775 



/•(..)= f^^p-di- (18) 



(,r-J)'-'" 

 and 



J {x - ;/) "(// — t) " 



Theorem A may again he applied to equation (17), as the kernel 

 L{x,ï) satisües already' for ii = the conditions written down for 



We may moreover observe that, as sufficient tiiongh not necessary 



X 



condition for the continuity of — I — ^ — f/i obtains: f^"\.v) con- 



(/.r J (,f— $)!-'„ 



tinuous and /'"+i)(.(') Unite with only a finite number of discontinuities 

 for a^ x< b. 



In the first place we siiall now prove lliat tlie special value (15) 

 for the kernel K{.v,ï,) satisfies the conditions mentioned in theorem 

 A or B and that in proportion as ). ^ — ?i or — n<^li{y)<^ — " + 1 

 {n positive integer or zero). 



a. Let ns suppose ). = — n, (J 5) passes into: 

 and we find: 



(-1) 



A' (,r-,5) — — -- ^' — n! 2 ^ — ^ — (*•— ï)'"+"-/' 



0x1' Hi=:0 '"■' ("i |-« — P)/ 



n—i> 



= nl (^- J (.r - i) -^ /,._, (. V.v-l\ 



SO that: 



/v„ (,«, §), /fi (,7', §) . . . /v'„-|-i (.I-, §) are continuous for a ^ | ^ ,x' ^ A, 



moreover /f,, (.(;, x) ^ 0, A', (.?;, .r) ^ A'„_i (,r, x) ^ 0, bnt 



A', Gi-, .r) =1= for a <. x ^ 6. 



Cohseqnently for A = — n the suppositions mentioned in theorem 

 A are satisfied by A'(.(', ï), also if z = 0. 



h. Let ns further sup[)Ose P. = — » -|- ;.„ (0 <^ /?(/„) <^ 1), (hen: 



50 

 Proceeflings Royal Acad. AmsferJnrn. Voi. XVIU. 



