48 KAEL PEAESON 



The Rayleigh solution value is easily found by putting Q t = 1 in any of the 

 forms of (lxxix), (lxxx), (lxxxi), (lxxxiii) or (lxxxvi). 



A case of peculiar interest arises when c = 0, or we take the density at the 

 centre of the clearance. In this instance we have : 



Now 



and e~ i( * /a * = 2*0*0,, 



therefore („•)• ^e^) _ w { *^ + . *£} (<,)• 



=e-w° 2 (-iy{ X2S -s X2is _ 1) }. 



Thus 



/„ (0) = iVe -* a2/<r2 {1 - 2v 4X2 + (v t - 3v 6 ) X4 + („, - 4r.) X6 + (v 8 - 5v M ) x , + . . .} 



= 27ro- 2 iV(w — 2v,w 2 + (v 4 — 3i/„) w 4 + (i> 6 - 4v 8 ) w 6 + (v 8 — 5^) w s +...).. . (lxxxvii). 



We are also able to consider the secondary problem : 



What is the distribution into unoccupied space surrounding a uniformly 

 occupied circular area due to a first migration ? 



Let the radius of the area be a and let the density at any distance c be 

 F n (c) after the first migration. Then clearly, if all space were uniformly filled, 

 we should have uniformity after the first migration, or : 



F n (c) + F n {c) = N, 



hence : F n (c)=N-F n (c) (lxxxviii). 



The solution is thus thrown back on the solution obtained for the previous 

 problem. In particular at the centre of the populated area we have : 



F n (0) = N-f n (0) (lxxxix). 



We are thus able to calculate the reduced central density due to a migration 

 from the area to the surrounding unoccupied district, i.e. the effect on population 

 of the spread outwards of a colony. 



(15) Problem VIII. Indirect solution of the General Problem of the 

 Random Walk. 



It may not be without interest to put on record the distribution density 

 after n flights in the case of a cleared circular area, if it be expressed in 

 Kluyver's manner by the integral of a Bessel's function product. 



