246 NATURAL INHERITANCE. 



(a-\-bx) q i.e. suppose that t. _„ t,=q and so on. 



v ' lif °={a + b) ql ^{a + bf 



XT f i \ (a + bx) q , a 9 



Here/, («)^^- and ^ = — - 



a + S^q 



l« 



2 " (a + b)* t ' " ) 



Generally r m a = — { a + b,.,m n > — 



J ° (a + b)*\ ] ° J 





(a + 6)* 1 '" J u J (» + 6) g U 



If, therefore, we wish to find the number of extinctions in any 

 generation, we have only to take the number in the preceding 



generation, add it to the constant fraction _ , raise the sum to the 



b 



b q 

 power of q, and multiply by 



(a + by 



With the aid of a table of logarithms, all this may be effected for 

 a great number of generations in a very few minutes. It is by no 

 means unlikely that when the true statistical data t , t v etc., t q are 

 ascertained, values of a, b, and q may be found, which shall render 

 the terms of the expansion (a + bx)' 1 approximately proportionate to 

 the terms of/ (as). If this can be done, we may app-oximate to 

 the determination of the rapidity of extinction with very great ease, 

 for any number of generations, however great. 



For example, it does not seem very unlikely that the value of q 

 might be 5, while t , t v ..t g might be -237, -396, 264, -088, -014, -001, 

 or nearly, 1 I, / T , ^ _i_ and _^. 



(3 + asV 5 3 5 



Should that be the case, we have, / (x) = j '- ^i^ = _ 



4 5 



If ) 5 



and generally r m = j 5 < 3 + ,.. l 7n [• 



Thus we easily get for the number of extinctions in the first ten 

 generations respectively. 



•237, -346, -410, -450, -477, -496, -510, -520, -527, -533. 



"We observe the same law noticed above in the case of 



3 



viz., that while 237 names out of a thousand disappear in the first 



