40 KAEL PEAKSON 



Hence the rule : If the solution can be found for a single migration, replace cr 2 

 by mcr 2 , and each v by the proper N, multiply by the factor (/x,A) m_1 , and the solution 

 for m migrations is deduced. 



It will thus be clear that, if the solution can in any case be found for one 

 migration fully, we can at once extend it to the case of any number of migrations, 

 with constant fertility-survival factor. 



(12) Problem V. To determine the distribution after a first migration into a 

 cleared rectangular area. 



Let the area be the rectangle 2a x 2b, and the origin be taken at its centre 



and axes of x and y parallel respectively to the sides 2a and 2b. Then the density 



at any point h, k, after a single migration F n (h, k) is given by the principle of 



the last problem by 



F n (h, k) = N-F n (h, k) (lxv), 



where F n (h, k) is the distribution from a uniformly occupied rectangular area 

 into surrounding unoccupied space. 



r+a r+b 

 But F n (h, k) = 2V <l> n {(x-hy + (y-ky}dxdy 



1 *™ 1 f +a f +b 2 \ <r 2 "'" n-2 J 



= k~NQ, 



2TT ™ o- 2 



1 [+ a f+* 2 \ „* * <r* J 



— \ e dxdy 



cr J _ a J -b 



2tt <r J - a J -b 



Let P (e) stand for the probability integral 



V 2tt J o 

 Then: * fY*<- h ^ dx= ' f^'V*"^ 



\I2tT<T J -a s/2tt J -{a+h)l<r 



' J2tt 



{a-h)l<7 r{a+h)l<r\ _ ^ 



+ ) e * dx 



o 



-r>J—)+P.' a+h 



Thus 



or 



F, (h, k) = NQ, P. {(5=*) + P. («±*) } {p. (^) + P, (»-±*)} . . .(Ixvi). 

 Now consider the differentiation of P (-] with regard to cr 2 . 



dcr 2 [ \o7 J 2crC^O- N/27T JO 2 O- V27T cr 2 



