Mathematical Contributions to the Theory of Evolution. 405 



" I doubt whether the correlation coefficients for collateral hered- 

 ity — at any rate in the middle classes — can be greater than 0*5." 



Strangely enough Mr. Galton's law of ancestral heredity which I 

 then rejected, while accepting a part of what I now consider his 

 erroneous theory of regression, gives just the link between linear 

 -and collateral heredity which I was then seeking ! 



Let Xi and x 2 he the deviation of two brothers forming part of the 

 array having 



k = 7/3' \ + 7/3% + 7/3% + . . . . 



for its mean, x x and x 2 being measured from the general population 

 mean. Let x x and x 2 differ from the mean of the array by x' and x" , 

 then 



Xl x 2 = (k + x')(k + x"). 



Now let ns sum x x x 2 for all pairs of brothers in a given array, then 

 since x' and x" correspond to any pair of chance deviations within 

 the array S («'»") = 0, S(x) = S(V') = 0. Hence for the array 

 S(a?iaj 2 ) = S(& 2 ) = 2nk 2 * where n is the number of pairs of brothers 

 in the population with the series of mid-parents hi, k 2 , k 3 , &c. Now 

 write 



lc Q = 7/3^ ki -h 7 /3 2 ~ & 2 + 7/3 3 



2*i 2< 2 



* The appearance of the 2 here requires notice. Let there be v brothers to the 

 array of any single series of mid-parents; then if s be the standard-deviation of 

 the array, the distribution of brothers corresponding to a given k 



= * e - mrn 



V2-irs 



Hence the frequency of a brother between x' and x' + ox' occurring with a 

 brother between x" and x" + Sx" 



= e clx> e-W^ clx". 



If we take as limits x' and x", both = + oo to — oo , we shall clearly take each 

 brother twice over with each other brother. Hence — 



= £ f + °° [ +C ° ^ (*o + O(*o + x")e-^<^'W 1 dx' dx" = iv%*. 



J _00 J -00 



Kow allow one pair of brothers to each system of mid-parents, and 



S(x 1 x. 2 )=27c Q 2 



for one mid-parental system, or if there be n such mid-parental systems, 



S(x l x 2 )=2n7c 2 . 



Actually the same mid-parental system may be repeated many times, only in this 

 •case the possible correlation of fertility with the character under discussion must 

 be guarded against. 



