( 739 ) 
the last being one of the equations (6) we find 
A p 0 Bn) Nd 
| jg a, p—l … 0 | 
OAN Tig a, a, eo | El : ae (13) 
Kp Ap—l Up? wold 1 
N Kp Ap Ap_l … A, | 
The formulae (12) and (18) agree with those of PLEMELJ. 
5. If the kernel A (xy) be defined throughout the square so that 
ET (# 
K (wy) = a 
(wy)? 
Me EE AE nl 
where (ry) is finite in the square and a < ——.,, then it may be 
n 
readily proved that the iterated kernels A, A, ..A,—1 are all infinite 
for z= y, and that the kernels K,, K,41,... are all finite in the 
whole square. Likewise all the gntegrals 
b 
[Ku (ayy (6) da On == kr) 
a 
are finite throughout the square. 
For this case it is shown by Porrcark *) that FReDHOLM’s solution 
still holds if the determinants 
and 
are modified in the following way. 
If by a cycle of & letters a, 8, y,d..u is meant the product 
K (a8) K (8y) K (yd). . K (ua) 
those products, in expanding the determinants, must be omitted which 
contain cycles of less than m letters. 
Now we wish to show that these modified coefficients may at 
once be obtained from those of PrrmerJ by substituting therein 
a, = a, ==. Anl = 0. 
For this purpose we note that the equation (4) still holds if A; be 
1) Act. Math. Bd. 33. 
