A MATHEMATICAL THEOEY OF RANDOM MIGRATION 45 



, Too [2ir 



F n {c) = N I I <f> n (<? + '>*~2rc cos d)rdOdr (lxxvi) 



Now cos m ddd = 0, if m be odd, and = 4 cos^ddO 



Jo , J 



_ 1 (26-l)(2.9-3)...l 7T_ 2;r 2s! 



2s(2s-2)...2 2 (2*s!) 2 ' 



if m be even and = 2s. 



Hence : 



,. w -*^/\h~4{®-(^}* w 



^-"efe 



= M u+1 (a/a) (lxxviii). 



-^»+i Wo") is thus the 2s + 1th moment of the ' tail ' of a normal or Gaussian 

 curve of errors (multiplied by \/2ir) about its axis. Its values have been tabled 

 for s=l, 2, 3 and 4. 



Thus we have : 



r.M-w-Wsffi*^ (faix). 



But it is easy to see that : 

 Accordingly : 



f n (c)=N Qt e ^y |i + __ + _y +...+_ y|_... ( i xxx) . 



The successive differentiations of this expression with regard to t = a- 2 . involved 

 in the operator Q t , which are needful if we wish to give the corrections to 

 the Rayleigh solution, are straightforward but extremely laborious. We can 

 throw the solution into other forms. 



Write: e 1 = ^c?/o*, e 2 = ^a 2 /<j s> , 



then we have : 



s! 



= NQ t e->S-£s fafe-dx •. (lxxxi). 



