APPENDIX F. 247 



step, and an additional 109 names in the second step, there are only 

 27 disappearances in the fifth step, and only six disappearances in the 

 tenth step. 



If the carves of surnames and of population were drawn from this 

 case, the former would resemble the corresponding curve in the case 

 last mentioned, while the latter would be a curve whose distance 

 from the axis of x increased indefinitely, inasmuch as the expression 



^+2*2+3*3 + 4^+5*5 

 is greater than one. 



Whenever f x (x) can be represented by a binomial, as above sug- 

 gested, we get the equation 



whence it follows that as r increases indefinitely the value of r m ap- 

 proaches indefinitely to the value y where 



y-u^D^^y) 



(o+4) 



that is where y = 1 . 



All the surnames, therefore, tend to extinction in an indefinite 

 time, and this result might have been anticipated generally, for a sur- 

 name once lost can never be recovered, and there is an additional 

 chance of loss in every successive generation. This result must not 

 be confounded with that of the extinction of the male population ■ for 

 in every binomial case where q is greater than 2, we have t T + 2* 2 + etc. 

 -\-qt q >l, and, therefore an indefinite increase of male population. 



The true interpretation is that each of the quantities, r m v r m. 2 , 

 etc., tends to become zero, as r is indefinitely increased, but that it 

 does not follow that the product of each by the infinitely large num- 

 ber N is also zero. 



As, therefore, time proceeds indefinitely, the number of surnames 

 extinguished becomes a number of the same order of magnitude as the 

 total number at first starting in N, while the number of surnames 

 represented by one, two, three, etc., representatives is some infinitely 

 smaller but finite number. When the finite numbers are multiplied 

 by the corresponding number of representatives, sometimes infinite in 

 number, and the products added together, the sum will generally ex- 

 ceed the original number N. In point of fact, just as in the cases 

 calculated above to five generations, we had a continual, and indeed 



