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result of this operation the order M, M, W,, the influences of Mt, 
neutralize each other, so that in fact we have only applied the 
double dilation M,M,M,M, *). Besides the intervals of time 4 (m,—m,) 
between the passages of particles by P, we find the intervals 
4(m,—m,) and 4(m,—m,) too. Now we come into conflict with the 
second restriction of § 2, unless the quantities m,—m, and m,—m, 
have a greatest common measure. This is the time of oscillation. 
We can easily prove that then all demands of § 2 are satisfied. 
By the investigation of AA’s of 9’s and space-A’s we shall find 
ia. that the paths may be closed in the same way as has been 
found for }, but that then half of the paths (chosen in a definite 
way) is described in the opposite direction. 
§ 9. There are no other time A's than ®, YP and A. For all 
AA’s of even numbers of M's the same considerations hold as the 
following for four W's. When at a moment ¢ the particle A is at 
P, and when we have to do with a AA of four Ds we must find at 
P also particles B, Cete. at the moments +2m,+2m,+2m,+ 2m,-++7, 
where the sign + has to be chosen for half of the +-signs, the 
sign — for the other half. This gives therefore more than one dila- 
tion, which together yield however (comp. the considerations on 9) 
only a dilation equal to their greatest common measure, which case 
is already comprised in . 
For all AA’s of uneven numbers of M's we can follow the 
reasoning on the case of 9. Thus this neither gives something new. 
Combinations of time-A’s yield nothing that has not yet been treated. 
§ 10. Symbols for the new symmetry-operations and symmetry 
elements. For shortness sake we shall give names and symbols to 
the s.-t.-A’s and symmetry-elements. As a preliminary system we 
propose the following: 
With a small change now and then we retain the names and 
symbols ot ScHosnriiss. When now a A of ScHornriizs is combined 
with a retroduction the name of the first A might be changed by 
joining to it the prefix reversal. The same may be done with the 
names of the symmetry-elements. Before the symbols of A’s and 
symmetry-elements we add M and m resp. When the change relates 
to a dilation the prefix is “time”, for the symbols this becomes 
1) In the here indicated way the treatment of AA’s of R's (and therefore of 
M's and E's) is much simplified. By applying one of the A's thus found to the 
symmetry-elements of another one we can see whether this brings us into conflict 
with the restrictions of § 2. A AA found in this way evidently forms a group 
of A's. 
