A MATHEMATICAL THEORY OF RANDOM MIGRATION 31 



This is integrable and gives : 



dc r vm j^j-^—z 



Integrate with regard to c, and remember that m 2 = if c=oo ; thus: 



= pWQ t Q t ' 1 Lr e-i**dx' (lii). 



V27rJ c /V<r 2 +(r' 2 



Comparing this with (li) we see that u 2 differs from v^ by (a) the introduction of 

 the factors Q{ and /aA and (&) the replacement of cr in the lower limit by v/V + o-' 2 . 

 The process can therefore be repeated as often as we please, and we have for 

 u m the value : 



u^bi&^NQtQ/Qt"... to m terms ^L f" e - ^ 2 ^', 



<j2irj~c[2 



where 2 2 = o- + o- /2 + cr" 2 + ... to m terms. 



After the operations indicated by the Q's are completed, we are to put 



Now it is clear that a differentiation with regard to any cr 2 is precisely the 

 same as one with regard to 2 2 . We can therefore write for all the Q's the 

 simple expression 



understanding that d/d(2?) operates only on % and that after the operation is 

 completed we can put % = Jma: Thus the complete solution is : 



~f e-^dx (liii). 



This is true for c positive or negative, i.e. whether the density be considered 

 at a point on the originally occupied or originally unoccupied side of the boundary. 



Up to terms of order 1/n 3 we have for the operator the value 



1 + m ( Vi q 2 - vrf + v s q* - v 10 q* + v u q*) 



+ m( ^~ l} (W ~ 2W2 1 + 2 v iVi <t) 



+ m(m-l)(m-2) v ^ ^^ ^ standg for ^^^ 



