APPENDIX F. 243 



a , a v a 2 , etc., remain the same in each succeeding generation. 

 We shall also, in what follows, neglect the overlapping of genera- 

 tions — that is to say, we shall treat the problem as if all the sons born 

 to any man in any generation came into being at one birth, and as 

 if every man's sons were born and died at the same time. Of course 

 it cannot be asserted that these assumptions are correct. Very 

 probably accurate statistics would discover variations in the values 

 of a , a v etc., as the nation progressed or retrograded ; but it is not 

 at all likely that this variation is so rapid as seriously to vitiate any 

 general conclusions arrived at on the assumption of the values 

 remaining the same through many successive generations. It is 

 obvious also that the generations must overlap, and the neglect to 

 take account of this fact is equivalent to saying, that at any given 

 time we leave out of consideration those male descendants, of any 

 original ancestor who are more than a certain average number of 

 generations removed from him, and compensate for this by giving 

 credit for such male descendants, not yet come into being, as are not 

 more than that same average number of generations removed from 

 the original ancestors. 



Let then —2-, — L, — ?-, etc., up to — ' , — be denoted by the sym- 

 100 100 100 ' * 100 J J 



bols t Q , t v t. 2 , etc., up to t g , in other words, let t , t v etc., be the 

 chances in the first and each succeeding generation of any individual 

 man, in any generation, having no son, one son, two sons, and so on, 

 who reach adult life. Let N be the original number of distinct sur- 

 names, and let r m s be the fraction of N which indicates the number 

 of such surnames with s representatives in the rth generation. 



Now, if any surname have p representatives in any generation, it 

 follows from the ordinary theory of chances that the chance of that 

 same surname having s representatives in the next succeeding gene- 

 ration is the coefficient of x s in the expansion of the multinomial 



(t + t x x + t. 2 x 2 + , etc. + t q x q ) p 



Let then the expression t -f- t x x + t 2 x 2 + etc. + t q x 7 be repre- 

 sented by the symbol T. 



Then since, by the assumption already made, the number of sur- 

 names with no representative in the r-lth generation is r . x m^ N, the 



R 2 



