Bauhaus-Universität Weimar

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 curves 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 
3<3+ 4i4+5f5 
is greater than one. 
Whenever fx (x) can be represented by a binomial, as above sug¬ 
gested, we get the equation 
-”*-(ïî6T*{ }* 
whence it follows that as r increases indefinitely the value of rm0 ap¬ 
proaches indefinitely to the value y where 
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 tx + 212 + etc. 
+qtq> 1, and, therefore an indefinite increase of male population. 
The true interpretation is that each of the quantities, rmv rm2, 
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
        

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