for Regression, #<?., in the case of Skew Correlation. 481 



square, r is Bravais' value of the coefficient of correlation. Be- 

 writing 6i in terms of these symbols, we have 



Similarly, 03 = 0, 



But the expressions on the right of (3) and (4) are the values 

 obtained by Bravais on the assumption of normal correlation for the 

 regression of x on y, and the regression of y on x. That is to say, 

 the Bravais values for the regressions are simply those values of b\ 

 and b z , which make 



S 6 2 and S- 



respectively minima, whatever be the form of the correlation between the 

 two variables. Again^ whatever the form of the correlation, if the 

 regression be really linear, the equations to the lines of regression are 

 those given above (as we pointed out in the introduction). This 

 theorem admits of a very simple and direct geometrical proof. 



Let n be the number of correlated pairs in any one array taken 

 parallel to the axis of x, and let be the angle that the line of 

 regression makes with the axis of y. Then, for a single array, 



or extending the significance of S to summation over the whole 

 surface, 



S(xy) = tf tan 0er 2 2 , 

 that is, 



tan 9 = r *- . 



ffZ 



In any case, then, where the regression appears to be linear, Bravais* 

 formulce may be used at once without troubling to investigate the 

 normality of the distribution. The exponential character of the surface 

 appears to have nothing whatever to do with the result. 



To return, again, to the most general case, we see that both 

 coefficients of regression must have the same sign, namely, the sign 

 of r. Hence, either regression will serve to indicate whether there is 

 correlation or no, for there is no reason, a priori, why the values of 

 61 and bz, as determined above, should be positive rather than 

 negative. But, nevertheless, the regressions are not convenient 

 measures of correlation, for, on comparing two similar cases, we may 

 find, say, 



bi > DU 62 < &' 



VOL. LX. 2 o 



