(346) 



Hence if only a be rather greater than unity, both integrals cannot 

 have an appreciable difference and we may put 



CO 



\e ^" du=l—e ''" = 1 - ^' 



■e 



IS 



From this result it is evident, that Tr„(c/Z) for n very large i 

 always nearly unity. The rambler, walking along a very great 

 number of very short stretches will almost certainly arrive in the 

 neighbourhood of his starting point. 



1 



Putting c=:-L, we find TfJc/Z) = 1 — e "= pr-^--, a result 



1 

 nearly equal to the true value — — . 



Returning to the general expression for Wn{c\aa^ . . «„—i) we observe 



the possibility of differentiating the integral with respect to c in the 



w + 1 

 usual way a number of 1m times, provided Im <^ — -— . 



Suppossing c > rt + «1 -1- . . . -f <^n-\ and putting 

 J^(ua)J^{ua^) . . . J^{uan^\) =f{u), 

 we deduce by differentiation 



1 = c I J^{cu)f{u)du, 

 



CO - 00 



0=1 iiJQ{cu)f{u)du , 0=1 u^Ji{cu)f{u)du, 







00 00 



=r I u^J ^{cu)f{u)du , 0=1 u*J^{cu)f{u)du, 



0= Cu'^^^-^J,{cii)f{ti)du , 0= Cu^'»J^{cn)f{u)du. 



y 



These equations allow us to introduce into the integral a new Bessel 

 function, the function J2m+i {u). For J2m+i (w) is connected with 

 Jo {u) and e/j {ii) by the relation 



J2m+l (w) = Po,2m (w) J^ (?<) — P\,2m-] (w) /„ (u), 



where 



