A MATHEMATICAL THEORY OF RANDOM MIGRATION 27 



Th™t-j«-(-i).-^^ 



+ {S ~ l)i l~* ] {S " 8) 1 . 3 ■ 5 A,-, + etc.) (xlvlii). 



Substituting in (xlvii) we have, if $ (■>?) = -= \° e'^dx (xlix), 



F n (c) = i\T [* (Q -J^ C - {a*a> (|v 4 + f „. + V-v 8 + W^ + W"») 



+ cr 2 ^ (v 4 + v 6 + 1 v, + -V-^ 10 + *#^ B ) 



(1). 



+ crV, (v 10 + fv u ) + cr'a)^ + . . .} 



as far as coefficients of the order v 12 and functions of order &> J0 . 



This is the solution in w-functions. Table III., p. 21, gives the values of the v's for 

 certain values of n, and Table II. , p. 20, is a preliminary table of the <u-functions. 

 These will enable us to readily find the values of F n (c). I have done this for the case 

 of n=% and n = 7, which will suffice to illustrate the character of these curves. 

 \jj (c/(r) can be found at once from Tables of the probability integral. It is drawn 

 with a broken line in Diagram VII. and is the Rayleigh solution for this case. I term 

 F n (c) an "infiltration curve" of the first order. 



Substituting the values of the v's from Table III., we have for n = 6 : 

 F 6 (c)/N=$ (-) +- {-026,712,414 (oX) + '053,325,539 (o- 2 o> 2 ) 



+ -002,114,303 (cr 2 ^) - -001,029, 898 (o- 2 w 6 ) 



- -000,134,978 (o- 2 a) 8 ) - -000,001,770 (o- 2 w 10 ) + ...}, 



and for n = 7 : 



F r (c)/N = 1t (-) + -{-022,850,925 (o'a t ) + "045,644,347 (cr 2 a> 2 ) 



+ -001,578,008 (o-X) - -000,758,570 (o- 2 o> 6 ) 



- -000,084,525 (o»co 6 ) + -000,000,178 (a'a n ) + ...}. 



The first term \p I - J is the ogive curve already drawn corresponding to the 



Rayleigh solution. We see at once that the term o*(ti w will not affect the 



fourth place of decimals. 



4—2 



