1922. No. 13. ox A SPECIAL POLYADIC OF ORDER )! — p. 



27 



§ 9. Expressions of the form 2' f , X f: and Jf e, X t". The symmetric 



differences of a matrix. 



Given in S„ two systems of // vectors : 



(A) 



fi. f.. 



f« 



Let the two conjugate systems of these be denoted bv y.' i and xi re- 

 spectively. 



We will find an expression for the quantity ^"f',Xf/, evidendv a 

 polyadic of order n — 2. It is a vector in three-space, the X then denoting 

 the ordinary vector product, and we know that this vector is expressible 

 in the form' (i, j, f being the unit vectors of Sg) : 



(B) 



f 



where the scalar product is to be taken of each two corresponding vectors 

 of the last two rows, i. e. : the dot is here written after the vector where 

 it is to be used in the developed determinant. 



We are going to show that we in S,, arrive at an analogous expression. 

 According to our definition we get the sum of ;/ determinants : 



(1 





ei e.3 



e« 



^1 e., . 





the last two rows being the components of f, and f, respectively. 



We develop each of these ;? determinants in terms of the (;/ — 2)- 

 rowed determinants of the first ;/ — 2 rows o( unit vectors. We get, / and / 

 being any two columns, j <C I'- 



(21 ^f',Xf, = ^ 



i yjb 



(-ir 



ei-.. .e. 



e/. .e„ 



«-'i 



e„ 



/'- 1 



/'7/'./ 



the sum Z being taken for all the rM possible sets of [j I). As before, 

 (_/V) \-' 



Xj and e'/ indicate that ey and e/ are stricken out. 



• See: Zur Theorie der Triaden von Almar N.ess. I5I and (61, p. 16. 



