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244   NATURAL INHERITANCE.

number with one representative ,._1m1.N, the number with two

'1m2. N and so on, it follows, from what we last stated, that the number of surnames with s representatives in the rth generation must be the coefficient of x' in the expression

I r-Imo+r-1m1T+,._1m2T2+ etc. +r_1m4,-1T9,-1 I N

If, therefore, the coefficient of N in this expression be denoted by

fr (x) it follows that *_19n1, ,`1m2 and so on, are the coefficients of x,

x2 and so on, in the expression f,-, (x).

If, therefore, a series of functions be found such that

fl(x)=to + tux + etc. + tQx4 and f, (x) =f,_1 (to+tlx etc. + tw)

then the proportional number of groups of surnames with 8 representatives in the rth generation will be the coefficient of x' in fr (x) and the actual number of such surnames will be found by multiplying this coefficient by N. The number of surnames unrepresented or become extinct in the rth generation will be found by multiplying the term independent of x in f, (x) by the number N.

The determination, therefore, of the rapidity of extinction of surnames, when the statistical data, to, t1, etc., are given, is reduced to the mechanical, but generally laborious process of successive substitution of to + tux + 12x2 + etc., for x in successively determined values of fr (x), and no further progress can be made with the problem until these statistical data are fixed ; the following illustrations of the application of our formula are, however, not without interest.

(1) The very simplest case by which the formula can be illustrated

is when q = 2 and to, tl, t2 are each equal to 1.

l

Here f1 (x)=1+3+x 2 f.2(x)_ 3 1+1 (l+x+x2)+ I l+x+x2)21

and so on.

Making the successive substitutions, we obtain

1 13 5x 6x3 2x x .t9(x)=319+9+9 +9+9}

f3(x)_1249+265x+343x3+166x8+109x4+34x° +16x8 + 4x7 + x8

2187 2187 2187 2187 2187 2187 2187 2187 2187

f,(x) = •63183 + '08306x+'10635x2+ '07804x8+'06489x4+ -05443x5+ '01437X6 +'01692x7+ •01144x8+'00367x0+'00104x1°+'00015x11+'00005x12

+'00001x13 + '00000x14 +'00000x15+ -000OOx'5