KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58- N:0 3. 5 



thetical foundation. His functions have in a very high degree the properties of 

 rather elastic developments of arbitrary functions, being in that respect in a way 

 analogous to the Fourier series. 



The Charlier functions lead to two different types of variation, not easily 

 distinguishable where they pass över into each other, but differing very widely in their 

 extreme forms. They are called type A and type B and embrace the väst majority 

 of all statistical series observed in nature. 



It is onlv the functions of type A that have been applied by Charlier to 

 correlation problems. 



The present treatise is confined to the study of these functions. It is espe- 

 cially the regression of the characteristics that is the object of our investigation. I 

 think the study will give ample illustration of the great elegance of application, 

 mathematically and numerically, of these very remarkable expressions. 



(1) The general correlation function of the A-ti/pe. The function of correlation 

 of type A by n variates a*x, x 2 , x s , . . . x n is expressed by 



/•'(.*',- ■'':• X s , ■ 



■>■„) 



7 (,r, . .r,, x 3 , . . . X n ) -1- 2: 2 2 . . 2. -I ,,/,,,. 



''" ! h + h + "' + i »<P(*i, X 2 , X 3 . . .X n) 



Ox^dx^dx 4 ». . . dx% 



where /,, i 2 , i 3 . . . i„ assume all values for which », + « 8 + / a H f-*«>3, and '/ is the so 



called normal correlation function, which, if .t,, x 2 , ';• • . . x„ are reckoned from their 

 means, is given by 



l / 



/l - ''•'-''•'■••• ••'"' (2./)" 



The parameters a n: and /J„, , , /( may be expressed in terms of the moments, and 



./ 



Further 



";. ■ »,2. «13> • • • »1»! 



a ?l , C . , , a .;;, . . . 'I „ 



Otjl , (!■■■, , ttj-| , . . . fl,,| 



@>H i (■In; > ^nj ) • • • H/m 



da ! = (l i ii ■ 



The parameters a nl of the normal function have been determined in terms of 

 the moments by several authors — first by Pearson in various papers, further by 

 Creiner. x For three variates the determination was first carried out by Edge- 



1 Greinek: tlber das Fehlersystem der Kollektivmasslehre. Zeitschr. i. Math. u. Phys. Bd 57. 1909. 



