275 
As 
ena n=(5 eee 8 
Fa gear sak!) wir 
we have 
Pp. 2 
A, Fm =! 9;,.-- i, SWN). (ai, ai, oi, Ws, PO 
; s l 
p. —1 NES ] 
(uy, Ws, Pig OKT (Oy, Ki) me 
As further 
= Pi, +1 Pn 
0. p. 
PA Pe A ty 
uz, V, Uijl ---Un ; 
| 
where 
— ee A 
Menn ey cued 
© e 
: ; : : Fi ien 
when the permutation region of ‚A contains of u,',..,u,* just 
dè 
Ss 
the last factor of each, we have also: 
Pp On ee Oi, 
A= war 
s 
P 
=, Am 
m ; 
where the summation has to be extended over all ( ) essentially 
: 8 
different permutation regions. This infers: 
P P MR 
A’) F m= (=) s! AF .m 
8 
viz. the CAPRLLIAN operator H is identical with the operator 
Ë = ates 
( \ sted The linear independency of the operators HS discovered 
8 
by Caprnii and their commutativity mutually and with other opera- 
tors composed of polar operators, is therefore a special case of 
the linear independency of the operators A, proved in #,, and their 
commutativity mutually and with all operators M,. A, M, K and 
mm ( 
P. As the operators A, M, A, M, dp K may all be written as 
sums of multiples of products of operators A,sail,....,n and 
the identical operator /, these operators bave for a form the signi- 
ficance of definite differential operators. When the sets of variables 
are of higher degree, these operators have the signiticance of differ- 
ential operators of more complicate character. The different kinds 
of forms mentioned on p. 273 may thus be distinguished by means 
of the definite differential operators by which they can be obtained 
and the other differential operators by which they are annihilated. 
