157 
6 
iT he’ hef hehe, =e IPH BPH (63) 
and 
Hel) ASD gu op 
Oae Ope = (— ie 2 g signe ae Pah (64) 
Hence the mean value of all quantities o,, Gg, Oxe Gee iS equal to 
gant ASD su en 
Oue ooo |) 2 2 g a JR (Hel), (65) 
in which AE is the mean value of all quantities EE 
In order to transform this expression, we consider the ideal quantity 
2u 
P= ((Pa.7..'a)(@b.~.-'b), (66) 
in which the a and b are a ideal factors of *g: 
vS ra Da == boh ij Obe. (67) 
This quantity arises from 
ar Ob (68) 
by replacing two definite systems of a, resp. 8 ideal factors by their 
alternating product. Such an operation is called a sample alternation *), 
In the general case a simple alternation nee replaces ¢ definite 
systems of s,,5,,..., 8, factors by their alternating product. s,,...,s 
is called the permutation-number of the alternation. Every alternation 
is apparently a multiple sum of permutations. The alternations with 
different permutation-number can be ordered according to their 
permutation-number and then get a suffix on the right. So e.g. we 
have for 2u =6: 
pA= Ai, 2A =Ae 5.224 = Aan 394 == Aas 34 == As , 32d = Ag 
23A— Ar ; A= As ; 429A = Ag 5 se ET ; GA = Aj} 
The number of the different possible permutation-numbers. be 4. 
For 2u= 6 we have therefore £ = 11. The different alternations with 
the same permutation-number can be distinguished by a suffix high 
on the right. So there are e.g. for 2u —6 15 different alternations A, : 
(69) 
The sum of all alternations A, with the same permutation-number 
divided by its number, is called the general alternation A,. Now 
we can show that there are & operators ,/ such that 
dl ifusv 
AFF (70) 
and 
| 1) For the development of. the theory of these operators and their application 
to the expansion of quantities and*forms in series see 19, 3, English 19, 4. 
