A MATHEMATICAL THEORY OF RANDOM MIGRATION 



N 



Now take n = 0, q = l, p = l + r and f(p) = 



2irh 



f°° N 

 then we have : ul -^—j- J (ul) J (ur) rdu = <^ (r 2 ), 



J 2iTTlT 



N ( m 



or, ( f> 1 (r i )=—\ u JAul) JAur) du (ii). 



An J o 



This determines the form of D and the summation of u ; for, if we take 



N f °° 

 ^(^) = 2^ uJ (ur){J (id)} n du (iii), 



we satisfy the general functional relation (i) and further the initial equation (ii). 



Let P n (r) be the probability that an individual after n flights will be a distance r 

 or less from the centre of dispersion. Then clearly 



P n (r) = 2ir^rdr<t> n+ ,(r>) 



= N \ rdrl uJ (ur) {</"„ (ul)\ n du. 

 Jo Jo 



t> .«. r / \ d{J.(ur)ur\ 



But* ur J„ (ur) = l \\ ' — J , 



v ' d (ur) 



hence P n (r) = N\ du d (ur) l '/. — { — i l oV /J 



w Jo Jo d(w) w 



= jRT J r J, (w) {J (ul)fdu, 



f °° flv\ n 

 oriiv = ur: =N ^(v) J l — j dv (iv). 



(iv) is Kluyver's fundamental solution, which he reaches by a very different 

 and more general analysis, (iii) is the form of it which best suits my present 

 investigation. 



(3) On an expansion in series of the expression for <f> n (r a ). By straight- 

 forward but somewhat laborious multiplication it can be shown that : 



{J (2jy) eT = 1 " W " W + ^"^ * 



(50n-57)n (1892-2125w + 270n 2 ) n 6 

 + ~ 1800 f 103,680 y ' etC - 



* Gray and Mathews, loc. cit. p. 13. 



