A MATHEMATICAL THEORY OF RANDOM MIGRATION 23 



minimum density r 1 = j2l, and the number on the zone r x to r^ + e,, is v i = Ne 3 /(irl'Jl/2). 

 Hence it follows that the numbers on narrow zones e u e 2 , e 3 in breadth, of equal areas 

 ire* = 7r4Ze 2 = v2 v 2Ze 3 , are given by 



NeJ{irV), Nj^HjrJl), and N<= 3 J2/(ttI), 

 or in the ratio 



Thus the total population on a small area at the centre of dispersion is twice that on 

 an equal area at the periphery of the distribution, and at both indefinitely greater 

 than on an equal belt at the distance of minimum density. The same point can 

 be indicated in another way. From r = to r = %l is ^ of the total area occupied 

 after dispersion, it contains "16.ZV or about -g- of the total population; from r = §l to 

 r = 2l is -£g of the total area, it contains "54 N. In other words the half of the 

 area nearest and farthest from the centre of dispersion contains -j^ of the dispersed 

 population ; the " middle " half of the area contains only ^ of the population. 

 The nature of the distribution is thus extremely different from that given by the 

 rotation of the Gaussian curve about its axis for this small number of flights. 

 For in the Gaussian case if the central area ■rre 1 2 = 2irr 1 e 2 , the area of the zone at 

 distance r x , the population on the centre patch is ^Ne'/a 9 and on the zone is 



which is always less and diminishes continuously with -increase of r x . Thus the 

 Rayleigh solution fails in this, as in the next three cases, not only to give the 

 form of the curve at dispersion, but to indicate that the dispersed populations on 

 zones of equal area round the centre do not decrease uniformly in number. 



Dispersal Curve for Three Flights (Diagram II.). 



The solution is discontinuous at r = l. The density is here infinite, but has 

 become finite at the origin. There is no discontinuity at r = 2l, but at the end 

 of the range the density drops suddenly from a finite value to zero. Thus the in- 

 tegral of the Bessel function product (see Eqn. (iii) ) is discontinuous at two points. 

 The Rayleigh solution is still widely divergent from the true curve of dispersal. 



Dispersal Curve for Four Flights (Diagram III.). 



By the rule already referred to (p. 18) the infinite density has returned to 

 the origin. There are only two points of discontinuity, i.e., at r = l and r = kl 

 the end of the range, at both of which there is an abrupt change in the slope of 

 the curve. The density at the end of the range is now zero and will remain so, 

 but the dispersal curve rises at right angles to the axis. The true dispersal curve is 

 bending round somewhat to the Rayleigh curve, but the latter is not even yet a 

 rough approximation to the facts. 



