8 WICKSELL. THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



CC ' 



Vij= i dx I dy(x — m l )'(y — m.,)i Fix, t/), 



r j: — rr.< 



we have 



Wj— the arithmetic mean of the variate x; 



m 2 = > ■» » > » i/ : 



o\ - r, ; r = . " ; a\ = i/ u2 ; 



O | Oj 



[34 30 = — v s0 ; [2/1,,- -r.,: [2^4 12 = - - r,,; l34 0S = r„; |4^ 40 =- r, — 3 r^,,: [3.4.,, - v Si — 3v n v 20 ; 



\%\2A 22 = i> 22 — v 20 v 02 — 2*;,; |3.4 13 = 1' 13 - Sr,,/;,,; [4^ = »»„< — 3*>J . 



Regarded as a function to give an arbitrary F{x,y) the above series is con- 

 vergent in all cases where application to statistics is in question. 

 The mathernatical conditions are chiefly these: 



1. F(x,y) must not, for any finite dominion of x and //, have an infinite 

 number of maxima and minima. 



2. The integral I i F(x, y)dxdy must be convergent. 



In th is ease the development will be convergent even if the parameters m lt Wo, 

 a lt a 2 and r be given other values than the above, only then differential coefficients 

 of the first and second order must also be included. 2 



If the variates be reckoned from their means and expressed in the dispersions 

 o, and o 2 as units, the correlation function will be brought into a simpler form. 



Putting 



■c 



= t and J2 — — - = ■»:, 

 a. o., ' 



the frequency function of ; and \ will be 



where now 



<Po(£ ,0- ff > °2<f (I , y) = t-ttt =% e 2( ' 



2rc V 1 — r- 



The parameters are here 





1 See Charlier, Loc. cit. Ser. I n:r 58. 



i For the proof compare Charuer: Loc. cit. Ser. 1 N:r 71. The proof given i" tliis paper niay easily 

 be extended to the case of two variates. 



