( 350 ) 



00_ 



^ = ( .;„ {va) J, {va,) . . . J„ (m„_i) c/y 



2 J V 



o 



In the case n^l, this is a known result to which another may 

 be added, if we take a ]> p. When the single stretch a is inclined 

 to the frontier under an -angle less than 



. P 

 arc si?i — 



a 



the rambler remains at the same side and, all directions of the stretch 

 being equally possible, we have 



hence 



1 / jr . p\ 



W,{F) = — i — -\- arcsin — J , 



. P Csmvp 



arc sm — = I J ^ (va) dv 



a J V 

 



Mathematics. — "A definite integral of Kvmmer" . By Prof. W. Kapteyn. 



In Crelle's Journal, Vol. 17, Kummer has determined the value oi 

 the integral 



00 b^ 



Up^ I e ^ xP dx , 







supposing 6" to represent a positive quantity and p not an integer. 

 He finds: 



Up = r{p^\)f{-p, h^) + r{-p-i) h^P+^^fipi-2, h% 



where 



__ X x"" x' 



y {p,x) = 1 + — + 2,^^^^^) + 3/^,(^^ i)(^,_|_2) + • • • 



-\- ... = 2 . 



s=osIp{p-\^l)..{p + S-l) 



In the following pages we propose to study this integral for the 

 case that p represents a positive integer, and at the same time to 

 show that there is a simple connection between this integral and 

 the integral 



00 62 



Vp=: I e ^ xP dx, 



h 

 where h is supposed to be positive. 



