263 
pt ci a; 
operator M; resp. A; we may easily calculate that ——-—~— 7 
viyj B Bj 
is the coefficient of A, in Ae MY and MD A™. The coefficient of 
Kin A; M; being one, this involves that 
di: Vi; Jj; B Pea 7 
ey ely = IN EE ĳ bj EE a le) 
p! ay aj 
Thus the expansion with regard to elementary affinors of the first 
kind is e.g. for p—=6: 
6 bb 0) 1,9 0) Lab yyy 
BAM AS ZA MH ES AM 42S A) ye 
110 (a) (a) acer (a) (a) 1a) (a) (a) 
de ede KANS ALM 4) Ne. (40) 
oS Ay a 12S gy 4e gy TN | 
+ 2 = A, M, re B aoe A, M, en = Ay, / 2 a ) u 
When an expansion of an affinor u is given with regard to ordered 
alternations or mixings, each of which is alternated by every higher 
alternation resp. mixing, then this expansion is identical with the 
indicated one. For on application the operator ¢;; AMM) resp. 
ej MM) A) all terms are annihilated except those which are 
derived from A™ resp. M;“. This one term only remains unaltered 
and is therefore equal to &; A‘) Mon resp. 6 MAO. The indi- 
cated expansion is therefore singly determined. 
So we have obtained the following principal theorem. 
Principal theorem C. Every affinor can be written in one and only 
one way as a sum of ordered alternations or mixings that are annihilated 
by every alternation resp. mixing with a higher permutation number. 
Expansion of an affinor with regard to reduceable covariants of 
different degree. 
When 4; and M; are conjugate, then for n >>5 it may be very 
well possible, that a general mixing lower than M; corresponds to a 
general alternation lower than A;. When however Arin Ai, then 
every general alternation lower than A; is of the form (2—8).2, 8.8/4: - 
The permutation number (@ + p —an), (n—1).e@ is conjugate to «.n 
and the number (« —8 +¢-+ p—(a—,)n—s,—.. .—S;,), ($y—1), 
(a—8 + 9, ...,(m—s,).(a—B) to (a—p).n,s,, .. ‚st. The second number 
