6 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSIOX OF 1TS CHAKACTERISTICS. 



worth. 1 See further the above mentioned Meddelande N:r 66 by Charlier. In the 

 following vve shall give the results as developed by a method which differs slightly 

 from that of the above authors. 2 



Defining as the moments about the mean of the second order of the y-function 



oo co -x cc 



la,! = \ (1*1 I dx 2 jdx 3 . . . \dx„ X a Xp(p{x t , X 2 , X 8 , . . . X n ), 



and employing the determinant 



M 



'"ii > ™12> '-i:i' • • • ^i» 

 '-21 5 '•;•.'' '■::;> ■ • • ™in 

 "31 > "< 3 j , A 3: . , . . . A,„ 



'•n I < ^nz i '"»3 > • • • Aih 



the parameters of the r/>-function may be expressed in terms of the moments by the 

 following equations 



i 



J - 



<i,i : ; 



M 

 1 dM 



Novv it may easily be shown that the moments of the second order of the 

 y-funetion are identical with the moments of the same order of the correlation fnnc- 

 tion of type A. Thns for X a p we may insert the moments of the observed correla- 

 tion scheme. 



For the parameters A hi ., h in of higher order than the second we will not give 

 the general expressions. They are expressed in terms of the moments of the observed 

 distribution of the orders below and equal to i\ -\- i 2 + h^ h ?"„• 



(2) Special case for one variate. In case of the variation of only one variate 

 x we obtain the frequency function by integrating for all vallies of x 2 , x z , x 4 , . . . x„. 



The result is, if x be reckoned from any origin, 



F(*) = r,(s) + 2^*> ( *> 



j=»3 



dx i 



w 



here 



1 Edgeworth: On correlated averages. Phil. Mag. 1892. 



2 Wicksell: Tlie general characteristics of the frequency function of stellar movements etc. Meddelanden 

 från Lunds Observ. Ser/ II, N:r 12. 



