A MATHEMATICAL THEORY OF RANDOM MIGRATION 13 



Thus far no choice has been made of cr 2 . If we take cr 2 = \K*, we have 6 2 = 0, 

 or if cr 2 be taken half the square of the swing-radius about the axis of the 

 solid of revolution z = F(r i ), that is if cr be the swing-radius of the solid about 

 any plane through its axis, then the second term in the expansion of F (r 2 ) 

 in cu-functions disappears. 



We are now able, I think, to grasp the relation of the Rayleigh solution 

 to the complete solution of the random scatter round a centre of dispersion. 

 If (f> n (r 2 ) be the function giving the distribution after n nights, then c/> a (r 3 ) 

 can be expanded in a series of co-functions, i.e. 



4>n ( r *) = b <»<> + h<»2 + b i O) i + ■ ■• + K<»M + ■• ■ ■ 



By choosing the cr 2 of the co-functions = \K\ this becomes, since b the 

 volume = N, 



^^ = 2^ e_i? ' 2/<r2 f 1 +b ^ + - + 6 -X*+ -}■ 



Lord Rayleigh's solution provides the first term of this series, or is the 

 correct solution to two terms in the expansion by co-functions. It possesses 

 the properties (a) that its volume is the same as that of the complete solution, 

 and (b) the mean square deviation from the centre of dispersion is the same, 

 i.e. 2o- 2 , as for the complete solution. 



The latter depends upon the fundamental property of the co-functions that 



o) M r 3 rfr = 0, if s be > 1. 

 o 



The expansion in co-functions shows us at once that, whatever be the magnitude 

 of n, the mean square deviation from the centre of dispersion is Jnl, and this 

 gives us readily a rough measure of the range of habitat of any species for 

 which n and I are approximately known. 



Another point may be noted here as to the Rayleigh solution. That solution 

 is the best fitting Gaussian error surface to the distribution, i.e. its volume and 

 its standard deviation are the same as those of the actual distribution, whatever 

 n may be. If we take the section, however, of the distribution through its 

 axis the standard deviation of this according to the Rayleigh solution is <r = J\nl, 

 but this is not the standard deviation of the section of the actual distribution, 

 i.e. the Rayleigh solution does not give the best fitting normal curve 'to the 

 section. It gives only the standard deviation corresponding to co . It is of 

 some value to note what .are the standard deviations of other component 

 <o, s terms. 



To obtain this we must determine the area and even moments corresponding 

 to any w.^ term. Let 



/. 



