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cause that permutation of two numbers G;;, which agree in the first 
or in the second indices, does not yield a different manner of calcu- 
lation of G. If a, and 6, e.g. are prime to each other and a, and 
6, likewise, it will be immaterial whether we first determine G,, 
and then G',, or vice versa, both greatest common divisors being 1 
and a division by G,, or G,, consequently not causing any alteration 
in the numbers. But we shall disregard such special cases, so that 
we only call two ways of calculation of G identical, if they are the 
same with any set of values of a,,...,@n,0,,..., On. 
4. The number G inquired into in N°. 2 then becomes equal to 
the number of ways, in which we can arrange the mn symbols 
CRO CO Er 7) 
in a sequence, if we consider two sequences identical, if these can 
be derived from each other by some permutations of two symbols, 
differing both in the first and second indices. 
5. If mm == 2 the number asked for will be the number A, of the 
ways, in which we can arrange the symbols 
Gia; Oi Mara: sG tone Re ee 
Gois Goos raa oars Gone ee eee 
in one sequence, only paying regard to the order of two of these 
symbols if in the above scheme they stand in the same row or in 
the same column. The number A, is the product of n! and the 
number B, of the arrangements, in which the symbols (2) follow each 
other in the manner indicated by (2). 
6. We can divide the B, arrangements into m groups, thus, that 
the zt group contains those arrangements, in which Gs; precedes 
the other symbols (3). That #* group can be subdivided into two — 
parts, the first part containing the arrangements in which Gs; precedes 
Gis, the second the arrangements in which Gs; comes after (4;. 
The first part of the z® group contains as many arrangements as 
when Gi; and Gs; are cancelled, hence B, 4 arrangements. 
In the arrangements occurring in the second part of the 2" group 
the symbols G1, Gie,...., Gi; precede all symbols (3), so that we 
may cancel the symbols G44, Giz,...., Gi. If we only pay attention 
to the arrangement of the symbols (721, Goe,...., Gaj mutually 
and towards the symbols . 
Ga pi) Gonars sn Gans hy se en ee oD 
; : es 
(considered as identical) we obtain En possible arrangements. 
n— 1) 
