332 



The associative Systems Ei and R4 

 If we call the unities of the sub-degrees'^yi— 1), (?i+l), and (2n— 1) 

 corresponding lo e, : e/, e, and e/ and the contragrediënt unities e,', 

 the rules of calculation for vi = 3 aie: 



/? 



ei = 



623 = 

 ei23 = I 



eil' = 



6231' = 



ir = 



en = k 



ei -le'i = 



e'231 

 e'll 



Rt 



ei = 



623 = 

 6123 = 

 61 1 = 

 623 I = 

 6123 I = 



611 =k 



— e'23 I = 



e'l = 



— e'l -23 1 = 

 e'23 = 



— ei 1 = 



— e'i23 = 

 e'll =- 



k2 



— 61 1 = 



62T = 



— 61V3 1 = 

 61 = 



— 623 I = 



— 6123 — 



en =k' 



6'23 I 

 6 1 I 



e'i23 



e'23 



e'l 



6'123 I 



e'll = 



(11) 



=r= 



((-y : — 

 a-ö: + 

 a-8 = a-ö 



_k'^ = -l-l cycl. 1,2,3 



k2 



61 = 



623 = 

 6123= 



611 = k 



623 = 



61' — 

 6123 = 



en' =-k2 



- k ei' 

 k'^ e'23 



n-i3 : + 

 =i=k3=-i,.-y:\^^y"-(12) 



t(~Ö=ft—^. 



ct-e=a-,i 

 cycl. 1,2,3. 



= - k3 = + 1 



a-y:-- (13) 



a-(f: iden- 



= I tity. 



(i-e : identity, 

 cycl. 1, 2, 3. 



1) "Cycl 1,2,3, . ..;n" means that the numbers 1, . . ., n may be substituted 

 by any even permutation of these numbers. 



