234 Novel Properties of Partial and Multiple Correlation 
Turning to the expression (ix) we may write it 
V2 , = A3 A , , ..A „ A ' ' 
1, 
^s"s"ss' 
VA,v.^s^A^' 
— ^s's's"s 
I, 
Aooo'o" 
^sss's" 
1 
The determinant on the right can be expressed in terms of partial correlation 
coefficients of one lower order, i.e. it equals 
Further : 
Also : 
1, 
ss'Pl-(s")-n) 
ss"Pl-ls'}-n 
ss'Pl-(s") 
— n ) 
1, 
s's"Pl- (sl-M 
ss"Pl-(s') 
-« ) 
s's"Pl— (s)— n ) 
1 
A " = A*^ 
A,, 
A,,',' A^..,.- 
As's's"s" As"s' 
SS 
■ A 
■ A ■ A 
A,',' ■ A,", 
A,, 
= A« 
As, 
A,'s' A,",.- 
As's's"s" Aj-'s' 
ss 
Agss's' 
■ A 
■ A • A 
Ai;".9" Ag 
■ 
As's' 
= A« 
1 
(1 - R\,-„) (1 - (1 - ^V.i-„) 
1 
(1 — 72^8. ^_ (,•',_„) (1 — R^s'.l-is)-n) (1 — -RV.l-(s')-n) 
Thus : 
(1 ~ -^"8. i-(s")-n) (1 ~ -^V. l-(s)-n) (1 ~ -RV. l-(s')-n) 
= (1 ~ -^"s. l-(s')-n) (1 ~ R^s'.l-{s"l~n) (1 ~ -^V'. l-(s)-n) (^i)' 
This is a relation between sets of three multiple correlation coefficients of the 
(n — 2)th order. 
Clearly V^.v = A^ ^ 
(1 - R%,,_„) (1 - (1 - i?V.i-„) 
1 
(1 — l-(s')-«) (1 — R^s'.l-(s")—n) (1 — -SV.l-(s)-n) 
^ 1» ss'Pl-(s")-n> ss"Pl-(s')-n 
ss'Pl-(s")-« J 1, 5's"Pl-(s)-n 
ss"Pl-(s')-n> s's"Pl-(s)-n» 1 
Squaring (x) and combining with (xii) we find 
' ss'Pl—n: ss"Pl-n 
ss'Pl—nt 1? s's"Pl—n 
~ ss'Pl— «■ s's"Pl-n; 1 
.(xii). 
(1 - (1 - (1 - i2V.l-n 
^' ssPl-Li'l-m ss"Pl-(s'}-n 
s'Pl-(s")-n^ 1, s's"Pl-(s)-n 
(s') — s's"^'l— (si— r 
1 
(1 - (1 - J (1 - R\".l-is^-n) 
.(xiii). 
