APPENDIX F. 243
a0, a1, a2, etc., remain the same in each succeeding generation. We shall also, in what follows, neglect the overlapping of generations-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 ao, a1, 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 I00' I OU' loo ' e c., up to l a0 be denoted by the sym
bols t0, t1, t2, etc;) up to t9, in other words, let to, t1, 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 surnames, and let ,.m, 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 generation is the coefficient of x' in the expansion of the multinomial
(to + t1x + t2x2 +, etc. + t5x°)P
Let then the expression to + tax + t2x2 + etc. + t,x' be represented by the symbol T.
Then since, by the assumption already made, the number of surnames with no representative in, the r-lth generation is '1m, N, the