OCR Rendition - approximate

WATSON and GALTON. Extinction of Families. 143 Should that be the case, we have fl(x)=(340 5 5 and generally rmo = { 3 +,._,m. } 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 1 + x + x2 3 viz., that while 237 names out of a thousand disappear in the first step, and an additional 109 names in the second step, there are only 27 disappearances in the fifth step, and only 6 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 t, + 2t$ + 3t3 + 4t, + 5t, is greater than one. Whenever f, (x) can be represented by a binomial, as above sug gested, we get the equation `m° Ca +b)q{ a+brim Iq whence it follows that as r increases indefinitely the value of ma approaches indefinitely to the value y where _ 1 y (a+b) a+by 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 surname 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, + 2t, + &c. + qtq > 1, and, therefore an indefinite increase of male population. The true interpretation is that each of the quantities, rm,, ,m,, &c., tends to become zero, as r is indefinitely increased, but that it does not follow that the product of each by the infinitely large number 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 35