C235") 
We will suppose that f(x. is continuous in the interval « Sr <5 
and that A (vy) is finite in the square a <a<b, a<y<hb. 
The method of NEUMANN gives then immediately the solution 
b 
ple) = fie) +A | P(es2) f(s) ds 
where 
Blei BEE AK (ay) HALE (fdan ns ss 2) 
and 
b 
K,, (wy) =k K (as) Ki (sy) ds . 
a 
The disadvantage of this solution is that it only converges for 
certain values of 2. But a much better solution was discovered by 
FREDHOLM in which the function [(@72) is exhibited as the ratio of 
two power series which are convergent for all values of 2. 
Our first object will be to show that the latter solution may be 
deduced from the former in a very simple way. Supposing that the 
finite function A (zy) can be expanded in a finite series of the form 
K (wy) = Xe) Fy) + Xa) Vy) +.» Xie) Vila)» (3) 
it may be shown that a linear relation with constant coefficients 
exists between 7 + 1 successive functions Aj;(xy) 
Gn Ky +1 (wy) — @n—1 Ky42 (wy) +. 
+ (—1)—! Ot ayy (Ey I ee ry) = Oe (4): 
Mire n= O, 12)... and A (ey) — K (ay). 
Thus it is evident that the series (2) is a reciprocal one which 
may be represented as the ratio of two polynomia 
eed, ji = =|)! By. an! 
a Nal te Gn (5) 
l—a@,2 + aA — Ae + (-—1)" On An 
where 
Ji = Vis 
B,= «,k,—kK, | 
ide 4 p ‚__ (6) 
B,=«,K,—a,K,+ K, | 
Bj = Ank, —an-2K, +... (1) Za, Kir (IK, | 
and considering the limit of this quotient for n= we obtain 
immediately the result of FREDHOLM. 
2. To prove the relation (4), we expand the determinant. 
48 
Proceedings Royal Acad, Amsterdam. Vol. XIII. 
