A MATHEMATICAL THEOEY OF RANDOM MIGRATION 37 



The distances are all given in yards. 



Thus we see that the least of these distances for 1 per cent, is greater than 

 half a mile, or, if an area be cleared but not rendered sterile, we might expect 

 within a year the mosquitoes to reappear within half a mile of the boundary, 

 and to reach an objectionable frequency even at this distance for most of the 

 cases considered. 



As far then as these rough numbers can be taken to indicate the state of 

 affairs, it is needful not only to clear an area but to maintain it sterile. The 

 clearance radius may be only -^ mile and is hardly likely to exceed a mile, and 

 the above results only mark the progress of immigration in the course of one 

 year after the clearance. Further the results, would be accentuated if the 

 boundary were curved or an approximately circular clearance made. 



It does not appear to me that any substantial difference would be made in the 

 main result by reducing n to 3 or 4, although some difference would occur if I 

 were reduced to 20 or 30 yards. 



(10) Problem III. To determine the distribution after m n- flight migrations 

 starting with a centre of population No.. 



The previous two problems indicate the nature of the general solutions to which 

 I now proceed. I shall adopt the longer process of proof in this first case as being 

 the more suggestive. 



By (xii) and (xlvi), calling the operator as before Q t , we have for the distribution 

 at X, Y due to a centre at the origin : 



4AX, *)-£ ft (!.-»<*+ w) (lT) . 



Hence the distribution at (h, k) after a second migration of n flights is 



^(h,h) = ^J J 4 n {X,Y)^ e - ^ dXdY. 



Call the Q t in this Q h and write the o- 2 on which it operates cr/ ; call the Q t in 

 !<£„ {X, Y), Q ti and the cr 2 on which it operates a-', we have : 



1 aX-Kf+(Y-kf X*+Y* \ 



2 <MM)=^ <k<fej_. J_.^V e dXdY. 



The integrations can be performed and give us 



, M h, t)_e^! Q,.Q,.~e- W - + ^" ,+ ^ (W). 



This only differs from ^> n (X, Y) by the introduction of of + o- 2 2 for a- 2 and of the 

 factor pAQ tl . 



