22 KAEL PEARSON 



Hence- f (0= —, ^Mlh (xxxix bis). 



Jn{ * ] (>/2^)" +1 P^2— r(±(»-l)) K ' 



This can be proved by induction. 



For most of the cases more approximate formulae still were deduced. Thus : 



*M- itfiwHf + hr) (xl) ' 



f^'^wM + lhWrS) w 



«^ab(!r( 1+ sT+-) (^ 



after which the first term only as given by (xxxix) is sufficient. It will be observed 

 that after <£ 6 (r 2 ), the curve touches at r = nl or f = 0, and the contact becomes higher 

 and higher as n increases. Thus, although short of n = oo , there is no real asymptot- 

 ing to the axis, still <f> n (r 2 ) for n > 5 not only vanishes for r = nl, but has increasingly 

 higher contact as n increases. This explains how the Gaussian curve can fairly well 

 represent the state of affairs towards the end of the dispersal range, if n is > 5. 



Mr Blakeman found that the ends of the range for the various cases ran closely 

 into the curves (xl) to (xliii), and they were tested and, if needful, corrected by 

 these formulae. 



Thus the whole graphical work was kept in check, and, I think, we may be 

 confident that the true forms of the dispersal curves for n = 4 to 7 are really 

 given by our diagrams and tables. 



(7) We may note a few features of these curves. 



Dispersal Curve for Two Flights (Diagram I.). 



There is no discontinuity in the solution from r = to 2l, the range within 

 which all individuals fall. The curve asymptotes to the vertical at the axis and 

 at r — 2l. Of course, while the density becomes infinite, the number on any small 

 area near r = or r = 2l, is finite. Thus the number between the circles of radii 

 ?\ and r 2 is 



2N ( . _,r t . _ 1 r 1 

 lT[ Sm 2l" Sin 21 



If r x = and r a = e v where e x is small, the number v t within the small circle of radius ^ 

 at the centre of dispersion = Nejfal). If r 2 = i\ + e 2 , the number lying on the zone of 



Ne I r 2 \ - ^ N [7 



breadth e, is — f f 1 - jr 2 J , and this if 1\ = 21 - e 2 , is v 2 = — / -j . At the position of 



