Mathematical Contributions to the Theory of Evolution. 391 



female variability. This does not theoretically agree with Mr. Galton's 

 definition, for he reduces the female to the male standard by multi- 

 plying them by the sexual ratio, or the ratio of the male to the 

 female mean for the organ under consideration. In order, therefore, 

 that my factor of reduction should agree with Mr. Galton's, it is 

 needful that the ratio of the standard deviations should be equal to 

 the ratio of the means, or, in other words, that the coefficient of 

 variation should be the same for the two sexes. Now for the stature 

 of men and women, I find for 1000 cases of each sex the coefficients 

 4*07 and 4*03 respectively, or the coefficient of variation is sensibly 

 equal for both sexes.* Mr. Galton found from his anthropometric 

 laboratory returns for somewhat fewer numbers, and probably for a 

 lower social class, values of 3*75 and 3*79, again sensibly equal. | 

 Hence the mid-parent, whether defined in my manner or in Mr. 

 Galton's, would have a sensibly equal value in the case of stature, 

 which is the one Mr. Galton dealt with in his ' Natural Inheritance.' 

 The coefficient of variation is, however, not the same for both sexes 

 in the case of all organs,^ hence for the purpose of simplifying the 

 formulae, I am inclined to think my modification of Mr. Galton's 

 original definition "will prove of service. 



(4) I shall now proceed to determine by the law of ancestral 

 heredity the correlation between an individual and any sth parent 

 from a knowledge of the regression between the individual and his 

 mid-sth parent. 



By the principles of multiple-correlation if a' , %i, x 2 , . . . . x n be n + 1 

 organs, with standard deviations <r , er l9 <r 2 . . . . o», and correlations 

 r oi ? ?'o2, ? 03 •••• r i2 , r )3 . . . . r«_j, n , then the frequency surface is given by 



z — const X e 2B 



where 



R 



1, r i, r 02 , 



roi, 1, 



?*12, ^13, 



and H pq is the minor of the constituent of pth row and qth column. 



* See 'The Chances of Death,' vol. 1, " Variation in Man and Woman," p. 294. 



f Ibid., p. 311. Mr. G-alton's family record data gave 1 *032 and 1 "005 for the 

 ratio of the coefficient of variation of sons to daughters and of fathers to mothers 

 respectively. See 'Phil. Trans.,' A, vol. 187, p. 271. 



X Many cases are given in the paper on " Variation in Man and Woman," cited 

 above. 



