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