Karl Pearson 
233 
Similarly we have by (iv) and (v) 
^sss's' ^sss'a" sss's" 
= A.. A,-,- - A ^,- A, ,A,-v- - A ^,» _ ( A,,A,y - A,, A, ,-)^ 
A A A2 
= ^ (A,, A,.,. A,.,. - A,, A^v - A,v AV. - A.^ A^ + 2A,v A,.,A,,) 
^2 ^sss ' 
which gives at once the result (vi). 
We shall now substitute values like (vi) and (vii) in our definition (viii) of 
V„'„" and so deduce (ix). 
We have 
Afi 
V - A2 . " 
( ^SSs's' ^s"s"ss' \ f Ass,"s" Aj's'ss" \ 
\+ A„,'vA,',v'J ' \+ A,,,v'A,'vvJ 
/ Asjj's' Aj"j 'ss' N / A^.','j"g" Aj's'ssN / Aj'j'j'-j" Ass,,j's" \ 
/ A,,,.v'A,vs«" \ ( A,.,vv'A,,,v' \ /A,.v'.sA,",'v.'\ 
V+ A,,,vA,.vvJ ' V+ A,vss"A,.v'«.'/ ' V - AV^'ss' / 
but the determinant on the right is the square of the determinant 
As's's"s" ' ^s's'ss' ; ~~ ^s's's" 
^s"s"ss' ? ^s"s"ss 3 As55's' 
; As's'^."jj , Ajj,.',j" , Ajss'j' i 
whence by taking the square root we have the result (ix). 
(3) Application to Multiple and Partial Correlation Coefficients. 
Equation (viii) may be put into the form 
1, 
^ss's" Ajj Aj'j' Aj'-j' 
A,,' 
1, 
A,'." 
A,,' ^ _A.v' 
\/a~"at,' 
A,,.' 
VA,,A,v'' Va.vA, 
A3 
1, 
(1 - R\.,_„) (1 - EV.i-„) (1 - ^V.i-„) - 
^' ss'Pl-n> ss'Pl-n 
ss'Pl-m 1; s's'Pl-n 
ss"Pl~ni s's"P\—nf 1 
(X). 
