12 



AI, MAK N/i:SS. 



M. N. Kl. 



I ) (■ fi n i t i o n : (al /// an ii-diuiru^ionnl space let thrre be given p linrarly 



iii(iipni(lciil vrclors a^ 0,^/,/, <X.^ ^l'i-,: ({/, -- ç,- a/,,-, (sum for / 



li"()iii 1 t(i //1, C], C._,, .... C;; /uii/i; (III orthogonal sys/nii of itiiit vectors. 



liV the space coiiipleineiil of those p vectors wc understand a determinant 

 whose ht st p rows are /ori/ird Jron/ the cümponerüs of the a's and whose 

 first II p rows are the unit vectors, i. e. : 



la>) 



As the vectors in these n — p rows are, of course, to be multipHed 

 indeterminately in the developed determinant, we see that the space comple- 

 ment of p vectors is a polyadic (tensor) of the (// — pf^ order. The simplest 

 and for our purpose most convenient way of expressing it as a sum of 

 (elementary) polyadics of the same order is by expanding it according to 

 the (n — />)-rowed determinants of the first n — p rows. 



What we in the following will try to show is that, by deriving the 

 fundamental properties of this space complement we arrive at equations 

 which can be regarded as generalized vector product equations of S.^, and 

 from which, therefore, we get the formulae of the Gibbsian cross product 

 as special results. 



We see that the space complement is a vector if and only if the 

 number of vectors is // — 1, and that this vector then is perpendicular to 

 each of the primary ones, i. e. : it is perpendicular to the hyperplane 

 containing the (// — 1) vectors iVom which it is derived. For the components 

 of the space complement are in this case the cofactors of the elements 

 (i. e. the unit vectors) of the first row. Hence the scalar product of the 

 vector CI,- and the space complement by definition is : 



(b) 



Oi^ Clio .... ai„ 



(hi (^12 ■ • • • ^1" 



(J,\ Oi 



fpl (Ipo .... ä/,,1 



p ^ 11 — 



which vanishes identically, two rows being equal. 



If // = 3 (i. e. : /) = // — 1 =; 2) we get the ordinary vector product 

 of two vectors. The space complement is a scalar if /> — //, viz. equal 

 to the determinant of the ;; vectors. 



