242 NATURAL INHERITANCE. 



very problem. I made certain very simple and not very inaccurate 

 suppositions concerning average fertility, and I worked to the 

 nearest integer, starting with 10,000 persons, but the computation 

 became intolerably tedious after a few steps, and I had to abandon 

 it. The Rev. H. W. Watson kindly, at my request, took the pro- 

 blem in hand, and his results form the subject of the following 

 paper. They do not give what can properly be called a general 

 solution, but they do give certain general results. They show (1) 

 how to compute, though with great labour, any special case ; (2) a 

 remarkably easy way of computing those special cases in which the 

 law of fertility approximates to a certain specified form ; and (3) 

 how all surnames tend to disappear. 



The form in which I originally stated the problem is as follows. 

 I purposely limited it in the hope that its solution might be more 

 practicable if unnecessary generalities were excluded : 



A large nation, of whom we will only concern ourselves with the 

 adult males, N in number, and who each bear separate surnames, 

 colonise a district. Their law of population is such that, in each 

 generation, per cent, of the adult males have no male children 

 who reach adult life ; a L have one such male child ; a 2 have two ; 

 and so on up to a 5 , who have five. Find (1) what proportion of the 

 surnames will have become extinct after r generations ; and (2) how 

 many instances there will be of the same surname being held by m 

 persons. 



Discussion of the problem by the Rev. H. W. WATSON, D. So., F.R.S., 

 formerly Fellow of Trinity College, Cambridge. 



Suppose that at any instant all the adult males of a large 

 nation have different surnames, it is required to find how many of 

 these surnames will have disappeared in a given number of genera- 

 tions upon any hypothesis, to be determined by statistical investiga- 

 tions, of the law of male population. 



Let, therefore, be the percentage of males in any generation 

 who have no sons reaching adult life, let a^ be the percentage that 

 have one such son, 2 the percentage that have two, and so on up to 

 a g , the percentage that have q such sons, q being so large that it is 

 not worth while to consider the chance of any man having more 

 than q adult sons our first hypothesis will be that the numbers 



