out THEORY OF STATISTICS. 
and possesses all the properties of, a correlation-coefficient ; the 
name may, however, be applied to it, pending the proof. A 
correlation-coefficient with » secondary subscripts will be termed 
a correlation of order p. Evidently, in the case of a correlation- 
coefficient, the order in which both primary and secondary 
subscripts is written is indifferent, for the right-hand side of 
equation (2) is unaltered by writing 2 for 1 and 1 for 2. The 
correlations 7, 74, etc., may be regarded as of order zero, and 
spoken of as total, as distinct from partial, correlations. 
6. If the regressions Bus... m bugs...» 6b. be assigned the 
“best ” values, as determined by the method of least squares, the 
difference between the actual value of x; and the value assigned 
by the right-hand side of the regression-equation (1), that is, the 
error of estimate, will be denoted by x5; ,.. . .; ¢.c as a defini- 
tion we have 
L193...n=% 7 Dinse) np 7 bis2s...n%s adler Diniz loin)" . (3) 
where x, x, . . . . x, are assigned any one set of observed values. 
Such an error (or residual, as it is sometimes called) denoted by a 
symbol with p secondary suffixes, will be termed a deviation of the 
pth order. Finally, we will define a generalised standard-deviation 
O13 . . . . » DY the equation 
N.ol, Sl aE Ns ih ) , - . (4) 
N being, as usual, the number of observations. A standard- 
deviation denoted by a symbol with p secondary suffixes will be 
termed a standard-deviation of the pth order, the standard- 
deviations oy a, etc., being regarded as of order zero, the standard- 
deviations ay, 04; €tc., (cf. eqn. (d) of § 3) of the first order, and 
0 on. 
7. From the reasoning of § 3 it follows that the “least-square” 
values of the partial regressions b;,3 . . . . a etc, will be given by 
equations of the form 
Sy —bm....a Xt... . + bins... ns Ta) 
= (mr, EE SE ETE IT 
§ being very small. That is, neglecting the term in &? 
Sao(2, x bios ..n-Tgt a bins . . .. m1 Tn) =0, 
or, more briefly, in terms of the notation of equation (3), 
(xy. 2195. ...2n)=0. : (5) 
There are a large number of these equations, (n — 1) for determin- 
ing the coefficients by, . . .. a» etc., (n— 1) again for determining 
FON