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of any of those three symmetry-elements; thus the combination of 
a symmetry-centre with a symmetry-plane having always the presence 
of a binary axis perpendicular to that plane as a consequence; and 
a centre combined with an axis of pair period always involving 
the existence of a symmetry-plane perpendicular to that axis. If 
now all hemibedrical and tetartohedrical crystals are considered as 
polyhedra, whose symmetry-groups correspond to complete secondary 
groups of the symmetry-complex of the holohedrical forms of the 
same system, then those secondary groups can be mathematically 
deduced from the primary groups, by suppression of definite sym- 
metry-properties from the primary groups; from a mathematical stand- 
point nothing can be objected to such a way of reasoning; only it 
is necessary to keep always in mind, that from a cristallogenetical 
standpoint the hemi- and tetartohedrical erystalforms have of course 
nothing to do with the holohedrical ones. 
Just because the centre, the plane of symmetry and the binary 
axis perpendicular to it, are always connected two and two in the 
way described before, it follows,: that the deduction of the hemi- 
hedrical and tetartohedrical secondary groups from the holohedrical 
ones, can occur only by simultaneous suppression of two of them, 
in the symmetrygroups of the holohedrical forms. This suppression 
‘an be made in three ways: 
a) So that one or more symmetry-planes + symmetry-centre are 
eliminated. 
6) So that one symmetry-plane + a binary axis perpendicular to 
it are eliminated. 
c) So that a binary axis + the symmetry-centre disappear. 
If now in a holohedrical crystal of any system, S, and S, are 
two secondary RONTGEN-rays, which will be equivalent by symmetry 
to a certain plane |, they will also be symmetrically situated with 
respect to the binary axis perpendicular to the plane J”; S, and 8, 
will moreover always be centrically symmetrical to themselves, 
because every particle of the space-lattice, if reached by the ether- 
motion, will start as a centre of a secondary radiation in all 
directions. 
If now in the holohedrical form of the system we imagine the 
centre of symmetry suppressed, then: 
in the case of a) S, and S, will still remain symmetrically arranged 
to the binary axis, perpendicular to the simultaneously disappearing 
plane; and: 
in the case of c), they will remain symmetrical with respect to 
BU 
