L. ISSERLIS 
411 
We may conclude then that in general the linearitij of any three of tlie six 
regression lilies involves that of the remaining three. 
(iv) If the regression surface of z on ?/ reduces to a plane, the regression 
curves of w on y and 1/ on x reduce to straight lines. 
We have as in § 7 
"^^d + a^ + ^ + c^- (58). 
But 4 = V- ± \^.^ - VA - l| , 
■■■ I' = ± " v^S^ ^ih ' " 
+ terms of higher order. 
Now the regression of z on y for a constant x is linear. Therefore the 
coefficient of y"^ is zero. To first order terms we may put /3i = 0 and /S., = 3, 
and thus 
Similarly cr,, i 6 ~ = 0, 
But c vanishes when = .ri/-Rz'' hy (52). 
Hence if x,,llz= xnT^i', 
it follows that x^n = itVx = '^'xu- 
We thus see that if the three generalised correlation ratios xiiHz^ zxH,i 
are equal to xyRz, yzRx, zxRy respectively, the six correlation ratios ^Vin nVx, 
xVz, zVx, zVn' ;iVz reduce to the corresponding correlation coefficients r,^, r,,, 
and that the "linearity" of the three regression surfaces involves the linearity of 
the six regression lines. 
