276 
The operator 22. For m=n: 
dt nd 
nl E » (a Re e=) 
ag.” dy, 
) 
‘(2-Prozesz ). According to the 
\ 
is the well-known operator { 
preceding we have: 
and the application of (? is therefore equivalent to the application 
P : . do : . 
of n! (5) „A combined with a division by the determinant of the 
sets of variables. We may therefore also say that a non reducible 
form in m sets of » variables is a form that is annihilated by 
sE ven: 
4 . . e ae 5 ee x; 
$'). A form in ” sets of m variables containing a factor ae 
can never be non-reducible. In fact, the corresponding affinor 
P 
m possesses a linear covariant of degree /—n. Such a form is 
therefore not annihilated by the operator £2.) 
Expansion in series of a form in sets of two variables. 
P P 
Let Fm be a form in m sets of 2 variables and m the corre- 
; P 
sponding affinor. As n= 2, the expansion in series of m with regard 
to elementary affinors is identical with that with regard to non- 
reducible covariants*). When we apply this expansion, we find 
iP 
for Fm the expansion: 
BN AON /m F 
= or m even 
P mn Vo WP ; 7 
’ me 
een he „am. Xf OR BL 
BIP Ae P [ml 
Be ad 
a 
where each term is a sum of products of one single non-redu- 
cible form with a certain number of determinants of the form 
Va hy @ Or shortly (@ y’) as written usually (Klammerfactoren), 
that is characteristic of that term. That such an expansion is pos- 
sible and singly determined, has first been proved by Gorpan. 
For the special case that there are only two sets of variables an 
application of permutation laws gives 
1) Comp. E, p. 264. 
3) Comp. p. 279. 
8) Comp. E, p. 265. 
