330 



others with tlie lin. homog. group too, and are called projective 

 quantities. Then they are of the sub-degree (Dutch : ondertrap, German : 

 ünterstufe) />, when the number of the factors of the units is />, 

 ;>=rl, . . . .,2fi, and we write them ,,a. The others are called 

 orthogonal quantities. All linear quantities maj be composed of 

 projective ones ami powers of k. 



When classifying up lo the special atHiu. group inclusive, for n 

 odd the system lln is obtained from the preceding one by the 

 identification : 



I = x. (5) 



The sub-degree p, p^n coincides with the sub-degi-ee {n-\-p) and forms 

 the degree (trap, Stufe) p. For n even no system is feasible here, because 



1^6,== — e/ 1 1, (6) 



hem'e identification of 1 with an ordinary number is impossible. 



When classifying up to llic (»rtli. group inclusive, R„ arises out 



of W^,, l>v the identification 



k = y. (7) 



The system makes no <lilferem*e between projective and non- 

 prqjective (piantities. The sub-degree />, p < n coincides with the 

 sub-degree {2n — p) and toinis the />//-(/<?</r^ö (neven (rap, Nebenstnfe) ;/. 



When classifying up lo (he rotational group iiiclusivo, for 7i odd, 

 R'i, arises out of //,', by the identification 



I = k = K . . (8) 



Neither does this systvMu make any diffei-ence between projective and 

 non-projective quantities. The suli-degrees /;, (n—p). {n - p) and {2n—p) 

 coincide and constitute the pr inci pa Idetj ree {UookUveip, Usiupistufe) p; 



p<n\ n' = for 7i odd and n' = ^ for n even. In all these 



systems the associative product of dissimilar fundamental units is 

 equal to the alternating one. 



The systems /i\ are the products of o ricjiual .systems i\m\ principal 

 roius ') according to the general formulae : 



;< — 1 



.1—1 



.^ ...... (9) 



n-l 



Fin — H^ O-iT 



Rl = /ƒ, //„ ChT 

 ') Cf. Grundl. pages 11-^18. 



