L. ISSERLIS 
399 
The regression surface of z on x, y is, with the values we have now obtained 
for the constants 
— — I ^ r 
1 - r '^ 
xu 
1 - r\,, 
^(^^5)(?-S)_,_, (53), 
The terms in the first line give the ordinary regression plane. In most cases 
the regression does not differ widely from linearity so that — is small. 
§ 6. We must now get some idea of the relative magnitudes of q_ci,-2, qx-i,,, qx-<r- 
and moments of lower order. 
First with regard to q^i,! whicii is equal to , 
_ S jn^iiX^-ii) _ S (»^/^.^^•-) 
If the regression of y on be linear 
qxh, = S ^ ^\ = r,,y \'/Si, 
so that q:^2,i is zero if the regression is linear and the frequency of x symmetrical. 
In fact q^iy — r^y V/3i = 0 is the same as Pearson's criterion for linear regression given 
by e = 0 {Skew Correlation and Non-linear Regression, p. 30, Eqn. (Ixix)). 
We may obtain a good approximation for qx-i,, by considering the regression of 
?/ on a; to be parabolic. This is a natural assumption to make if the regression 
surfaces of x on s, y and y on z,x be also of the hyperboloid type we are discussing 
for the regression of z on x, y. 
For with origin at mean we may write 
O'x <^il (^ii'^z 
Hence keeping y constant and summing for the ^■'s 
— = e i 1 1 . 
0"x O".-/ <^z C/yCz 
But ii^t:i^U)i._^ji^R^-A*. 
* See Pearson: I.e. Eqn. (Ixv) (wliere Y,,-^ is a misprint for A',,), and /3./', /3i" refer to the distribution 
