OCR Rendition - approximate

WATSON and GALTON.-Extinction of Families. 139 mathematicians, I put the problem into a shape suited to mathematical treatment, and proposed it in the pages of a well-known mathematical periodical of a high class, the "Educational Times." It met with poor success at first, because the answer it received was from a correspondent who wholly failed to perceive its intricacy, and his results were totally erroneous. My friend the Rev. H. W. Watson then kindly, at my request, took the problem in hand, and published his first results in the above-mentioned periodical. These have since been considerably extended, and 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. I therefore feel sure that Mr. Watson's memoir will be of interest to the Anthropological Institute, and I beg to submit it to their notice, both for its intrinsic value and in hopes that other mathematicians may pursue the inquiry and attain still nearer to a complete solution of this very important problem. 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, ao per cent. of the adult males have no male children who reach adult life ; al have one such male child ; a2 have two; and so on up to a5 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. 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 generations upon any hypothesis, to be determined by statistical investigations, of the law of male population. Let, therefore, as 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, a, the percentage that have two, and so on up to a0, 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