Miscellanea 181 
V. On certain Errors with regard to Multiple Correlation occasionally 
made by those who have not adequately studied this Subject. 
By KARL PEARSON, F.R.S. 
(1) It is well-known* that if we endeavour to predict the value (jf a variate from n 
correlated variates Ai, x-^,, ... .(,■„, by determining a linear function of .fj, .to, ... which has 
the maximum correlation with .r,,, then the value of is given by 
where A is the determinant 
/4- = l-A/Aoo. 
A= 1 , 
'"20) 
'■(11, >M: 
r-n, 1 : 
'■(Id) ''/ii) I'liii ••• 1 
and Apg is the minor corresponding to the constituent of the ^th column and ^tli row. 
The system I propose to consider is that in wliich all correlations like ^q,, are equal, whatever 
p be, to a constant p, and all correlations where p and q may take any values from 1 to n, 
are the same and equal to t. We now have for the value of A the expression 
1, p, p, ... /) 
/}, 1, f, ... 6 
p, f, 1, ... e 
p, f, e, ... 1 
To evaluate this determinant add all the rows but the first together, giving 
np, \+{n-\)e, l+(/;-l)e, ... l+(?l-l)e, 
multiply the result by p/(l + - 1) e) and subtract from tlie first row. We have 
1 - 
np'- 
<->, 0, 
0 
Hence 
l+(n-l)6 
p, 1, e, ... e 
p, f, 1, ... ( 
p, e, f, ... 1 
np'' 
]+(n-V 
X A,,,,. 
np^ 
l+(:n-l)fj l+{n-l)e 
■(i), 
(ii). 
Hence proceeding to the limit we have 
(n-l) 
li^^plJ^ (iii) 
* Biometrika, Vol. viii. p. 439. 
t The sign of iJ„ must be determined from other considerations. 
