Î922. No. 13 ox A SPECIAL POLVADIC OF ORDER //—/>. 



45 



Therefore : 

 (10) <PX'P' = 2-2-+ >i,Eu\; 





>' + '■ 





The sum in the brackets is equal to the following determinant of order 

 (« — 1 ), multiplied by ( — 1 )" : 



(A) 



e^. . . . e,_i e, + i. . 



e« 





n~.n+ 



. . . . e„ 



1 . . . f. 



For all the (// — 2)-ro\ved determinants of the first // — 2 rows are 

 of the form E:j, where y = 1, 2 . . . . / — 1, / + 1 . . . . ;/. The plain 

 complement of Eij is f/'. It must be noticed that fy' stands in the /''' column 

 of this determinant if i ^ j\ but in the (/ — 1)'*' column if i J. Hence 

 the algebraic complement of Eij in the first case is: 



1) 



II — \ -T i c ' 



f;'--(- 1)"(- iVf/, for i>j 



(11) 



but = (— 1)"(— n'f/, for i<j 



But then (101 readilv shows that: 



(12) 



øX'^' =(- 1)" 



x^ y.., 



ei e, 



yii 



Ci e, 



f 1 f . 



e„ 



which gives the formula for the vector product of two dyadics in three- 

 space"!* as a particular case. 



By comparing (12) with § 4 (aM we observe that (12), as in S.^, holds 

 good also if (P and 0' are vectors, i. e.: if xi and f^' are scalars. 



t Almar Næss, loc. cit., § 37 (7). 



