1886.] 



Family Likeness in Stature. 



57 



intersection of the lines that separated the vertical columns with 

 those that separated the horizontal lines, the sums of the four adja- 

 cent entries. Then I drew lines with a free hand through all entries, 

 or interpolations between entries, that were of the same value. These 

 lines formed a concentric series of elliptical figures, passing through 

 values of z that diminished, going outwards. Their common centre 

 at which z was the greatest, and which therefore was the portion of 

 maximum frequency, lay at the point where both x and y were of the 

 same value of 68^ inches, that is, of the value of the mean stature of 

 the population. The line in which the major axes of the ellipses lay 

 was inclined nearer to the axis of x than that of y. It was evident 

 from the construction that the median value of the entries, whether 

 in each line or in each column of the table, must lie at the point 

 where that line or column was touched by the projection of one of these 

 ellipses. It was easy also to believe that the equation to the surface 

 of frequency and the lines of loci of the above-mentioned points of 

 contact, admitted of mathematical expression. Y Also that the problem 

 to be solved might be expressed in a form that had no reference to 

 heredity. In such a form I submitted it to Mr. J. Hamilton Dickson, 

 who very kindly undertook its solution, which appears as an Appendix 

 to this paper, and which helps in various ways to test and confirm 

 the approximate and uncertain conclusious suggested by the statis- 

 tical treatment of the observations themselves. I shall make frequent 

 use of his mathematical results, both in respect to this problem and 

 to another one (also given in the Appendix), in the course of my 

 further remarks. 



As regards the present subject of the connexion between the regres- 

 sion in direct and in converse kinships, it appears that it wholly 

 depends on the relation between the quartiles of the two series of 

 "arguments," and is expressed by the formula chv=p~w'. In this 

 case c s =(l*21) 2 = l'46, and p 2 =2"89 ; also w=§; therefore w'=% 

 nearly. 



It will be observed that in all cases of converse kinship, from man 

 to man — as from man to brother, and conversely ; from man to nephew, 

 and conversely ; from father to son, and conversely ; c=p, therefore 

 in these the ratio of regression is the same in the converse as in the 

 direct kinship. 



Brotherly Variability. — The size of human families is much too 

 small to admit of the quartile of brotherly variability being deter- 

 mined in the same way as that of the population, namely, by finding 

 the quartiles in single families, but there are four indirect ways of 

 finding its value, which I will call b. 



(1.) A collection of differences (see Table VIII) between the statures 

 of individual brothers, in families of n brothers, and the mean of all 

 the n statures in the same family, gives a quartile value, which I will 



