277 
Tele 
«2A ue yo = ——— ernie NT): 
(20)(e) 
so that for this special case the expansion in series becomes: 
rl ee As 0 
5 P—a+1\? 
bais 10 
This expansion of Huf v? remains applicable for n > 2, because, 
there being only two sets of variables, only alternations of the form 
22A give not identically zero. This is the so-called second expansion 
in series of GORDAN |). 
The terms of the expansion with regard to non-reducible covariants 
may now be further deeomposed in different ways. First each operator 
m 
22A can be decomposed into simple mixed alternations. Then an expan- 
OF Vive) KP yet 
a Je 
P 
sion of Fm is obtained with regard to locally alternating forms 
for which in each term the power of a determinant of the variables 
is the same as the in the same term occurring power of the deter- 
minant of the ideal factors of m corresponding to those sets of 
variables. That such an expansion is possible and singly determined 
has been first proved by A. REISsINGER *). 
Secondly each elementary affinor may be decomposed in ordered 
elementary affinors of the first kind. With this decomposition corre- 
sponds an expansion of the form: 
iz On et) zo Es 41 mn 
7 x — 4d 
Fm= = D2 97 Per a, oA | Ops, MO) 8 Bo dab Ae af ") 
g ) P 
The factors of the form (a, y’) occurring in each term satisfy the 
condition that they belong to the permutation regions of a definite 
ordered alternation ,2A acting on xX’ y’? and characteristic of that 
1) Srupy, Methoden zur Theorie der ternären Formen § 3 and § 4. The so-called 
first expansion in series of Gorpan corresponds to an expansion in series of a 
mixed affinor and is not discussed here. 
2) Ausgezeichnete Form der Polaren-Entwicklung eines symbolischen Produktes. 
Progr. Realsch. Kempten 1906 —07. 
3) See Ey p. 263. 
