A MATHEMATICAL THEOEY OF RANDOM MIGRATION 41 



Writing <^ = t as before, we find 



aH)}-- 1 *^ ^ 



Hence 

 dt° \ p ° Wj = - 2 ^ U d^ \Jt) 



In/W-IVm/ , ,. 1 (s-l)(s-2) 1.3 



2 £ s * \ 2 1.2 2.2 



the expression being the same as that on p. 32. 



Now let us write the following for brevity where r) = u/a- : 



11 _1 2 



A (v) = ~ j^ V e *' M, (v) + v ^ (v) + ---+ v 2s*l>2 (s-D (v) + ■■■)> 

 A (v) = 2 j^ ye~ W fa* + 3v.& (v)+-+ sv^hs-2) (v) + - •)- 



A(^)=j|;^e-^(4v a +10^ 2 (>7)^ 



and so on. All these functions are directly expressible in w-functions as 

 on p. 27. 



Further let P s ( v ) = ( - 1)^ Jlp.fo)-! J^-^-uM (1*4 



Then we have, if 



7^ = (a — h)/a; rj 2 = (a + h)/a; Cj = (a — &)/cr, e 2 = (a + A)/o% 



F n {h,h) = N [{P fa,) + P fa 2 )} {P (0 + P (e 2 )} + {A (r,,) + A fa,)} {P ( Cl ) + P (e 2 )} 



+ {P (*) + P (a {A (0 + A (e 2 )} + {A (0 + P, (e 2 )} {A (*) + A (v*)} 



+ ... + {P s (e 1 )+P s (e 2 )}{A + i(^) + A + iW}+-] (Ixxi). 



The .//-functions involve the rapidly converging v-coefficients, and the first few 



terms will suffice to get an idea of the distribution. If we retain only the Rayleigh 



terms we find : 



f n (h,k)=N[l- {P ( Vl ) + P fa)} {P (0 + Po (*)}] (Ixxii), 



which can be ascertained for given values of a, b, h, k and o- from the ordinary 



tables of the probability integral. 



