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rations in question there belong those that are combined with trans- 
lations. Thus we have only to consider rotations, rotatory reflections 
(including reflections and inversions) and combinations of these two 
with retroductions, dilations and reversal dilations. 
§ 3. We need not consider other configurations of space symmetry 
elements than those considered in the theory of space-symmetry. 
Evidently the configuration of the space symmetry elements (including 
the geometrical parts of the s.-t.-A) is a figure at rest, so that when 
one of the s.-t.-A, composed of a s.- and a t.-A, is applied to it, it 
will give a congruent or symmetrical configuration, though at a 
different moment, but also at the same moment. The s.-A alone 
being then a A of the configuration '), this latter must correspond 
with one of those obtained in the theory of space-symmetry ’). 
§ 4. Introduction of the generative operations of a group. That 
the method indicated in communication N°. 7a Le. for the searching 
of all groups of s.-t.-symmetry-operations is valid has been proved in 
the preceding. This method however may be much simplified by 
making use of the generative operations of the point-groups *) viz. of 
operations chosen in such a way, that each operation of the group 
can be considered as a product of some of those generative opera- 
tions (the order in which the factors are taken having influence). 
We shall indicate each group by placing the symbols of the gene- 
rative operations, separated by comma’s, between broken brackets. 
When to each of the generative operations of the group G= 
(U, D,E, we add a time operation (including the identity) we obtain 
the group T= {O, HB, KE. When moreover pure time operations 
are added as generative operations, so that we obtain lr, =;... , 
M, GA, HB, KC}, we need only vary the operations G, , KX, ¥, M... 
in all possible ways to find all groups of s.-t.-symmetry operations, 
that can be derived from G. Now though it is possible that in 
a group IT, which has been found in some other way and which 
1) This holds also for translations. As we have confined the number of nuclei 
to 1, no A’s composed by means of translations can occur in the groups sought for. 
*) It might seem doubtful, whether for time rotations the restriction to those 
through rae radians, remains valid. In fact in these considerations there is no 
objection to time rotations with infinitesimal rotation. This has a meaning only 
when besides the time period is infinitesimal. Then this symmetry element indicates, 
a uniform circular motion. But for the rest this does not change the followin 
considerations. 
5) A. ScHOENFLIES, Krystallsysteme und Krystallstructur, Leipzig 1891. 
