CHAPTER I. 



The frequency function of the A-type for three variables. 



1. According to the theory of mathematical statistics as developed principally 

 by Bruns and Chakliek the frequency- function of a unitary statistical population 

 with three variâtes x, y, z generally has the form 



( I ) ifc », s) = ,(.,, ,) + S B m ËÛ^lfeyl . 



K ! T\ > 9> i \ » ^ g^- 



If x, y, z are the deviations from the means we have 



'i (x, y, z) = He - Vä./' 



/= >Lc 2 + + Cfc 2 + 2Dys + 2fe + 2i*b# 



and the quantities .4, 5, C, Z), -E, .F may be determined in such a way that 



»+i + *>-3. 



The function (1) is called the correlation function of the yl-type, and the 

 coefficient H is so chosen that: 



00 



(2) f[[dxdyd0<p(x,y,z) = l, 



- 00 



If v rj o., are the moments about the mean of the function F{x, y, z) we have 



00 



(3) fff dx dy dz x a y§ *T F{x, y, z) = v aBr . 



— 00 



Now it may be shown that 



GO 



4 \dx dy Uz x L y> J . . , = — 1 )'+•'+* . , r - , K, i r - i -, k 



' J J J 9 J dx*dy>dz k 1 ; \r-Jc ' r'3> Y « 



CO 



when simultaneously a > i ß > y y > Here we denote with the moments 



of the function <p(x, y, z). Whenever either a or ß << y or ■( <C k the integral 

 becomes equal to zero. As \ rj o. { = 0 when a -f- ß + T is an odd number we see 

 that the integral also disappears whenever (a — i) -j- (ß — .y) + (t — ^) is an odd 

 number. 



