280 
This expansion is singly determined, the corresponding expansion of 
À p p 
P 
m being singly determined *) and an extension of that given by Gopr 
for n= 2. We have thus found the theorem: 
Principal theorem B. Every algebraic form, homogeneous and of the 
degrees 0, 6,... in m sets of n variables x’, y’.... can be expanded 
in one and only one way in a series of terms, each of which consists of a 
product of a number « of determinants that are formed from the vart- 
ables of n of the sets with a non-reducible form, in such a way that 
the deternunants in each term belong to the permutation regions of a 
definite ordered alternation „A, characteristic of that term and acting 
OANA NON x hog tah ALA the number a being characteristic of a 
definite group of those terms. 
B 
Thirdly we can proceed so far with the division, that m becomes 
a sum of ordered elementary affinors all of the first or all of the 
P 
second kind. With this corresponds an expansion of Fm in ordered 
elementary forms of the first resp. of the second kind, which may 
be characterized in the following way : 
Principal theorem GC. Every algebraic form, homogeneous and of 
the degrees 9, 6,... in m sets of n variables x’, y’,..., can be expanded 
in one and only one way ina series of ordered forms of the first resp. of 
the second kind. 
Examples: 
The 6-linear form 
6 
fm = m).. oie Xi. Ke 
6 
can be decomposed into 76 ordered elementary forms of the first 
kind corresponding with the affinors. 
6 
1) 6Ant oM! m 
N : 6 
2,..+; 6) 5A0@Me om, As=1,..., 5 
Dee ea Shoe dan eh le me 
{oel Kan em a Peed REN 
26: 2111p BONA eA, ani Aen 
Bloem AO) snare Ed am tude ailing eae an 
AT 50) AE Mae mn le a 
NEET 
