A MATHEMATICAL THEORY OF RANDOM MIGRATION 17 



of the plotted curve which gives any new ordinate can be found mechanically. 

 It will be seen that the process is theoretically straightforward, but very laborious. 

 Thus for the dispersion curve after the fourth flight some 43 points had to be 

 found, and this involved the construction of 43 subsidiary curves and their integration. 



There were, of course, graphical difficulties in the construction of the subsidiary 

 curves in the neighbourhood of the asymptotes and variolas devices had to be used, but 

 at almost every point there were tests of the accuracy of the work. Some of these 

 I shall now notice. 



Case (i). The solution for two flights is : 





. (xxxvii). 



= r>2l 



The reader will find no difficulty in deducing this directly from the case of n=l, 

 which corresponds to a narrow zone of radius r = l, the rest of the plane being 



unoccupied. Thus : 



N \ 

 <£, = t— rfrom r = l — le to r = l + ie . .. , . . 



r 2irk 4 4 I (xxxvii Us), 



= from r = to I — \e and r= I + ^e to qo J 



e being taken indefinitely small. 



By distributing each element of 4>i on the zone round a circle of radius I we 

 obtain (xxxvii). 



The result may be obtained also from (iii) by putting n = 2, i.e. 



& (f) =o~\ uJ o ( ur ) K (ul)Y du, 

 Z77-J0 



= V[ : (2l + r)r(2l-r)r]-* ^ ^ Q ^ 



2tt Vir2- 1 n(--|) 



= from r = 2l to 00 , 



from a theorem of de Sonin by putting a = r, b = c = l. Compare Gray and 



Mathews, p. 239, Ex. 52. 



Case (ii). The solution for three flights may be obtained from that for two, 



by distributing analytically the density given by <£ 2 round circles of radius I about 



each point. The resulting double integral is then expressible in elliptic integrals*. 



We find : 



where k 2 = 1 6l s r/{(r + If (Si - r)}, 



r > and < I ; i 



nlj^ f (z \ y (xxxviii). 



WlJVl \2'7' 

 where k 2 = (r + Z) 3 (3Z - r)/( 1 6?r), 

 r > I and < Si ; 

 = r>Sltor=co 



* This solution, or its equivalent, was first sent me by Mr Geoffrey T. Bennett. 



