K. Pkakson 223 



Asfaras tlie ava'duhle dnta <it present (ju inhevitance coeßcients for ascending 

 ancestry are within the liiuits of observatioH<d crnir represented bij a geoinetncal 

 series and by the saiiie series. 



Froiii this it fi)Ilt)\vs that * : The contribntioiis of the ancestry also follow a 

 geometrical series, althovglt not that originaUy projiosed by Mr Galton. 



{(i) Mr Galton has assuineil, that ii' l.lie relatives include all tlu^ ancestry, and 

 if all these ancestry had the sanie doviation /;, the offspring will have a probable 

 deviation of //. This is really the introdiiction of a biologieal hypothesis, the 

 truth of which can only be tested by Observation. Mr Galton deduces this result 

 in the following manner: he supposes a stable population, i.e. one in which the 

 niean and variability of each generation remain the .same, and the parents in each 

 generation are the whole or at any rate a random sample of that generation ; 

 there must also be no reproductive selection, or fertility niust not be correlated 

 with the charactcr of which the inheritance is under consideration. Further 

 there must be no a.ssortative mating. Under these circunistauces we have the 

 following form of (i) p. 217: 



p,j={J, +J., + J,+ ...)h, 

 and since Mr Galton holds that p,i will then equal h, we have 



J, + -L + Js + ... = l (2). 



Any geometrical .series e{l + p + p- + ...) for the J's will satisfy this conditiou 

 if e = 1 — p ; the series i + i + ^ + t'it + • • • '^ not the only one satisfying (2)f . 



But it is doubtfulj how far this conclusion is justifiable. Statistically it is of 

 cour.se unnecessary. (i) is a relation between the probable deviation of Q from 

 its mean and the actual deviations of each P from their individual mean.s. 

 There is no reason why the means of Q and of all the P's should be the same. 

 They may be different owing to enviroiiment or to selection. Further there is 

 no reason why the variabilities should be the same ; parents may be a selection 

 out of the geiieral Community in each generation. As soon as we realise that the 

 /t's are deviations from the generation means, and these are not all the same, and 

 that the variability in each generation differs, the need for the relation 



J, + J, + J,+ ... = 1 

 ceases to be apparent. 



In data like eye-colour in man and coat-colour in thoroughbred horses there 

 has been undoubtedly a secular change going on; the proportion of blue eyes in 

 the latest offspriug is considerably less than in the great-great-grandparents, 

 while the early grey horses have largely disappeared from the stud-book. 

 Further we find each ancestral generation is roughly spcaking less variable than 



* R. S. Proc. Vol. G2, p. 394. 

 + R. S. Proc. Vol. U2, p. 402, and Vol. fiü, p. 147. 



t It is hardly cousoiiant, for example, with what we know of eye-colour, where there is an assorta- 

 tive mating coefficieut of 1002: see Pliil. Tiniis. A, Vol. 19-5, p. 113. 



