1280 



consider (18) as valid also in iliiit case, hi I lie same way [ISa] is 

 valid ir /X^) = 0, and pia) ^ 0. 



If p{a) = pib) = 0, a kernel tor wliicli the functions P{.r), Q(x), 

 >Si.v) and T(x) remain tinite, will always satisfj (14) and so its charac- 

 teristic functions will always be solutions of (8;. 



§ 6. From the conditions (18) for the kernel il is not difllicult to 

 get conditions foi- the characteristic functions. If 7,(^1 be such a 



function we have 



b 



(fi{x) = /, I K(,c,y)(fi{y)dy, 



and so 



Hence 



r dK{.c,!/) 





''-dK{y,ri) 



ƒ. ( OK{y,ri) 

 K{ij.a)((;{v)d'j , (fi{a) = /.ij ^ tf^(y)<it/. 



a '^ 



I, U 



r , . rdK{y,riy 



<l Ab)=z /, I K[ v.b)tfi {y)dy , ff ; {h) = /./ I -^ - 7 ,• {y)dy. 



a n 



From (18) we now get 



q i (6) = (Cf i (tt) i ^(p'i (a), I 

 ff'i(b) = y<fi(a) I fi(f'i{a).\ 



§ 7. To conclude, we will tind the conditions (19) for the ortho- 

 gonality of the solutions of (18) in a direct way, independent of a 

 kernel whatever. We suppose y(.») and ip (.i-)to be two solutions of (8) : 



Multiplying tlie former of these equations by i|' (.'•). the latter by ^(.r 

 and subtracting we tind 



dx 



and from this 



b 



