335 



(17) 



with non-coiïi|)lieated duality. This system may also be obtained 

 from the preceding system R'l (page 334) by the transition e^ —^ So, 

 — i e^ ^ e.^, etc., e^ — ^e^, ie^—^e^, etc. It it noteworthy that, for 

 /i = 4 the theory of relativity (Cor the space-element) exactly corre- 

 sponds to this more simple system. 



For non-homogeneous rectangular interpretation of the fundamental 

 variables e^, and ej^^ are a vector, resp. a trivector of the first 

 kind and le,, and Ie,.,3 are the corre.s|)onding quantities of the 

 second kind ^). I is a projective and k an orthogonal pseudoscalar. 



R'l contains and distinguishes all these quantities. rI identifies a 

 vector resp. a trivector of the first kind with a trivector resp. a 

 vector of the second kind and k with an ordinary number. 



Decomposition of the Associative Product. 



The associative product of two projective (piantities of the sub- 

 degrees // and </' and the principal degrees p and q, p', q' ^n, p ^ q, 

 consists in the most general case of p -\- 1 parts, each of which 

 being a product of a projective quantity with a certain numl)er of 

 factors k. As a distributive combination each of these parts is a 

 product itself. The number of factors k is called the transvection- 

 number of this product and this numbei- is at most equal to the 

 smallest of the numbers // and q' . We call these products, if p' 

 and q' are both < or both '^n' , beginning from the lowest and 

 otherwise beginning from the highest in sequence: 



1) The customary distinction for n odd between polar and axial quantities does 

 not hold good for n even. 



22* 



