52 KARL PEARSON 



Illustration II. Let us assume precisely the same conditions as in the previous 

 illustration, except that the area shall be supposed sterile, and we will consider 

 what happens at the end of the first migration. 

 At the centre we have by Equation (lxxxvii) : 



x F e (0) = Ne- {1 - 2v iX2 + ( Vt - 3*,) x , + („, - 4v 8 ) X . + • • •}■ 

 But -2 Vi = -083,333, xA*>) = l ~ e *= -2-227,000, 



v t -Sv 6 = --032,407, x*(0 = 2 -4e 2 + e 2 2 = --494,471, 

 v 6 -4i> 8 = --005,498, x,(« 1 ) = 6-18e 1 + 9e 1 ,, -e, i, = 8-031,303, 

 v 8 - 5v 10 = -000,082, X8 (e 2 ) = 24 - 96e 2 + 72e 2 2 - 1 6e/ + e 2 4 , 



e 2 = 3-227, =34-752,347. 



Hence: J 6 (0) = Ne~ rw (l - -185,583 + -016,024- -044,156 + -002,850) 



= -03l2V. 



This three per cent, of the density in uncleared area might possibly prove 

 a trouble and on our assumptions it may be doubted whether the half-mile 

 radius is sufficient. If we take the first term only, we find - 040iV, or four 

 per cent., not an important practical difference. 



The introduction of even the first modifying term when c is not zero appears 

 to lead to such complexity that I content myself with calculating the approximate 

 value given by the Rayleigh solution for distances of £ and ^ mile from the 

 centre of the clearance. In this ca,se e 2 = 3*227, e, = '807 and 3 - 227 respectively 

 half-way to and at the boundary. I proceed just as before and deduce the 

 following approximate value for 1 f e (c), i.e. 



/ , (c) = (fiAfN {1 - e-> + e- fe+€ «> (1 - 20-9769e t - 7-8851^ - -23556/ 



- -0228e x 4 - -0016c/ - -OOOle, 6 )}. 

 Hence 



^(440) = -179 (/xA)W, corresponding to e^'807 



and 



1 f,(880) = "709 ()u.A) 9 iV, corresponding to Cl =3-227. 



Thus the density at ^ of a mile from the centre of the cleared patch would 

 be some 18 per cent, of the density in uncleared country. In other words on 

 our assumptions a clearance of one mile diameter, if kept sterile, would hardly 

 suffice to keep an area of ^ mile diameter free of mosquitoes. 



Compared with a straight boundary, where the density falls to about one 

 half that of uncleared country at the boundary, we see that the bending of 

 the boundary has a most marked effect in its neighbourhood, the curvature 

 raising the boundary density from about 50 to 71 per cent, of the uncleared 

 density. In fact the density is almost equal to the 75 per cent, in the boundary 

 angle of a square clearance. 



