1430 
the dilations. Let us denote this group by MZ. ScnornriiEes denotes 
some space groups by placing between broken 
brackets the symbol of the translation group 
used for the formation of the group and sepa- 
rated from it by a comma, the symbol of the 
isomorphous point group. Now the group of 
symmetry operations of the above mentioned 
atom model with 24 electrons will evidently 
aN et be represented by { 7’, 1}. The group contains 
Fig. 2. therefore the symmetry operations I and Il and 
moreover the same symmetry-operations, each multiplied by Pm, 
where for m must be taken each positive or negative whole number. 
Fig. 2 shows possible paths of 12 ‘of the 24 electrons. 
§ 4. The model with 12 electrons. Let us suppose now that only 
12 electrons are circulating with as much tetrahedrical symmetry 
as is possible. Of the above 24 electrons each pair must then coin- 
cide. In this case an electron must be able to cross the boundary 
of its domain, but for the trivial case that it is always moving in 
a &. When only space symmetry existed this crossing would be 
impossible. At the moment of the crossing of a 8 the two electrons 
coincide, but the velocity of one of them is symmetrical to that of 
the other one with respect to the plane. The collision caused by 
such velocities might be avoided when one electron reverted in its 
path. From comm. Nr. 7a this evidently happens, when the reflection 
in the 8 is accompanied by a retroduction 9 or a reversal dilation 
D viz. a retroduction M multiplied by a dilation Y. 
For causing each two electrons to coincide we have only to choose 
for the symmetry moment of the retroduction that of the crossing 
of one of the six s’s. We may form the group of the symmetry 
operations for this case in the following way. Replace II from 
§ 2 by: 
MEg MUE | MBE MWE 
HIT < MAS = MA'Sy MA"Ey MUN ay 
MU Ez 2) Melee coe Sd MA" Eq 
number of symmetry operations of a group equals that of the particles. In the 
group in question however each operation transforms a particle into the same 
particle at a different moment. By this the number of particles is no longer equal 
to that of the operations in the group. This last number is equal now to that of 
the positions of all the particles required for the definition of the model. 
