



Problem of Random Flights. 



Th( 



ise si 



mplify to 











* i= 6i^ bl ~ 



-3>-), 



0<r<2l 





or 













*«=ew (4i - 



-r)\ 



n<r<u 



435 



(1-4) 



agreeing with (62) of Lord Rayleigh's paper. 



I do not write out the corresponding calculations for 

 n = 6 ; the results depend on the symbolical formulae 



1^=, * if 5 > ; - 4 ^ = 0, if s<0. 

 jr 4 ! p 



The results are stated by Lord Rayleigh in formula (64) 

 of his paper. 



In dealing with the specially interesting problem when 

 n is large, the calculation by symbolical methods gives no 

 substantial variation from the treatment by Lord Rayleigh: 

 see (65)-(70) of his paper. 



2. The problem of Random Flights in two dimensions. 



After the previous account, it will suffice to treat the 

 present problem rather more briefly. Lord Rayleigh 

 obtains in his formula (32) the result 



*»(0 = i ^£ , _yiyJo(»y){Jo(^)}*. • • t*-i) 



Just as in § 1, this may be transformed to the complex 

 ^>' 2 ) = 4^ I 



mtegra 



XdXl (r\){l (l\)y\ . (2-11) 



where 



z 2 z l £ 6 



M*) = 1 + 22 + 2^74^ "*" 2 2 . 4 2 



2 



+ .. 



is the modified Bessel-function. 



Tiie integral (2*11) can be replaced by a symbolical for- 

 mula, but the result is not capable of any simple evaluation. 

 It should be noted, however, that when the integral is 

 written in the form (2*11) the questions raised by Lord 

 Rayleigh (/. c. pp. 333-336) as to the legitimacy of dif- 

 ferentiation, applied to the integral (2*1), are easily settled. 



The discussion of the integral (2*11) for large values 

 of n follows the same lines as on p. 337 of Lord Rayleigh's 

 paper. 



2G2 



