14 \\ ICKSELL, THE CORKELATION PUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERISTICS. 



As we have already found that 



we need only consider the quantities 



l ( i] = when i>s, 



7(0). 



■loj, 











;(1) 

 •'Oj, 



/(l). 









7 (2 ! 



*0j, 



;(2) 

 il}, 



/ (2) ■• 







7(3) 



Hit 



7(3) 



7(3) 



7(3). 





7(4) 



lOjt 



7(4) 



7 1./'• 



7(4) 



f(4) 

 • / 3j. 



/ ,4) . 

 7 4.7 



We have 



— CO — CO 



Integrating per partes we find 



ne 



d; r 



/$ = (- 1)M(*-1)(5-2)...(S i + 1)^1 ri|<jP (|, /)^-', 



CO 



that is, completing by (1), 



/i s j= when i>s. 



Using the above equation, we find with the aid of (1*), (8), (9), (10), (11), 

 the following five systems of formulae. 



n (13) ? "i = - r ^r-=- ff ^^ 1 W' 



(13*) / 8 = _^3) --0, t (,)Ä,fo). 



Hl (U) /g - ^) + r" ^J# = [Äy(??) + ,-^, + 2 (, ; )]T„(,;), 



(15) I?} = i **^ = 2rÄ, + 1 (, -).,■„(.,), 



(16) 7 8 ^ 2 ^2) = 2^CM(<;). 



