400 
On the Partial Correlation Ratio 
. 5^ is a quadratic expression in y if we remember that h being of order 
\/ y.Hj" - y^Rx^ is of the same order as \/ yVz^ - r^yz*- 
But the relation ^.{xVii' -'>''xy) -'^ = ^ is satisfied when the regression of y on 
X is parabolic f. 
Here (^2 = iS„ - /3i - 1, e = e -r,,,\/'^„ e = (/^.y|. 
Hence qx^y - r^y = V - /3i - 1 (^S). 
Similarly (/.,„2 - r^y V^/ = \/,,%^-r%, ^//3,'-A'- l (^6). 
The use of these approximations will save the direct calculation of qx^y and 
(fxyi provided we can determine the signs to be attached to xiiy—n'^xy and 
'^^ yVx — t^'^xir This is often easily done by inspection of the regression curve whose 
equation is 
ay V,±V^,-/3,-lK'^ ""^V, 
We can approximate to qx-^y^ as follows ||: 
qx'y"' — 
where Yx is the value given by the regression straight line and o-^y,. x 71^ the 
second moment of the array of y's for a given x, about the point Yx. 
But S{nxaHYxjY} ^ ^ ^nxXh\,ay^)l<Jx^ 
Jya-x-a-y- 
~ ^2^'''xy 
Thus qx-.y. = + /32» 
Ox^y- 
xy> 
Sitnilarly q^.y = '%^Jf ^ + AV^.,, 
or, so far without approximation 
'X "y 
2NaJa„^ +i(/y2 + /:^2)r.,. 
* It is noteworthy that the hypothesis that regression of z on x, y although of 2nd degree is such 
that regression of z on x for a constant y is linear leads to the result that the total regression of « on a; 
is parabolic. 
t Pearson, I.e. p. 28, Eqn. (Ixiii). 
+ Pearson, I.e. Eqns. (li), (xlv) and (Ixiii). 
§ Pearson, I.e. Eqn. (Ixv). 
II It can be found fairly directly by tabling to the squares of the variates, when we need a simple 
product moment. In a later part of this paper some comparisons of actual and approximate values for 
numerical cases will be found. 
