 
/•oo J /g\ 
cos nx = n / cos (s sin x) ^ ds n even 
Jo s 
J is) 
sin (s sin x) — n odd 
. (23a) 
§ 8. Solution in Finite Form. 
An important case arises when the nucleus K(a;, s) can be decomposed 
by elementary methods into a series of products of the form * 
K(x,s) = 2Csf/,(a:)Ws) (24) 
the series being either finite or uniformly convergent in the domain 
considered. Then the relation (15), for example, gives an immediate 
solution of the corresponding equation (13), thus 
u{s) = I K(,7j, s)v(s)dt> 
Jy 
= • 
where 
A^- = XCi I ki(s)v(s)ds. 
Jy 
If now K.{x, s) is decomposable into a finite number 7i of products, then 
* It is not here a question of the development of the nucleus according to its funda- 
mental functions. 
