for Regression, fyc., in the case of Skew Correlation. 487 



that is, 



2 2r 12 r 13 r 23 > r 12 a -r 12 2 r 13 2 , 

 > 0. ; 



or 



But (r 13 ri 2 r 23 ) is the numerator of / H , the net coefficient of corre- 

 lation between x\ and # 3 . Hence the standard error in the second 

 case will be always less than in the first, so long as p 13 is not zero. 

 The condition is somewhat interesting. 



To take an arithmetical example, suppose one had in some actual 

 case > r<*> 'Jo $ 



r 12 = +0-8 >i t - 

 r 23 = + 0-5 r 13 = +0-4. 



One might very naturally imagine that the introduction of the third 

 variable- with a fairly high correlation coefficient (0*4) would con- 

 siderably lessen the standard deviation of the x^- array ; but this is 

 not so, for 



0-4 (0-5X0-8) 



/>13 ~ -/0-75XO-86" : ' 

 sb the third variable would be of no assistance. 



III. Case of Four Variables. 



This case is, perhaps, of sufficient practical importance to warrant 

 our developing the results at length as in the last. 



If a?i, %2, it's, a' 4 , be the associated deviations of the four variables 

 from their respective means, the characteristic equation will be of the 

 form 



(14). 



The normal equations for the fr's are, in our previous notation, 



Hence 



r 12 r, 3 r 24 

 r 13 1 r si 



r 24 



r 23 1 



(15), 



and so on for the others, b^ 6 13 , &c., we may call the net regressions 

 of xi on ajz, ajj on a? 3 , &c., as before. By parity of notation^we have 



