261 
have only factors of „u in the region p. Therefore i may be written 
as a sum of terms which are all ordered with respect to the first 
and second factors. Proceeding in this way we obtain a sum of 
perfectly ordered terms. In each of these terms the region p contains 
only factors u and the other region only factors v. 
P 
8 d d d ; 
Now we consider the affinor m= ,uc,vo,wo.... with Q alter- 
nating regions of p,q,r etc. factors, p2>q2r2..., which may be 
annihilated by every alternation with Q alternating regions higher 
P 
than „4. When e.g. the first factor of m is w, then we apply 
a ILA, the permutation region of which contains w and pu. As 
Pp PR 
this ,414 annihilates m, m is a sum of terms all beginning with a 
u. Let now the second factor most to the left be e.g. a v, then we 
apply a ‚m4, the permutation region of which contains the two 
first v and the p—1 last u. Proceeding in this way we obtain a 
sum of terms which are all ordered as to the position of the u 
with respect to the remaining factors. Now we continue with alter- 
nations „714, the region p of which always contains the u that are 
already ordered; thus we obtain ordering of the v ete. until perfect 
ordering is reached. At eacli passage and therefore also in the final 
result the factors u of all terms remain in the region p, the v in 
the region q etc. 
We shall apply the theorem just proved to the elementary affinor 
P 
jfu and we shall prove that the result is singly determined and 
identical with 
PD. 1,,...,07 p 
um Sey A MP)y. ED 
the summation being extended over all ordered alternations and 
mixings, the number of which is just dj. For this purpose we use 
the wellknown property that the numbersystem of the permutations 
P is an associative system, which may be resolved into & “original” 
systems with J units. The units of such an original system may 
be chosen in such a way that 
Jos for q—r 
O.. ys Joa 
Such a system contains at the most dj idempotent principal units, 
the sum of which is the modulus of the system 7/. Let now for a 
Ig In = (33) 
p p 
definite value of À for any affinor v be AD MO v AO where 4,© 
