256 
Expansion of an affinor in alternations or mixings. 
Thus we have especially for the sum of all classes M/, and for A; : 
ee iof ae 
My = — & yi Ai 
3 
ee ee =» (2) 
A7 Vki M; 
i 
By the application of MM, to the upper equation we learn that 
Yik = 1, so that when we consider My and A, resp. as o-th general 
alternation, A, and as o-th general mixing M, we have, / being 
the identical operator, and because A, = M,= 1: 
Cee) ea Ol ke ca 
= = Vii Aji > yu Mi. ee ae © SR (35) 
1 t 
In the same way we can prove: 
: m 0, igh 
U == >3 Yk A= = ee MIEREN ee ee ap (14) 
a 
For a permutation number s we have in these expansions : 
vei == (Ii (p—s+)). « - . + (15) 
and for a permutation number a. 2: 
ve) 
Thus we have obtained the theorem: 
Principal theorem A: Every non special affinor can be expanded in 
one and only one way into general alternations, into general mixed 
alternations, into general mixings and into general alternated mixings 
with the indices 0,2,3....,k. 
So we deduce e.g. from table (10) for p= 6: 
(16) 
6 a — En == = En == = 
i= ( Ae eae 4102 Ae Gare (45> ye 6 alee oa, aa “ail 
ao Sees A, eeN), u (17) 
6 = ee on el ume = as = 
u=(‘M,,—2'M,,—2**M,+3'M,—**M,+6*M,—4°7,+ MM, — 
En =e b= iG 
—6*°M,+5°M, HM) u 
Decomposing the ordinary alternations and mixings into mixed 
resp. alternated ones we can calculate: 
