264 
is doubtlessly higher than the first, all s,,...... ‚Sn being <n. 
Therefore all general alternations lower than „A; are annihilated 
by all mixings with a permutation number higher than M;. W hen 
thus the whole number of times that p contains is r, the operators 
jl can be arranged in r+ Ll sets. When the sum of operators 
Pp 
in the (a + 1)-th group is ,/’, then ,/’u is a sum of quantities that 
may arise from the application of alternations ‚„Â and that are annihilated 
P 
by every (41nd and by all higher alternations. Each term of ,/’u is 
a penetrating general product of a factors E, E = e, © en, and a not-ideal 
affinor of degree p—an, which is a covariant of u and is evidently 
annihilated by each operator „4. Such an affinor will be called non 
veduceable under the linear homogeneous group and the indicated 
expansion the expansion with regard to linear homogeneous non 
reduceable covariants. 
Now we shall prove that there exists only one expansion 
in non reduceable covariants. For this purpose it suffices 
P a 
to prove, that a penetrating general product r of E and a non 
q 
reduceable affinor y, q =p —an, is annihilated by all operators 
p 
I’ except by ant’. As r can arise from the application of a .,A 
and is therefore annihilated by every mixing with a permu- 
tation number higher than the one conjugate to a.n, this is evident 
for all gl’ for which 8<{e. When 8 >>a@ we may remark that 
al’ is a sum of multiples of operators M; Aj, in which the alter- 
nations always have more than «a permutation regions with m factors. 
The desired proof will thus be given, when we have shown that 
p 
r is annihilated by every operator 2,4, B >a. 
Thereto we make use of the theorem that a non reduceable 
quantity possesses no linear covariants of a lower degree '). By 
p 
means of an operator 2,4 we may derive from r a penetrating 
r 
general product of E? and an affinor w, r= p— Bn. However, 
r q 
w would then be a linear covariant of v of degree r <q. This is 
k 
impossible, so that w is zero. 
ae ; : 
Every term of ./’u is alternating in « different regions of n 
factors and is annihilated by every alternation with an alternation 
1) The proof of this theorem will be given separately in another paper. 
