265 
region of more than a factors. From theorem II we conclude there- 
fore that ee can be reduced to a sum of ordered alternations 
with the permutation number a.m. This decomposition is singly 
determined, each of these alternations being a sum of the ordered 
alternations from the expansion according to C with permutation 
numbers from «.n up to the highest number below (a+ 1).n that 
have the same « alternation regions of n factors. 
Thus we have obtained the theorem. 
Principal theorem D. Every affinor can be written in one and 
only one way as a sum of terms each consisting of penetrating 
general products of a number «, characteristic for this term, of 
factors E, with a linear homogeneous non reduceable affinor of 
degree p—en and forming an ordered alternation with the per- 
mutation number «.n. This expansion may be obtained by arranging 
in groups the terms of the expansion with regard to elementary 
affinors. 
B . 
u may therefore be written 
P ORE Leds, (2) 
tae Se ee 
a 
where an Ls?) is the sum of the ordered elementary operators of the 
first kind which have the same alternation regions with n factors. 
For n=? the expansion in mixed alternations according to 
the principal theorem A is at the same time an expansion in 
non reduceable covariants. It is therefore singly determined for 
every affinor and containing & terms it is also identical with the 
expansion with regard to elementary affinors. From (16) we tind 
therefore for this case 
p \ (2a 
k—a—1 Ee ( 2 ) = 
Lf = Se ON (42) 
From the deduced expansions in series we may derive in a simple 
way very general expansions in series for algebraic forms in m 
rows of n variables as will be shown in the next paper. At the 
same time we will mention how the above is connected with known 
expansions in series of algebraic forms. 
Expansion of the affinor of Riemann-CuristorFEL with regard to 
ordered elementary affinors. 
