253 
When s,,...,s;, is the permutation number of A;, 4; contains 
de ag aaa 
simple alternations A;. The same holds for mixings. For a non 
special affinor all classes and likewise all simple and general alter- 
nations resp. mixings are linearly independent. 
When successively , 
..,s,4 and "eM are applied to a non 
special affinor, the result is then and only then zero when more 
than one factor of a region s belongs also to a region s’. The 
highest permutation number for which 
rn P 
u 
515-054 
is not always equal to zero is called conjugate to s,,...,s;. From 
this follows that 
sne = (t+ p—s, —...—8;) , (8;—1) .t , (Sti —8) . (t—1),...,(8,-8,).2. (5) 
This relation is a reciprocal one. When the permutation regions 
of an alternation or mixing are numbered in such a way that a 
greater region has always a lower number than a smaller one, it is 
possible, that for all values of « the et? factors of each of the 
regions are placed in the order of this numbering. In this case the 
alternation or mixing is called ordered. To an ordered alternation 
there evidently belongs only one ordered mixing with conjugate 
permutation number, such that these two operators do not annihilate 
each other. Then these two are called conjugate. 
A general alternation and a general mixing with conjugate per- 
mutation number are called conjugate. Every general alternation or 
mixing is annihilated by all simple and general mixings resp. alter- 
nations with a permutation number that is higher than its own 
conjugate one. For p > 5 the order of the general mixings conjugate 
to Ag A is not the same as of Mi, M,, for p==0; 6:8. 
2Ae ’ 2,243 ’ 3.244 ’ 345 ’ 3,2A6 ’ 23A7 ’ ‚As 14,2Ag ’ 5A10 
5 M19 ’ 421 ’ 23M, ’ +Ms ’ 32M ’ 32M, ’ 3Ms ’ 22M ’ 2M: 
Wiens in … …, ,,Au Dhaene Mu the wnaltered factors are replaced 
by the ideal factors of their symmetrical resp. alternating product, 
we obtain a mized alternation resp. an alternated mixing with the 
same permutation number, written: 
m p Sips, A p 
ss 40 resp. Mu. 
1 We 
