M.-N. Kl. 



Therefore, we can write : 



^\'.x\. 



e, e. 



e„ 



y, y. 



y I. 

 y,i 



One special case of this formula is of particular interest. 



We know from three-space, that the vector of a dyadic (GiBBSt is 

 obtained by insertion of the cross between each pair of its vectors. The 

 dyadic be* '/' = t a + ) b + f C. Then '/',. (Gibbs writes W^) = \X<\. 

 + i X b + f X C. We also know that the components of this vector are 

 the so-called symmetric differences of the matrix of a, b, C,**. They play 

 a rôle in the theory of triadics in S.^***. In any square matrix there are 



in general 



pairs of elements such that the elements of each pair 



are symmetric with respect to the principal diagonal of the matrix. We 



thus can form 



;;(// 



differences („the symmetric differences") by sub- 

 tracting one of these two elements (a definite one I from the other. The 

 number of symmetric differences is equal to // if and only if n = 3. Of the 

 matrix of a, b, C they aret 



^z — ^2> ^1 — ^3- ^2 — ^1- 



We observe that the minuend is chosen in a definite way, alternately 

 in the upper and lower half of the matrix tt. 



In order to be able to tell at a glance, whether we are speaking of three-space or 

 «-space, we will in the following (usually) denote a dyadic in Sa by ï' = t U -f j b — f C, 

 in Sn by 0. 

 •* Zur Theorie der Triaden von Almar Næss, § 4. 

 •" loc. cit. § 33 & § 45- 



t loc. cit. § 4 (i) or p. 71. 

 tt loc. cit. p. 70, footnote. 



