A MATHEMATICAL THEORY OF RANDOM MIGRATION 15 



This is also true for s = 0, as well as any integer value. It follows accordingly 

 that while the total area of any w-function from to oo is positive, its k is 

 negative for values of s>l. In other words the negative parts of co are on the 

 whole furthest from the axis. Again the absolute value of k M decreases as 



- j- when s increases, or the higher the cu-function the less it contributes relative 



to its area to the total mean square deviation of the curve. 



Applying these results to the curve of scatter given by (x), i.e. 



m i %\ \r( l l ,6ft- 11 , 50n-57 



1892 -2125ft+ 270ft 



w 13 — etc. j (xxxiv), 



103,680ft* 



we have if A be the whole area and k the radius, 



- N 1 f 3 1 5 1 35 6ft- 11 21 50ft- 57 

 V27ro-2l 16 ft 24 ft 2 1024 ft 3 1280 ft 4 



77 1892-2125ft + 270n : 



etc. V ,...(xxxv), 



n/2tto- 2 I 16 ft 24 ft 2 1 



49152 ft 5 



5 6ft- 11 7 50ft -57 



024 ft 3 3840 ft 



1892- 2125ft + 270ft 2 



49152 ft 6 



+ etc. V (xxxvi). 



Hence if we even neglect terms of order — , we see that the Rayleigh solution 



lb 



gives too large an area for the curve of section and too small a swing-radius ; 



these values are 



1 iV" 

 Rayleigh area, - -t=- , Rayleigh swing-radius, o-, 



2v27rcr 



. 1 1 N I 3 1\ m . ,. 1 / 1 



Irue area to - , — j= — ( 1 ; Irue swing-radius to - , <r 1 -I 



ft 2V27ro-\ 16 ft/ ft \ 8ft 



Accordingly for n small the graph of the Rayleigh solution tends to exaggerate 

 the concentration, i.e. using it as an approximation we shall somewhat reduce the 

 extreme parts of the curve at the expense of exaggerating those near the centre 

 of dispersion. 



While there is no difficulty about determining the curve of distribution when 

 ft is large from (xxxiv), beyond the great labour of dealing with hitherto untabled 

 functions, the investigation becomes very troublesome when n is small. The 

 functions <a are suited in this case to represent the discontinuous functions which 

 actually form the values of <f> n (r"), but the extreme discontinuity of <£„(r 2 ) for n 



