262 
P 
and My) are conjugate, then A‘) Mi v is an elementary affinor and 
therefore according to theorem IL we have: 
p p 
APD MO y= AD MAS A wo. or er BABA) 
a 9 z 7 3 i ê 
P : 5 
where the w; are elementary affinors, while the summation has to 
be extended over all ordered alternations A. As Me A= 0 for 
a B we have: 
rap \ Ee 
AOM Dn EMA Dre a 
Repeated application teaches thus that the operator 4© M® never 
can be nilpotent. According to a wellknown axiom from the theory 
of the higher complex numbersystems we conclude from this, that 
there exists an idempotent number of the form 
a, (AL? MOD) at a, AG MEO et SE 
For every ordered alternation there thus exists such an idem- 
potent number and the products of these numbers being zero, they 
form a series of idempotent principal units. The number of ordered 
operators A; and J is therefore d;; or less. It cannot be less, however, 
as a repeated application of theorem Il teaches that every elementary 
A Rae 
affinor may be written in the form > A) u m. If u were 1, then 
4 
the powers of A‘) M's) would belong to the first class of Prtrcn 
with respect to one of the idempotent principal units and to the fourth 
class with respect to the other ones. As in an original number- 
system such numbers (nilpotent by-units (Nebeneinheiten)) cannot 
occur, 4 = 1 and the operators 
d 
ji =e; Al uy kh eee i 
are therefore idempotent principal units. In the same way 
dy ' ) À 
TS eee sae Re 1G A ig a 
form a similar system. These operators are called ordered elementary 
operators of the first resp. second kind and the affinors which may 
be derived from them, ordered elementary affinors of the first resp. 
second kind. 
When y; is the number of permutations in A; or M; (hence for the 
permutation number s,,...,s; equal to s,/...s,/) and when 8; is the 
number of operators A; or M; that do not annihilate a definite 
