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Polar. operators. Let be p==q...=1. As 
’ d x’ ’ 
._— SN, 
‘ dx’ 8 
we have 
,’ d x’ ’ (x’F 1 ) , 
sa P cf S00 SSS ror ss re 
y dx’ y Q YY 
and 
re deep P | . 
a (y . 5) m.xPy?...==m. (xr + yr = 
o! dx P P 
is . 
KEN EE 
P 
ei, dV . ogee 
Ba es therefore the th polar operator of x with 
Q! 
P 
respect to y’. By application of this operator the form #m changes 
into a form with the sets of variables x’°~', y’t. The corresponding 
P P 
affinor of this form is no longer m, but is derived from m by 
application of an operator ,4;/, the permutation region of which 
P 
contains the ideal factors of m corresponding to y'+. Applica- 
tion of this operator is therefore equivalent with application of 
*+i Mf combined with a change of the sets of variables. 
SAPELLI's operators HS). Let again be p=q=...1 and let us 
P 
call the sets of variables Fm x’,,...,x’, and the corresponding 
exponents @,,-.---.,@ ; 50 that @, Ht. +0, =F, then the diffe- 
rential operator H‘) introduced by Capri is written in our notation: 
Que HO 
s U is 
m ‘ Xe 
where the summation has to be extended over al ( ) combinations 
8 
P 
of s of the numbers 1,....,m. By application of H® to Fm we 
find: 
P 
Hm or’ = HO (a, x,’ … (am Km) = x8 8! BCT Wi, )- 
P s 
0 | 
: EN ; = NPs) EN sie p . 
Oxi,’ . Jee “Ox; ‚wi, - Xi’) sj (Uy Xj, ) oe eas (u; _. a xj) Jen 
where’ 9, Imes are the indices of 1,2) nete de mot belong 
. . cr 5 m . 
to 2,,...,t%;. The summation has to be extended over all ( ) possible 
8 
combinations. 
