A MATHEMATICAL THEORY OF RANDOM MIGRATION 



1 /r\ 12 _ r 2/2o-2 / r 2 r 4 r 6 25 r 8 ] r 10 \ 



?"Y 4 -j- 2 /2o-2 / r 2 r 4 r 6 225 r 8 9 /r\ 10 1 /r\ 12 ' 



- e '<*" 720-2160- 2 +1350- 1 -300- 6 + ^3-^ - + 



128 W \' v " (^"""V """o- 6 ^ 8 o- 8 8W "64 W. 



Remembering that by (vii) P = 2ar 2 /n, we have from (ix) 



6n-ll / r 2 ?* 4 r 6 1 r 8 



+ w( 24 - 48 ;? +18 ;?- 2 < ?+T6;/ 



50n — 57/ , r 2 r 4 r 6 2*i r 8 I r 10 '' 



18 92 -2125» + 270»V 720 _ 21( . ^ +1350 ^_ 300 r; + ?pr;_9 r ; 1 f 



103,680n 5 \ cr 2 o- 4 o- 6 ' 8 cr 8 8 <r 10 64cr 

 - etc. [ (x). 



This is the general expansion for the distribution of the individuals emerging 

 from a centre of dispersion after n random flights. Clearly if we want to go as 



far as — we must retain terms up to (r^/o- 2 ) 2 *, and the convergence is small for n 



small. Thus for the first two or three flights, (x) as far as I have calculated the 

 terms gives poor results, even if they are notwithstanding better than the Rayleigh 

 solution. The arithmetical work required to calculate the ordinates is also severe. 

 "If we put ri = oo, we have 



<M r ) = 2^ i ( xl )' 



Lord Rayleigh's expression. Now cr* = ^nl 2 , hence unless I becomes indefinitely small 

 as n becomes indefinitely large the population becomes widely scattered. If the 

 • • single flight I be very small, but the total flight nl be finite, then \nl 2 tends to 

 become vanishingly small, or the population remains close to the centre of dispersion. 

 This is really the "flitter" as distinct from the flight. 



Examining the solution found it is clear that it may be looked upon as the 

 sum of products of two factors, one series of factors not involving r/a- but only n 

 and the other not involving n but only r/a: Thus we may write 



<j> n (r 2 ) = A r (v w + v 2 o> 2 + v i o) i + v 6 w s + . . . ), 



where 



2iro*' 





27r<T* e V 1 2o> ' * ' 



