331 



for Ji odd and 



Rn 



Oo2 



(10) 



for n even, where Oi denotes an orif^inal system of the order /and 

 Hi a pi'incipal row of the order /. Bnt for some divei-gence in -|- and — 

 signs the systems R^ are identical with Clifford's yi-way algebras 'j. 

 If none of the nnits is privileged the choice of the numbers 

 occurring in the identifications is altogether determined by the 

 dualities existing in the different groups. There are four altogether, 

 and we shall call them : 



From the mode of transformation we conclude for the existence 

 of these dualities as subjoined: 



1) Clifford's systems have been worked out by J. Joly, Proc. Roy. Ir. Acad. 

 5 (98) 73-123, A manual of quaternions (05) 303—309. He gives geometrical 

 applications after the manner of the quaternion theory wilhout decomposition of the 

 product. A. M'AuLAY has elaborated this -matter as well, Proc. Roy. Soc Edinb. 

 28 (07) 503—585. These papers do not aim at a foundation on the theory of 

 invariants or a closer investigation of the fundamental groups. 



2) The squares of the dualities not founded on contragredience have been indicated 

 by blacker demarcation. These dualities only exist when n is even. 



