44 KAEL PEARSON 



On the other hand a cleared but not permanently sterile square mile would 

 after a year have a density at the same point — \ mile from the centre — of: 



]0 / 6 (440, 0) = ( M A)W[1 -{P (-402) + P (1-205)} {2P (-803)}] = (/.A) 8 -69iV, 



or of 69 per cent, of that in uncleared country. 



Another point seems of some interest. What is the density at the boundary- 

 after the first migration ? 



At the middle point of the edge it is 



^,(880, 0) = iV[l-{P (0) + P (5-08)}{2P„ (2-54)}] 



= N(\- -5000 x -9889) 



= -5062V. 



This is almost the ^N of an indefinitely long straight boundary. 



At the corner it is 



^(880, 880)=iV^[l-{P (0) + P (5-08)} 2 ]=-75i^ nearly, 



or, as we should expect, has risen much beyond the \N value. 



There is no difficulty in tracing the contour lines of the population density 

 in this case. 



If we consider a cycle of 10 breedings in a non-sterile patch we have : 

 10 / 8 (880 ) 0) = ( / ,A) 9 iv-[l-{P (0) + P (l-607)}{2P (-808)}] 

 = -7422V (/aA) 9 , 

 and M /.(880, 880) = (/*A)W[l -{P (0) + P (l-607)} 2 ] 



= -8012V (^A) 9 . 



Thus if the patch were not sterile, the effect of the clearance would at the 

 boundary after the lapse of a year be marked by a 20 to 25 per cent, reduction. 

 The illustrations I have given are of course dependent on the values of the 

 constants selected. Such constants have at present been little studied, and 

 accordingly small weight can be laid on the actual numerical results. But the 

 theory appears to indicate useful lines of inquiry, even if its results will of 

 course need to be controlled everywhere by local facts. In a general way there 

 can be little doubt that a theory like the present will not only lead to a more 

 systematic classification of local facts and to fuller observation of the habits of 

 local species, but that this knowledge itself will in its turn test the applicability 

 of the theory, or suggest the directions in which it may need modification. 



(14) Problem VII. To determine the distribution after a first migration into 

 a cleared circular area. 



Let the radius of the cleared area be a. Then at distance c from the centre, 

 inside or outside the circle of radius a, the distribution F n (c) is given by : 



