392 



Prof. Karl Pearson. 



Putting as 2 , . • . . a*« constants, say & l5 7c 2 , .... we have for the 

 mean value x of the corresponding array of #'s, 



R 



7<a + 



Ro2°0 7 , Ro3°"0 



&3 + «... + 



^RoO^l RoO°2 " Rood's 



The standard deviation of x Q for this array is 



2 = ffo^R/Koo ( vii )« 



These results (vi) and (vii) are the regression formulae.* 



Now let oc h % 2 , . , . x n he the mid-parental values of the 1st, 2nd, 



3rd, .... nth order, and x = A* the mean value of the organ in the 



offspring. 



Then the value of R is given by 



R == I 1, pn Pn P2> Pi*"- Pn 

 pi, ; 1, pi, pz, pz • • • • pn-i 



P2, Pi, 1 S Pit P2 • • • • Pn-1 



(viii). 



Pn, pn-i 1 



and the regression formula is : 



/Bin 



U= - — — — & 2 + .... +_._&„), 



X-D-'OO -*1 J^OO ^2 ■"'00 ^-a / 



if we stop at the nth mid-parent. 



Comparing this result with the analytical statement of Mr. Galton's 

 law of ancestral heredity given on p. 388, we see that we must have 

 from (v) : 



Roi/Roo = 



_1 Si _ 

 2 a x 



1 



~~2~72~ 



R02/R00 = 



__1 2* _ 



4 <7 2 



-(— v 



Ro^/Roo = 



_ 



2!<r q ~ 



/ 1 y 



(ix). 



There will be n such equations, if we go to the mid-nth parent, 

 and there are n quantities p u p% . . . . p n to find. Thus Mr. Galton's 

 statement that the 'partial regression coefficients are J, 5, 



* See ' Phil. Trans.,' A, vol. 187, p. 302. 



