333 



^3 



«-/? identity 



«-y „ (14) 



a-ö „ 



cycl. 1,2,3. 



Witli a non-liomogeneons reotangular interpretation of the funda- 

 mental variables e^ is a polar veetor, e', an axial bivector, e, an 

 axial vector, e', a polar bivector ^), I a projective, and k an ortlio- 

 gonal "pseudoscalar", ke^ a polar, and k^e,, an axial versor (qua- 

 ternion with tensor i) without scalar part. /^.^ includes and discri- 

 minates all these quantities, Rl identities polar quantities with axial 



ones and I with an ordinary number, Rl identities all the polar 

 quantities and all the axial ones as well, and k with a common 



number, whereas in R'^ only the difference between vectors and 

 ordinary numbers exists. 



The rules of calculation for n = 4 are: 



R[ 



(t - Y : — 

 «-(ƒ: — 

 f( - 8 : -{- 



>c= + 1 

 ^j cycl. 1,2, 3,4. 



+ 1 



1) In space these quantities have the symmetry-pruperlies of a line-part with 

 direction, a plane-part with rotative <iirection, a line-part with rotative direction 

 and a plane-part with -f and — side, all conceived as parallel removable with 

 respect to themselves. For n odd it holds good that po'ar quantities change their 

 sign, when the + direction of all axes is inverted, and that axial ones do not 

 change their signs. 



22 



Proceedings Royal Acad. Amsterdam. Vol. XXI, 



