KUNGL. SV. VET. AKADEMIENS HANDLINGAR. BAND 58- NIO 3. 23 



(7) The regression of the means. The mean of the i":s in a r-array are given by 



CC CO 



— 00 XI 



Thus. in cases of correlation of type A, we have 



>'/ 



f/i' 





i +j 3 



Hence, by the definition of the symbols 1® and IfJ we find 



""' \/ (0 > + /? T 70, + V 7<°> + ••• 



which by aid of system I, II gives 



(32) 





This is the exact formula of regression of the mean of | on r\. Mulatis mu- 

 tandis it gives also the regression of v on i'. 



Utilising the conclusions arrived at in paragraph (6) we are led to the following 

 regression formulae: 



Case I. Correct to the fourth order. 



(32*) I 



■c 

 Sv 



r>, — rpoiRt W — pu R^i }) — rp ot R t (i]) — p ls R 3 {rj) 

 l+p 03 R 3 (rj) + p u B t {ti) 



Case II. Correot to the fourth order. 



1 



(32*) II I, 



rri-r^R^-^R^-r^R^-^BÅv)-^^^^-^^^^) 



l+p 0i B 3 {r j ) + p ot R i (r ] ) + -pl 3 R e (r ] ) 



Mutatis mutandis the formulae are valid also for 7 (| . 



The regression curves (32*) I, II may be directly constructed. When the 

 characteristics G l ,G 2 ,r,[} 30 ,p 2l , i 3 l2 ,p 03 ,p iii ,(i 3l ,f} l3 ,{i 0i , are found, each term in the nume- 

 rator, as well as in the denominator, may be rapidly computed if we have a short 

 table of the polynoms Bi(rj). Of course both numerator and denominator may be 



