34*2 TAfe and Letters of Francis Galton 



was {K»rsua(itMl to take it up and sent his discussion of it to the above 

 Journal'. His discussion witli certain preliminary remarks by Galton was 

 also published in the Journal of the Anthropological Institute*. The kernel of 

 Watson's paper is as follows: The symlxjls I,, t,, ... t,, ... t,^ denote the chances 

 of a man luiving no children or one. ... s, ... q children. Then the chance of 

 a surname iiaving s repi-esentatives in the next succeeding generation, if it 

 has p in any generation, will be the coefficient of .t** in the multinomial 



(/, + l,x + /,x-" + . . . + l^x'')" = T>\ say. 



Let ,«i, be the fraction of N, the original number of distinct surnames, 

 which in the »/th generation have v representatives, then the number of 

 surnames with « representatives in the v generation must be the coefficient 

 of x* in 



{,..m, + ,..m.7'+,..»«.r +...+,. ,7V-. ^''"'}^=/r(^)^. say. 

 It follows that r_,m,, ,.,m,, etc. are the coefficients of x, x', etc. in the ex- 



f>ression /,_, (x). As soon as the t's are known, it should l)e possible, although 

 aborious, to find the succession of functions given by 



fr{x)=f,.,{t, + t,x + ... + t,af). 



As the numerical values of the ^'s are not known, Watson takes two hjrpo- 

 thetical systems. In the first he takes '/ = '.i and 'o = <, = ^ = ^. He finds by 

 a brute-force expansion that out of a million distinct surnames 333,333 will 

 disappear in the first, 148,147 in the second, 89,620 in the third, 70,030 in 

 the fourth, and only 34,150 in the fifth generation. In this ca.se the total 

 male pipulation is clearly constant and two-thirds of the surnames have 

 disappeitred in five generations. Watson's second hypothesis is that the I'b 

 are the successive terms in the binomial 



where X, -|- X, = 1 . In this case 



/, (x) = (\, -h X, x)«, and ,?n, = X,'', 



and ,Tn, = (X,-|-X.,X,»)"' 



= (X, + X„m.)' = X,'(^-t-,m.y. 



and generally rW, = X,* ( -^ -f- r. i "i, ) • 



The extinctions in each generation can then be easily cjilculated. Watson 

 takes the case of X, = |, X, = ^ and q = 5. In this case the t's are 



<.= -237, «, = -39G, <, = '264, /,= -088, /.= -014, t, = -00l, 



and the extinctions in the first ten generations of 1000 original distinct 

 surnames : 



237, 109, (Jo, 40, 27, 18, 14, 10, 7, G, 



' Educatwuil Tiiitei, Vol. xxvi, 187.3, p. 17 and p. 115. 

 '' Vol. IV, pj.. 1.38-44, 1874. 



