572 



Journal of Agricultural Research 



Vol. XX, No. 7 



In this expression 2r 2 XA means the sum of squares of the six known 

 correlations. 2r XA r AB r BX means the sum of the products of the groups of 

 three correlations, corresponding to the sides of triangles. There are four 

 of these triangles, XAC, XAB, XCB, ABC. 2r XA r AB r BC »'' cx means the 

 sum of the three products of the groups of correlations which are 

 arranged in closed quadrilaterals, and 2r 2 XA r 2 BC means the sum of the 

 product of squared correlations in pairs which involve no common vari- 



able (r 2 xx r 2 BC , r 2 xc r 



:) (%. 13)- 



The formula for four known causes is easily found by a continuation 

 of the methods used to find the others if attention is paid to the sym- 



metry of the expressions. Since, how- 

 *^* - - .£"""* ever, this formula, as well as that just 

 given for the case of three causes, is some- 

 what cumbersome, it will be convenient 

 to use a more condensed notation. 

 <j>(XABC . . .) may be used for a func- 

 tion involving all possible correlations 

 among the variables (XABC . . .). In 

 the definitions 2r 2 means the sum of the 

 Fig. 13.— Effect and four correlated known squares of all correlations ; 2rV 2 , the sum 



of the product of all pairs of squared 

 correlations which involve no variables in common; Zrrr, Zrrrr, and 

 Xrrrrr are the sums of the products of all groups of correlations which, 

 represented by paths, form closed figures, triangles, quadrilaterals, and 

 pentagons, respectively. "St^rrr is the sum of the products made by 

 multiplying each triangle of correlations in the sense above by the sec- 

 ond power of those correlations which do not involve any of the vari- 

 ables in the triangle. The number of terms of each kind is given above 

 the brace, where it is more than one. 



<t>(AB) 



1 — r 2 (2 terms). 



<t>(ABC) = 1 - 2r*+ 22m- (5 terms). 



<I>(ABCD) =1— 2r 2 + 22m - — 22mr-t-2rV (17 terms). 



10 10 15 12 15 10 



<j>(ABCDE) = i— 'Lr 2j r2Lrrr—2'Zrrrr-\-2Zrrrrr-\-'Lr 2 r 2 —2'S l r 2 rrr (73 terms). 



