1274 



If' ill tins forniiila we i-ead ^{ik + , instead of ^fi/,-, we gel 

 1 



(I 



From these considerations it appears that 



1 



i /C(.r,v) ros ((2A— l).Tv— if^;L-Kv = hidk-tck) cos \2/c~ 1 )nx- ^(ik\, 



(I 

 1 



I /Tla-.y) sin \(2/c - 1 )jff^-^Ji,\dy = h,idk-c„) ■^'In \(Ü-\)ni/ - ^Jl] 



(I 

 and tiiis proves, tlial the fii nations (7) are characteristic functions of 

 A'(.t',//j belonging to the characterislic nnmhers 2/(^4 -\- r/,) an<l 



There exist no other characteristic fnnctions, for the functions (7) 

 form a complete system ; so we have found the characteristic fnnctions 

 and nnmbers of the kernel (5). 



Every symmetrical kernel of the form 



is the Slim of a kernel such as f2' and a kernel such as (5). For 

 D[z) and L{z) being deliiied in (0,2) and (—1, + 1), one may put 

 2F{z)=zD{z) '• l>{[~{-z) , 2 f (2) -- D{z) — D{\ f^) for 0^ c < 1. 

 2r[z) = n{z) y D{z—\) , 2f{z) D{z) -D{z—\) for 1 <c<2, 

 2<I>{z) -r /\{z) 4- A(^ ^ 1) , 2<f{z) — L{z)-L{z^\) for -I <^<0, 

 2'P{z) = L{z) \ L{z^\) , 2<f{z)z= L{z)-L(z-\) \ox 0^^<^, 

 and so K{x,y) becomes the sum of a kernel such as (2) and a 

 kernel such as (5). But from this there is little to conclude with 

 respect to the characteristic fnnctions of K{.r,y). 



§ 4. We now consider a much more general case. The equation 



d { dz\ 

 ^^.(^/>(*^')^ji |9(4 + /*I^^=0 ..... (8) 



be given; we put for brevity 



d { dz\ , 



so that the differential equation becomes 



Lj-z -f fi^ = . . . . . . . (8a) 



We suppose the function /y(.f) to ha\e a continuous differential 

 coefficient in the inter\'al [aj)) and the function r/(.r) to be continuous 

 in that interval. 



