L. ISSERLIS 393 
Subtract (11) from (7) after replacing ) 
'zx and ')'zy by the appropriate sums, and 
we obtain 
= ss 
nxy {Z - Z^yf - S ^(Z-Z){X- X) 73 j - *S' ^l:c,,z ~ {z - z) {ij ~ y) 
n^y {z - Zocyf - n^y — (ixy " z) {x - 7, - J -z){ij- y) y,' 
y 
Using (8) this can be written 
IxyJIi - xy^i\ N<Ji = S8 [n^y (Zxy - Z) (Zxy - Z)] 
= SS {7lxy {Zxy - + SS [Uxy {Z - Z) (Zxy - Z )} . . . ( 1 8). 
But SS{nxy{Z-z){zxy-Z}] 
{ I x — x ,y — y \ f ~ - x — x ,y — y W 
= Na^ (ys<Tzr,x - 73 - jsysr^yCTz + y^o-zVy, - jsys'i^yO-z - 73'- o-^) 
= No-z" {73 (r^x - 73 - Ji'^'xy) + Js i^'yz - y-i - Ti^'xii)] (14*)- 
Using the values of 73 and 7,' given by (9) and (10) we see immediately that 
'Tzx - 73 - 73'?"^,./ = 1'yz - 73' - 73 1'xy = 0. 
Hence (18) becomes 
(xyHz' - x,A') iVcr/ = SS [Uxy (Zxy - Zf] (15). 
This is the generalisation of (4). We deduce that xyH^ = xyRz if and only if the 
regression is strictly linear, that otherwise xy^z^ > xyRz' and by (6) that xyHz^< 1. 
I 3. These properties and definitions can be extended to the case of m 
variables x-^, x^, ... Xm. We now use ^..n^i foi" the mean of x^ when x.,, x^, ... x^ 
are given and denote by S a summation extending to the variables x^, x^, ... x,n. 
l..,Vl 
If we define the correlation ratio of by the equation 
i\^cri2(l -2,3 mHi-)= S (X,-.^^ mXif lh.,M (16). 
We can deduce in the same way as in § 2, the relation 
Na,',^, ,„H,'= S {(x,-,^, ,,,x,yv, (17). 
2.,.m 
In order to generalise (15) we recall that the "best fitting" linear function of 
the variables x^, x^, ... Xm to the mean 2,3. ..m^i is 
