A MATHEMATICAL THEOEY OF EANDOM MIGRATION 51 



A(c) = e-<^> 0*A)-W(l + e, (l + O+l! ( X + i2 + 2l) 



The successive bracketted terms in e 2 are 



1-3227, 1-3748, 1*3804, 1-3809 and 1-3809, 



which is equal to e +H to our number of decimal places. Hence we may put 



«/e (c) = (fiAYNe-^^ jl + e, (e ; ° - "0582) + ^ (e* - '0061) 



3'. ' 4 s\ 



= ([xAyNe-^+'J (1 - e'* + e^ + ^ - -0582^ - -0030^ - -OOOle/) 

 = (/xA^ll - e""' 1 4- e" (i ' +i2) (1 - •0582e 1 - -0030^ - -000 le x 3 )}. 



At ppntyf 1 



10 f e (0) = (pAYN{e-™ (l)} = ( f tA)W724. 



We can test the accuracy of this result by using Equation (lxxvii) which, 

 if we put v 4 — N 4 , gives: 



„/, (0) = (pAYNe--<* (1 - 2N iX2 +...) 



and X2 = 1 - i, = 0*A)»iVe— ' (l + ^ + . . . 



= (/AA) 9 i^-730. 



The agreement is accordingly good enough for practical purposes, and we 

 may say that within a year the mosquitoes would at the centre of the patch 

 have a density 73 per cent, of what they would have in uncleared country. 



I now consider the density at a quarter of a mile from the centre, e 1 ="0807, 

 and using the above formula we find : 



W F, (440) = (fjLAyN(l-e-' m7 + e-' mi x -9953) 



= ( f iA) s N-75, 



or, we see that at a quarter mile, midway between centre and boundary of 

 the patch, the density is only 2 per cent, more than at the centre. 



Finally, at the boundary itself, e 1 ='3227 = e 2 , 



„/, (880) = (fiA) s N(l _e" w + «-■•« x -9809) 



= (lxA) 9 N-79. 



Thus the cleared patch would within the year have filled up with a population 

 of mosquitoes varying in density from 73 per cent, at the centre to about 80 per 

 cent, at the boundary, or the clearance without permanent sterility would have 

 been quite ineffectual with the assumed values of the constants. 



7—2 



