244 NATURAL INHERITANCE. 



number with one representative r _ 1 ??i 1 .N, the number with two 

 r^m^ N and so on, it follows, from what we last stated, that the 

 number of surnames with s representatives in the rth generation 

 must be the coefficient of x s in the expression 



| ,_ 1 m + ,_ 1 m 1 T + ,_ 1 TO 2 T 2 + etc. + ^m^T^ 1 |n 



If, therefore, the coefficient of N in this expression be denoted by 

 f r (x) it follows that r _ x m v r _ x m 2 and so on, are the coefficients of x, 

 x 2 and so on, in the expression f r _ x (x). 



If, therefore, a series of functions be found such that 



f l {x)—t () + t 1 x+ etc. +t q x Q and/,. (x)=f r _ 1 (f^ + t^x etc. +t 9 x g ) 



then the proportional number of groups of surnames with s represen- 

 tatives in the rth generation will be the coefficient of x s mf r (x) 

 and the actual number of such surnames will be found by multiplying 

 this coefficient by N. The number of surnames unrepresented or be- 

 come extinct in the rth generation will be found by multiplying 

 the term independent of x in/,, (x) by the number N. 



The determination, therefore, of the rapidity of extinction of sur- 

 names, when the statistical data, t , t v etc., are given, is reduced to 

 the mechanical, but generally laborious process of successive substi- 

 tution of t + t x x + t 2 x 2 + etc., for x in successively determined values 

 of/ (x), and no further progress can be made with the problem until 

 these statistical data are fixed ; the following illustrations of the ap- 

 plication of our formula are, however, not without interest. 



(1) The very simplest case by which the formula can be illustrated 

 is when q = 2 and t , t v t 2 are each equal to -|. 



Here / (x)- 1 + \ +X \ {x) = 1 j 1 + i (1+* + ^) + * 1+x + x^f 



and so on. 



Making the successive substitutions, we obtain 



, , . 1 I 13 5x Sx 1 2x x \ 



,, v 1249 + 265£ 343a; 3 166a? loOr" 34a^ 16a£ + J^.4- x * 

 ' A ' ' 2187 2187 2187 2187 2187 2187 2187 2187 2187 



= -63183+ -0S306£+ -10635a; 2 + -07804a?+ , 06489rc 4 + -05443a; 5 + '01437a; 6 

 + -01692a- 7 + -01144a,- 3 + -00367a? + '00104a; 10 + -00015a; 11 + -00005a; 12 

 + -OOOOla; 13 + -00000^ 14 + -00000a; 15 + -OOOOOay 6 



