273 
When now for s’,t’,... are substituted here all possible combinations of 
P 
Gu, ap wae Cu we find that each characteristic number of m is zero. 
Of the proved property we make use by calling an algebraic 
form non special, alternating, symmetrical, locally alternating, sym- 
metrical or permutable, an elementary form or an ordered elementary 
form of the first or second kind, when corresponding names 
are used for the SOE uS affinor. By application of an operator 
m m 
K, A, M, A, u. A, M, Tm ae yee or “L will be understood applica- 
tion of that operator to itie barredpandi affinor of that form. By 
means of the corresponding affinor we are now able to reduce a 
great part of the properties of forms to the formal properties of the 
operators K, A, ete, treated in Zj, which simplifies the treatment 
of forms considerably. 
The characteristic numbers occurring in the linear factors are ideal 
identical with the symbols of AroNnorp and CreBsen. When one of 
) 
p ie 
the sets e.g. x is symmetrical, then u is also symmetrical and both 
may be written as the p-th power of an ideal fundamental element : 
Paes Oy SS? EP = (U )P: 
p p 
Also in this case the occurring characteristic numbers are symbols 
, p 
of ARONHOLD and CuuBscu. When x is alternating, then u too is 
alternating and also in this case both may be written as p-th powers: 
pile pa OP RP. 
p p 
In this latter case the occurring characteristic numbers are identical 
with the complex symbols introduced by Waruscn and WerrzeNBöck, 
P 
the multiplication of which is anticommutative. When x’ and there- 
Pp N 
fore u too is more general, then the notation in the form of powers 
may be still useful sometimes. Then however, the ideal roots x’ and u 
do not determine any longer the isomers of x and u. Both characteristic 
numbers are ideal numbers of complicated character in the products 
of which no commutation whatever is allowed any longer. By 
means of complex symbols WerrzeNBöck *) has proved the first part 
of the principal theorem A for forms in sets of variables that are 
all alternating. The above proof is an extension to forms with sets 
of variables of more general character. 
1) Beweis des ersten Fundamentalsatzes der symbolischen Methode. Sitzungsber. 
der Wiener Akad. 122 (13) 153—168, p. 155 etc. 
