260 
in one and only one way in a series of k elementary affinors. An 
elementary affinor with the lower index i and the upper j is a quantity 
that is annihilated by all alternations A', 1 > i, 1 > 1 and by all mixings 
M,,h>j,h >I. Such a quantity is also annihilated by all A, and M, 5e 
it can arise from the application of il, A, „As Ms; and M, and it is 
invariant by application of ‘I, 9;; Apt, p M, and d,, M,. For a definite 
value of n (number of fundamental elements @) all elementary affinors, 
for which the permutation number of the A contains permutation regions 
> n, are zero. 
For p= 6 e.g. the expansion is: 
ree. Like fete he ene ee: 
u—(4,i7,,+254,M,,+814,M,+254,M,+1004,M,+2564,M,+| nk 
+254,M,+1004,M,+814,M,+25A,,M,+A,,M,) 
Expansion of an elementary afinor in ordered elementary affinors 
of the first or of the second kind. 
Theorem II. Every simple alternation or mixing that is annihilated 
by all higher ones with the same number of permutation regions of 
more than one factor, can be written as the sum of multiples of 
ordered alternations or mixings with the same permutation number that 
contain in each of their regions only factors from the corresponding 
regions of the original alternation or mixing. 
Let us prove this first for an affinor: 
0 Pp d as y d mn, 
m=pgÂn= pleo gv = (U, Up) o (V‚ vg) 
P 
When the first factor of m is no u, we may apply a „14, the 
permutation region of which contains this factor and „u. Then we 
= 
may write ,4:4m as the sum of p +1 terms, the first factor of 
P P 
only one of which (namely of m itself) is a v. Therefore m may 
be written as a sum of terms the first factor of which is u. When 
P 
now in one of those terms m, the second factor v precedes the 
second factor u, then we may apply a ‚+14, the permutation region 
P 
of which contains the two first v and the p—1 lastu. Tben „+: Am, 
may be written as a sum of p+1 terms in only one of which 
P 
(namely in m, itself) the second and third factors are v and which 
