(343 ) 



2ir 



J"o (WS,._2) J^ {uan-\) = — j /„ (m5„_i) c;y„_2 , 



O 



271 



J„ (?^<ï„_3) J, ('^«-2) = 5~ I "^o ("■""- 2) %n-l , 



O 

 • ••••••••••••••• 



27r 



J„ [ua) J„ (?/aJ = — l J, {us,) dep , 



o 

 and consequently 



W„ 



^n{c ; a«j . . . a„_i) = c 1 Jj (uc) J^ {ua) /„ (t^aj) . . . Jg {uan-\) du. 

 o 



From this result we infer, that the probability sought for is of a 

 rather intricate character. The u -\- 1 functions J are oscillating 

 functions, and have their signs altering in an irregular manner as 

 the variable u increases. Hence even an approximation of the integral 

 is not easily found, and as a solution of Pearson's problem it is 

 little apt to meet the requirements of the proposer. 



From a mathematical point of view the integral presents some 

 interest. In fact, if we consider it as a function of c, it is readily 

 seen to be continuous and finite for all real values of c, and the 

 same holds for a certain number of derivatives with respect to c, 

 but a closer inspection shows, that this analytic expression, regularly 

 built up as it is, represents in different intervals different analytic 

 functions. To make good this assertion, we have only to remember 

 that the integral stands for the probability required in Pearson's 

 problem. Hence we know beforehand, that it always must be positive 

 and increasing with c, but that it never surpasses 1, this upper 

 limit being actually reached as soon as c becomes greater than 

 a-\-a,-\-.. .-{-an-i- Moreover, if we suppose a^a,-\-a^-\- . . .-{- a„_i, 

 the inequality a "^ c -\- a, -\- a^ . . . -\- a„_i is possible for small values 

 of c. And if the latter inequality holds, the rambler of Pearson's 

 problem necessarily arrives outside the circle with radius c, and the 

 probability is zero. 



Thus, by solving the problem, we have found 



00 

 c"^ a -\- a, ... -\- Un—i ,.•• ^ } r 



.=ic \ J^ {uc) /„ {ua) J„ {ua,) . . . /„ (m««-i) du , 

 a>c + ai ... -f a„_i ,... 0) J 







quite independently of the number of the /„-functions, showing 



