486 Mr. G. U. Yule. On the Significance of Bravais* Formula: 



zero if both net regressions are zero; (2) it is a symmetrical func- 

 tion of the variables ; (3) it cannot be greater than unity ; for, 

 by (13), 



or adding r^V^ 3 to both sides, and transferring r 13 2 to the right-hand 

 side 



< (l-r 13 2 )(l-r 23 2 ). 



If any two coefficients, say r 12 r 13 , be supposed known, the inequality 

 we have used above will give us limits for the value of the third,. 

 Throwing it into the form 



. 

 we have r^ must lie between the limits 



\/na 2 r 13 2 -^ r 12 2 - r 13 2 + 1. 



The values of these limits for some special cases are collected in 

 the following table : 



Yalues of r 13 and r 13 . Limits of rs 



fia = **i3 = 

 r = ?*i3 = 1 

 r = + 1> r is = 

 r 12 = 0, r 13 = 



r = 0, r 13 = 

 r 12 = r 13 = r 



r 13 = r 13 = v/05= 0'707 

 r 12 = + v/0'5 r 12 .= 

 V 







+1 

 1 







1 and 2r z 1 

 2r z 1 and 1 

 and 1 



., 1 



One is rather prone to argue that if A be correlated with B, and B 

 with C, A will be correlated with C. Evidently this is not necessary. 

 A may be positively correlated with B, and B positively correlated 

 with C, but yet A may, in general, be negatively correlated with C. 

 Only, if the coefficients (AB) and (BC) are both numerically greater 

 thanO'707, can one even ascribe the correct sign to the (AC) corre- 

 lation. 



It is evident that one would, in general, expect to make a smaller 

 standard error in estimating x\ from the two associated variables # 2 

 and a' 3 , than in estimating it from one only, say a 2 . But it seems 

 desirable to prove v this specifically, and to investigate under what 

 conditions it will hold good. The necessary condition is 



ri2 2 + r 13 8 2r 12 r 23 r 13 2 



