A MATHEMATICAL THEORY OF RANDOM MIGRATION 39 



by other constants N of known value, and in every w-function we are to replace 

 ct* by mcr 2 or T, that is to say a uniform stretch in the ratio of Jm to I is 

 given to any surface z = co M parallel to the axes of x and y. This is denoted by 

 writing fl M for <o M . 



If we confine our attention to the Rayleigh part of the solution — which will be 

 more and more nearly exact as m increases, for the N'& rapidly decrease in value — 

 then we have 



J> n {h, k) = (nA) m -WaCl (lxi), 



and we see that every density gradient curve for the successive migrations is to 

 be obtained by a stretch from the first migration density curve. 



In general, however, this result is not absolutely true because the different 

 components of the true solution are mixed in different proportions, the iV's being 

 functions of m. We see, however, that the stretching rule becomes more and 

 more accurate, as we increase either the number of flights or the number of 

 migrations. 



(11) Problem IV. To find the form of the general solution for the distribution 

 into surrounding space after rn migrations of any population initially spread 

 uniformly over any given patch with density N. 



The density at h, k, after m migrations due to a centre Ndxdy, is by (lvii) 

 above 



= (^a)™- 1 Ndxd y q » L g-H^-^+^-m/^ 



To give the patch let x be integrated from v 1 to v t , where v 1 and u 2 will 

 usually be functions of y, and then let y be integrated from u Y to u t . 

 We find: 



This is the general form of the solution when the population spreads from a 

 uniform patch into non-sterile surrounding country. 



If on the other hand we want the distribution after m migrations starting 

 with a cleared patch, which is not kept sterile, we have 



m F n {h, k) = (^) m -*N- m F n (K, k) (lxiii), 



for the whole district would have had a uniform density of (/iA) m_1 iV had there 

 been no clearance. Hence 



now A { h, q.w (x - i £ £ i «-* «*- i >' +6 '- «*■*.*) . 



