Karl Pearson 
235 
The denominator on the right-hand side of (xiii) can be replaced by the left-hand side 
of (xi), or by the square root of the product of both sides of (xi). This is a relation 
between partial correlation coefficients of the (n — 2)th order with multiple coeffi- 
cients of the (n— l)th order and partial correlation coefficients of the (^^— 3)th 
order with multiple coefficients of the {n — 2)th order. In the particular case of 
three variates 1, 2, 3 with total correlations fj2, u-i, r..^ (xiii) reduces to 
1, 
31^2' 
■ 12^3' I3P2 
1> - 23P1 
■32P1 , 1 
{1-R\,,,){1-R% ,,,) (l-jg^3.12 ) 
(I - T%,) (1 - r\,) (1 - r\,) 
I, 
^2-1 
r.: 
31 > 
1, 
''32 ) 
^23 
1 
(xiv), 
a result which might possibly be of practical value in testing the accuracy of 
the determination of the first order partial and multiple correlation coefficients. 
I now return to equations (vi) and (vii), and I divide (vii) by the square root 
of the product of A^v and Aj"," obtained by cyclical interchange from (vi). 
We find 
A..." 
^s's's"s" ^sss's" 
."A, 
V A,.,. A,"," V A,-,.,..," A,',',, - A2,.,,,.., V A,",;.,, " 
Av. 
^s"$"ss' 
'^^.s's'ys ^s"s"ss ^s's'ss ^s's's's" ^ ^s"s"ss ^s'V's'.s' 
1 - 
A^, 
^s''s"ss ^s"s"s's' 
or, changing sign throughout, 
i'.s"Pl- 
s's"Pl-(s)-n ss"Pl-(s'}-n ^ ss'Pl-(s")-n 
VT 
(XV). 
s'P l-(s")-n 
(xv) is the familiar result for obtaining a partial correlation coefficient of the 
(w - 2)th order from those of the {n - 3)th order, but the proofs usually given of 
it seem to be based on some appeal to general analogy rather than to the definite 
algebraical form of the coefficients concerned. It was, indeed, a lecture proof of 
the relation (xv) from the determinantal forms of the coefficients which led me to 
the results (vi) and (vii), as apparently novel determinantal relations. 
We can next consider results (vi) and (vii) individually. From (vi) we find 
A V,,v A A A,v A,"... V A;,,-,.A,,,-v7 ' 
or, converting into partials and multiples, and writing 
