Natural inheritance / by Francis Galton.

  • Galton, Francis, 1822-1911.
Date:
1889
Catalogue details

Licence: Attribution-NonCommercial 4.0 International (CC BY-NC 4.0)

Credit: Natural inheritance / by Francis Galton. Source: Wellcome Collection.

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    244 NATURAL INHERITANCE. number with one representative the number with two 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 ic* in the expression I + + etc. + If, therefore, the coeffi,cient of N in this expression be denoted by fr{ x) it follows that r-i^) r-1^2 coefficients of x, and so on, in the expression f^._^ (ж). If, therefore, a series of functions be found such that yj(£c)r=¿Q ++ etc. + t^x^ and {x) =/,._j {t^-k-t-^x etc. + t^x'^) then the proportional number of groups of surnames with s represen¬ tatives in the rth generation will be the coefficient of x* in {x) and the actual number of such surnames will be found by multiplying this coefficient by N. The number of surnames unrepresented or be¬ come extinct in the rth generation will be found by multiplying the term independent of x in (x) by the number N. The determination, therefore, of the rapidity of extinction of sur¬ names, when the statistical data, Íq, etc., are given, is reduced to the mechanical, but generally laborious process of successive substi¬ tution of ¿0 + t-^^x + t.^x^ + etc., for X in successively determined values of/. ( x), and no further progress can be made with the problem until these statistical data are fixed ; the following illustrations of the ap¬ plication of our formula are, however, not without interest. (1) The very simplest case by which the formula can be illustrated is when 5'= 2 and ¿i? ¿2 each equal to -J-. Here /1 {x) = ^ = ^ l+x + and so on. Making the successive substitutions, we obtain ^ / Ч 1 ( 13 5ж 6,2'^ 2x X л h+^ + 9 + 9 + 9} ,, , 124Э^265аз ^343ж^ , 166»^^ , , 34^5 16х^ , ix^ J ( I — I Г Ч~ - + 4- и - + -U ^ 2187 2187 2187 2187 2187 2187 2187 2187 2187 /4(л;)= -63183+ -08306^0+ *10635^2+ •07804^3+ -06489.^4+ •05443аз-'5+ -01437^« + •01692:^'' + *01144л;8+ •003б7а;9 + •00104ж^® + -00015^ + •00005ж''2 + •00001ж+ -OOOOOojiH •00000д;15+ 'OOOOOJC^^