281 
Wabeke sede AMS m, Benen 
62,...,70) ssd emd om, 21e 
A Mn 
76) 041 SM Fi 
for n = 5, 1 becomes zero, forn—4:1,...., G"fOk im = oe oe 
25 and for n=2: 1, ...., 56. The expansion in elementary 
6 
forms corresponds to an expansion of m which is found from 
the preceding one by taking together the horizontal rows 1; 
2,.-..,6; 7...,15; etc. From this can be deduced again ‘the 
expansion in non-reducible covariants. For n > 6 : 1..., 
BENTO Ore ee TO Or Woe. ee Oe. TY eee, 76: for 
a ot AO Oe 76; for AO Pe”: 30; ee 
Nei FO aS EAA Neat Ole an re TO dn EEN 
75; 76. As to the expansion of a form of the sixth degree in a 
number of sets of variables less than 6 e.g. 
6 9 9 2 12 2 9 
Fu=nj fiem3 Xix2xs 
6 
we may remark, that x’,’x’,?x’’, can be obtained by a definite 
2 
simple mixing *°M,®. Hence, in the expansion Hee all ordered 
elementary operators of the first kind vanish, when their first factor 
is an alternation that is annihilated by 3?1/,@. In the first place 
fuse to... 25:-0f 26—30 remains one term, of 31—46 there 
remain nine terms, among which three different ones, of 47—56 
four equal terms, of 57—60 two equal terms, of 61—69 six terms, 
among which three different ones, of 70-—75 four terms, among 
which two different ones, while 76 remains. In total there remain 
therefore for n<3 twelve terms and for n = 2 seven terms. This 
last number gives also the number of terms in the expansion of 
W AELSCH '). 
Expansion of a form in m sets of n variables of arbitrary degree. 
P 
Principal theorem D. Every algebraic form Fm, homogeneous 
; a : : Po 
and of the degrees 0,¢,... in m different sets of variables ®’,y’,... 
can be expanded in one and only one way in a series of ordered 
elementary forms of the first resp. of the second kind. 
') Comp. p. 278. 
