244 NATURAL INHERITANCE. 



number with one representative r . l m l .'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 rib. generation 

 must be the coefficient of ** in the expression 



etc. + r 



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

 f r (x) it follows that r .-^n-^ ^-pn^ and so on, are the coefficients of x, 

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



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



fi(x)=t + tjX+ etc. +t q x q and f r (*)=/._! (t + t^x etc. +t q x 9 ) 



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

 tatives in the rth generation will be the coefficient of x'inf 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 inf r (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 + t. 2 x* + etc., for x in successively determined values 

 of f r (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 } , t 2 are each equal to |. 



Here /, (*) = I|^/ = 1 1 1 + 2 (1 +* + **) 



and so on. 



Making the successive substitutions, we obtain 



13 5x 6ar 2 2x 



2187 2187 2187 2187 2187 2187 2187 2187 2187 



/ 4 (a;)= -63183+ -08306a:+ -10635x 2 + '07804x 3 + > 06489x 4 + -05443x5+ '01437a; 6 

 -01144Z 8 + -00367a; 9 + '00104a; 10 + '00015a; 11 + 

 00001a; 13 + '00000x 14 + '00000i; 15 + 'OOOOOa;' 6 



