[ 57 ] 

 VIII. Note oii a Definite Integral. By I. J. Schwatt *. 



rpofind 



C afidx. 



p, g, and n being positive integers. 



Let g = en+oL (a<ii), 



then m a x en . (dJ» + l — 1> 



(1) 



(2) 



(3) 



0^ + i) p (x n +i)p ^ o»+i) p • 

 (^+l^l)?==(-l)«S(-l>('^(^+l)^ 



therefore 



If c<jo, (3) may be written 



Since the Integral under the second summation may be 

 readily obtained, we shall consider only the case when 

 c < J). 



(g" + l)r-g-* ^ ta-h'LP-fi-h, ... (5) 



then 



I a - hn , p -^ h =- n{i) _ /3 _ li _ 1) + n ^_£_ A _ 1} W+i)»,*-e-^i- (6) 



IE in (6) to It are assigned the values 0, 1, 2, . . . . , p — f$ — 2, 

 and tbe resulting Integrals combined, we obtain 



V 7 — x 



9 .^-a-l^ninp-jS-if 



7=1 

 p-0-1 



n ( a - 7 , i+ l) /., 



* Communicated bv the Author. 



