12 WICKSELL, THE CORRELATION FUNCTION OF TYPE A, AND THE REGRESSION OF ITS CHARACTERIST1CS. 



(5) Maihemathical developments necessary for the derivation of the formulae of 

 regression of the characteristics. Before proceeding to the development of the regres- 

 sion formulae we shall put together the mathematical formulae necessary for the 

 development. 



Employing the notation 



— 00 



we find at once (remembering that the </>-function and all its derivatives become 

 zero when £=±») by partial integration, that 



(1) lij = when i>s. 



Further 



— 00 



But we may write, completing the squares in the exponent, 



,K ' Vin V2fc(i-r i ) 



Introducing the symbol 



i _!Z: 



we find 



V2n 



''<• ( - *> = 7fb* f/, ° {,]) ■ v ° (IS) 



Thus, as 



CO 



(2) JV,(^=sl) «-*-=?, 



00 



we have 



Further 



(3) 



(4) 



— 00 



00 



jä=? /«* (ö) " - ' "'" + rV " ' + '""' ~ "• 



