APPENDIX P. 
243 
a0, av a2, 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 «0, av 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 etc., up to be denoted by the sym- 
bols <0, tv <2i etc,, up to <s, in other words, let <0, tv 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 rm, 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 xf in the expansion of the multinomial 
(<0 + <!* + <2x2 +, etc. + t,xq)p 
Let then the expression <0 + tjX -f <2æ2 4- etc. -f tqxq 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 ,.^0 N, the 
R 2