A MATHEMATICAL THEOEY OF RANDOM MIGRATION 

 We Lave : 



49 



and 



fa /"2ir 



F n (c) = N V <f> n (c 2 + r 2 - Ire cos 6) rdOdr, 



F n (c) = N-F n (c) 



= N y~Y \ ]' \ uJ o i u ^ c ' + i*-%er cos 0) J (ulfdurdddrX , 

 by (iii), 



= N 1 — J u J (ur) J (uc) {J (ul)Y durdr , 



by Neumann's Theorem (see p. 6) 

 = i^Tl- f°° £^M£ j ( a j ( ur ) urd (ur)\ J Q (uc) du~\ 



= N \l - ^ HiMl! f a d {J, (ur) ur} J (uc) du\ , 



by the theorem cited on p. 7, 



= N 

 = N 



I _ f " W^)Y L rJi ( ur )\ a j (uc) du\ 

 1 — J {J (ul)} n aJ^ (ua) J (uc) du . 



Or, writing v = au, we have : 



M°)=iv 



-/V.«j-.(.|) {'.H)}'*] (»)• 



This expression is concise. The integral expresses the probability that if an 

 individual start from the origin and take (n + 1) flights, the first of magnitude 

 c and the remaining n of magnitude I, at random, he will find himself within 

 a distance a of his starting point. But there does not seem any convenient 

 method of evaluating the integral. Comparing with (lxxxiii) we have the curious 

 identity : 



[>■ w * (» -:) {* (• SF * - * - ft*-"^! (•• -:p. (« „t.)-<-)- 



Write c = l, a = r and w — 1 for n, then 



where Q n ^ is the operator, 



dri 



l+v i (n-ir^-v s (n-ir^ i +... + (-^^{n-iy^- s+ ..., 



dn 3 



dri* 



