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APPENDIX F.   245

and the constant term inf5 (x) or 5m,, is therefore

6.8183 +'08306 + •10635 + •07804 + •06489 + •05443 + •01437 + •01692 + •01144 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 f1, f2 up to f5 when reduced to decimals, are in this case •33333, •48148, •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 respectively.

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 tl + 2t2, and this quantity is in the present case equal to one.

If axes Ox and Oy be drawn, and equal distances along Ox represent 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 expressionsfl (x) does not follow some assigned law. The calculation 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 t5+tlx+ etc.+t,x, are proportional to the terms of the expanded binomial