APPENDIX F. 245 



and the constant term in,/* 5 (x) or 5 m is therefore 



„. 01QO , "08306 , -10635 , '07804 , '06489 , '05443 , '01437 , '01692 , '01144 



o 3183 + + + + + + + + 



3 9 27 81 243 729 2187 6561 



•00367 - 00104 '00015 

 19683 59049 177147 



The value of which to five places of decimals is '67528. 



The constant terms, therefore, in f v f 2 up to f 5 when reduced to 

 decimals, are in this case '33333, -48148, r 57110, '64113, and -65628 

 respectively. That is to say, out of a million surnames at starting, 

 there have disappeared in the course of one, two, etc., up to five 

 generations, 333333, 481480, 571100, 641130, and 675280 re- 

 spectively. 



The disappearances are much more rapid in the earlier than in the 

 later generations. Three hundred thousand disappear in the first 

 generation, one hundred and fifty thousand more in the second, and 

 so on, while in passing from the fourth to the fifth, not more than 

 thirty thousand surnames disappear. 



All this time the male population remains constant. For it is 

 evident that the male population of any generation is to be found by 

 multiplying that of the preceding generation, by t x + 2t 2 , and this 

 quantity is in the present case equal to one. 



If axes Ox and Oy be drawn, and equal distances along Ox repre- 

 sent generations from starting, while two distances are marked 

 along every ordinate, the one representing the total male population 

 in any generation, and the other the number of remaining surnames 

 in that generation, of the two curves passing through the extremities 

 of these ordinates, the population curve will, in this case, be a straight 

 line parallel to Ox, while the surname curve will intersect the popu- 

 lation curve on the axis of y, will proceed always convex to the axis 

 of x, and will have the positive part of that axis for an asymptote. 



The case just discussed illustrates the use to be made of the general 

 formula, as well as the labour of successive substitutions, when the 

 expressions/^ (x) does not follow some assigned law. The calcula- 

 tion may be infinitely simplified when such a law can be found.; 

 especially if that law be the expansion of a binomial, and only the 

 extinctions are required. 



For example, suppose that the terms of the expression t + t x x + 

 etc. +t rj x 7 are proportional to the terms of the expanded binomial 



