( 737 ) 
Remarking now that 
VAE (a) Ko (as) Ar) 1) Pay) Yay) Nul) 0 
| | | 
, . | b a : 
K ay ei |X, (#7 )A,(#1)-X n(@,) 1 5 (Pea EF nw, )O 
|X, (en) A(n) A(n) 1 | Pen) Fo (an)e ¥n(#,)0 
it follows that the first number of (7) is zero. This proves the 
equation (4) when p— 0. Writing this result 
an K (es) — Gn—1K, (#8) + … + (—1)?—! a, Kres) + (—1)"Kriles) = 0, 
multiplying by A (sy) ds and integrating between the limits a and 
b, we get 
enk (ay) — dn—1K,(ay) +... + (—1)"—la, Kiley) + (—1)"Ap42(zy) = 9 
Repeating this process it is evident that equation (4) holds for 
all values of p. 
3. If now n is infinite, the equation (5) may be written 
D Xe 2 
(aya) = 262) 
D(a) 
where 
» (—l)pâr ; 
Daya) SEN) = fe dada. «iy (3) 
Ve vpr 
and 
bb 
a (DEP (oep 
DA L = . ING de dep. sen 
iep? Deep 
a 
a 
For the proof that the first of these series converges absolutely 
and uniformly in the square, and that the second converges absolutely 
for all values of 4 we refer to the original memoir of FREDHOLM. 
4. The preceding method enables us also to obtain the coefficients 
of both series in the form which has been discovered by PLEMELJ. 
Ea ane in the same way as before we have 
ap 
‘ > > je U ge Uip 
de .dep= Kle) | -FÁ dader) 
a es « Verd 
a a 
ef K (wr )dr, i J (7 Tt iy 
Pp) 
45* 
