\LIV| lntr<xln<-ti<>i< ccix 



TAUI.K XL1V. 



Tii hlc of fttitoftOM to tent Geometrical I 1 <; in tJie caw of tlte Variate 



Ihli't'rence Method. (I'earson, Klderton and H.-nderson, lliontetrika, Vol. XIV. 

 pp. -294297, 310.) 



We suppose X v and Y v to !>< tin- deviations of two variatcH, each from tln-ir 

 own secular trends, arid we; drsiiv to find r XY , the correlation of A' with Y. The 

 subscript p does not denote that A' and Y an- taki-n at the same time, actually 

 there might be a lag if p denotes time ; it marks that X and Fare the corresponding 

 values we wish to discuss. 



There are three kinds of correlations which may arise among these fluctuations 

 from the secular trends : 



(a) X p and X P T may be correlated, p' T . 



(b) Y p and Y p r may be correlated, p" t - 



(c) X p and Y pT may be correlated, p T . 



Now let A n Jf p and A n F p denote the nth differences of X v and F p respectively, 

 and let. curled brackets { } denote that the mean values of the whole series of 

 observed X p> Y y 's have been taken. Then the following results may be deduced 

 from the finite difference values of A n AT p , A n 7 P * : 



nln! n (n - 1) n, 



iyt 



} 



n (n 1) 



, n I ti ! n(n- 1) .. 



(2;t)!^ ~" (w + l)(" ' ox ^ ~* " ' 1 



n n(n 1) 



T Pi +7" 7~KT P2 ... 



!L ln! n(n-l) n 



r 2n! P ^w+ln + 2 P n+l p 



..... 



- 



Now it is clear that if all the correlations of X p and X pr , and of Y p and Y PT , 

 were zero, and Z p correlated only with Y p and not with Y pr> then we should have 



and thus we should obtain po = ^jcr> tne correlation of corresponding fluctuations 

 from the secular trends of X p and F p . This was "Student's" original version of 

 the Variate Difference Method. It assumed that a fluctuation had no association 

 with adjacent fluctuations, but solely with the corresponding fluctuation of the 



* E. S. Pearson, Biometrika, Vol. xrv. pp. 8739. 



** 



