Correlated Averages. 201 



But, since by convention the modulus of the variable is unity, 

 A = A ( = B = C) ; a result which continues to hold as x is 

 supposed to vanish. 



This reasoning is quite general ; and accordingly, re- 

 placing the symbols x x , x 2 , #3, we may extend to four and 

 higher numbers of variables the solution which has been given 

 above for the case of three variables. In the case of four 

 variables p 12 , /o 13 , &c. p 24 . . . being as before the coefficients of 

 correlation for each pair, the reciprocal of the discriminant 



A'= A, A Pl2 , Ap 1B , Ap u . 

 Afts, A, A P& , Ap 24 . 

 Ap ls , Ap 23 , A, Ap 34 . 



&PU, A P24, Ap 34 , A. 



We have merely to border with a new row and column the 

 determinant used for the case of three variables. But in 

 forming the first minors of the reciprocal in the case of four 

 (and similarly for any even number) it must be observed that, 

 according to the rule of signs *, the minor of the reciprocal 

 which is equated to A 2 q 12 (in general A n ~ 2 q 12 ) is not now 

 A 3 (/>23/>3 4 /9i4),/but — A 3 (p 2S p M p u ) ; or, as it might be more 

 elegant to write, — A* (p 23 p u p u ) . By the same rule, 



Ap i3 =+A 3 (/? 24 p 31 p 42 ), 



^Vl4=— & d (P21p32p4Z), 



Example. — To exhibit the correlation between four quan- 

 tities, of which the first, a\, is formed by taking at random the 

 sum often digits (say from a page of mathematical tables); 

 the second, a 2 , is formed by adding to the first x x another 

 random decade; .v 3 = ,v 2 + another random decade; # 4 =,2? 3 + 

 another random decade. 



The coefficients of correlation between the pairs, which have 

 usually to be ascertained by observation, are here deducible 



a priori. E. g. p 12 = a / . For, putting f x and f 2 for the 



actual deviations of the first and second quantities (the sum 

 of ten and the sum of twenty digits), we may regard f x and 

 (?2~ ?i) as fluctuating independently, according to a modulus 

 which is the same for both, say c ; being that which apper- 



* Cf. Salmon's 'Higher Algebra/ chap. 1, arts. 6 and 8. 



