15^ 



^Vjjnkk 



This correlation-coefficient can therefore also be written in the 

 following form 



2 an aki 



or 



Tik = 



r.ik 



)/:Saji\ 2 ajct' 





VBjj.Buk 

 Introducing the coefficients Uji, we find 



^ f /' Ctjl ftJcl 



''jk = 



We now will imagine the variable m to be connected with some 

 cause Qi. To express our meaning more clearly: we suppose the 

 quantity .vj to be built up of some variables ui, viz. as the sum 

 of these vai-iables, in such a way, that in this sum the term ?/-/ is 

 lacking if Xj is not subject to the influence of the cause Qi. 



So in the relation 



.vj = ctjl Ui -{- etj2 w-2 + . . . + «j/''/ + • • • + «i7 u, 

 we have 



aji=l, when Qi does act upon Xj, 



(rji=zO, when Qi does not act upon a^j. 



Thus in 2! Er itjC' only those terms f,/, f,j% . . . f,..,^ occur which 



correspond to the variables 11,.^, Ur.,, . . . u,-, due to the causes 

 Qri, Qr2> • • ' Qr„ actually influencing a'j ; on the other hand those 

 terms are lacking, which owe their existence to the causes jiot con- 

 tributing to Xj. 



In the sum ^ ei^ ajictki onlv those terms f/^ occur, for which both 



«ƒ/!= 1 and ajci^= 1, that is to say : the terms, which derive from 

 the causes Qi, acting both upon Xj and xk- 



The expression iyk^= ^ ^i^f(ji((ki therefore may be called the 



1=1 



square of the mean value of those elements of .rj and xk, which are 



due to the common causes. 



Introducing for ^^jjm.k the name: "mean error common to .<j and 



Xk\ we may define the corielation-coeflicient of the quantities .rj and 



