Correlated Averages* 



195 



the coefficients of x^, x 1 x 2 , and x 2 2 , in the quadratic above 

 written considered as derived directly from the Galtonian 

 coefficient p 12 . Employing the quadratic to determine the 

 most probable Xi corresponding to any assigned x 2 {of. above, 

 p. 191) we have 



Pn- 



This may be written 



P2P3 — $l* 



„ _ Ql2 



where Q 12 and P x are certain minors of the determinant A, 

 which forms the discriminant of R ; 



A = pu qu, q u 



qn, P2, q 2 s 



^135 ^23j ^3 



If we carry the integration one step further, integrating 

 K^~ s with respect to x 2 between extreme limits, we shall find 

 for the probability of any particular x 1 the expression 



Le-^dxi. 



But by convention the modulus of the probability- curve under 

 which the values of x ± range is unity (above, p. 194). There- 

 fore A = P 2 . 



By parity of reasoning, 



A = P 3 = P,. 



Pl3 



Ql3 

 Pi 



Q, 





~ P ~~ P 

 r 2 r 3 



23 



Therefore 



Qi2 = Api2 ; Qi3 = A/? 13 ; Q 23 = A/) 23 . 



Thus we may write the reciprocal A' of the determinant A :— ■ 



A' = 



A, A/? 12 , 



Api3, Ap 2 3? 



But, by a well-known theorem, each first minor of A'=A 

 multiplied by the corresponding constituent of A. Thus 



Ms 



A?23 



A 



(A 2 -Ay) = & Pl ; (AV 23 Prs - a%) =Ay u . 

 2 



