AI.MAK NÆSS. 



M.N. Kl. 



(I, (1., r7y 

 /Vj //, Ä,, 



thus giving as tlio components of tin- product the thr**,- fjuantitles 



which also is written 



(1) 



a Xb 



i J f 



c?j r?._, (^?., 

 />, /a, /a. 



i, j, f, being the unit vectors of S.^. 



If we in this way shall obtain a vector, it is, of course, necessary 

 that the number of determinants which can be picked out of the matrix, 

 is equal to the number of dimensions of the space concerned. Since this only 

 is the case when n — 3, the operation of forming the vector product from 

 two given vectors has been considered to be unique for S.^, without any 

 possibilit}' of generalizing to Sn. But, of course, it is not obviously given 

 beforehand, that such a generalized product — giving in S.^ the Gibbsian 

 vector product as a particular case — necessarily shall be a vector, nor that 

 it shall be derived from two given vectors. On the contrary, we will show 

 by an example that we even in elementary vector analysis may meet with 

 quantities, deriving from another number of vectors than two, which with 

 respect to fundamental properties must be considered to be analogous to 

 the vector product. 



Let in three-space a system of three vectors be given: a, b, C. To 

 this system there corresponds one, and onl}' one, definite system of vectors, 

 say cC, b*, C*, called the reciprocal to the first, such that 



(2) a a* + b b* + c c* = / = a* a + b* b + c* c. 



The starred system is easily determined by elementary matrix opera- 

 tions. Let t, j, f be the unit vectors in S.^, and 



(3) 



(4> 



T =\a + \h + U. 



ly* 



I a' 



j b* + f c*. 



Then 



a* a + b* b + c* C = W\- ■ W. 



And since this shall be equal to the idemfactor, the matrix of W*c must 

 be the inverse of that of W, and the matrix of W*, accordingly, the conjugate 

 to the inverse of that of W. Then we get from this immediately : 



