254 
m a 
From the operators A and M/ we can form in the same way as 
m a 
above general mixed alternations A and general alternated mixings M. 
As to the order 1,...,%, the independency and the property of 
being ordered, the same laws hold for these operators as for A, 
A, M and M. 
When an A; and a M; with conjugate permutation number do not 
annihilate each other, the unaltered factors of My; are alternated by 
A; and those of A; are mixed by M;, so that: 
m m a 
Aa A 
and likewise: 
A; Mj = Ai Mj = A; Mj = Ai Mj. . oy aH as (7) 
Expansion of the general alternations and mixings in classes. 
Theorem I. A general alternation resp. mixing of a non special 
afinor can be written in one and only one way as the sum of mul- 
tiples of all classes with the same or a lower permutation number. 
For a mixing the coefficients of all classes are positive, for an alter- 
nation those of the even classes positive, the other ones negative. For 
the same permutation uli their absolute values are Gag 
A; == =2 J; aj; K; 
_$+1 for Kj even 
me sb) da =a 1 gaen odd. (8) 
Me = S cij K; 
J 
The very simple proof of this theorem may be omitted. In order 
to determine the coefficients «;j we must investigate in how many 
ways it is possible to choose the ¢ permutation regions s,,..., 5, 
of an A; or of a M; thus, that each of the f regions s,,.…..,s' 
of a permutation of a definite A; falls quite within one of those 
regions. When mij is this number, we have 
mij p! my p! 
(a (P—% p—s, — 81-1 Sil eS ile 
8,/...81! zn : 
fy Ba : 
For p=6 is eg. (see table following page) 
All A, all M and all A’ being linearly independent, we have 
inversely : 
(9) 
Uij — 
\ 
1,...52 
di = By A i 
; pias, up! mec apse aera 
DAE 
Krt M; 
; 
