ON SOME NOVEL PROPERTIES OF PARTIAL AND 
MULTIPLE CORRELATION COEFFICIENTS IN A 
UNIVERSE OF MANIFOLD CHARACTERISTICS. 
By KARL PEARSON, F.R.S. 
(1) Let the universe consist of N individuals each the bearer of n characteristics 
symbolised by the numbers 1, 2, 3, ... s, s', s" , ... n respectively. Let r^s' be the 
correlation coefficient of the s and s' characteristics, and let the whole system of 
total correlations be provided by the determinant A, where 
1, 
•(i) 
1, 
1 
This determinant being symmetrical because rss' = ?'s's- 
We shall use A^^' for the first minor corresponding to the constituent in the 
sth row and s'th column, and A^^.'s"^" for the second minor corresponding to the 
first minor of A^^' which is associated with the constituent in the s"th row and 
s"'th column. 
Then if 
i2;3 -. (s) ... n denote the multiple correlation coefficient of the sth 
characteristic on the other n — 1 characteristics, i.e. the n without s, and 
ss'Pm... (s)... (s') ... n denote the partial correlation coefficient of the sth and s'th 
characteristics for the remaining n — 2 characteristics constant, the following 
results are fundamental and well-known : 
A 
I. 123... (s) . 
= 1 - 
.f.9'Pl23... (.<) ... (.s'l . 
A,, 
A., 
,.(ii), 
.(iii). 
Va,„a,v 
To abbreviate the subscripts we shall write these 
R,.i-n and ,vPi-n, 
but where others of the variates are to be left out of account we shall be obliged 
to introduce the bracket system to mark partial or multiple correlation coefficients 
