1423 
Because of the third restriction introduced in § 2 we must suppose 
ls 24, and „a, to equal zero. Substituting this in (6) and (7) we find: 
4224 
Bee eee OO Ee t(D) 
while (6) and (7) are then reduced to: 
ie + 3) sai’ elli 1% 1% 3%, ada Hdi an — 0 | 
nt a sas — | am (9) and 14, 14, + 9%, 44, + 44, 2, = a tie (10) 
rde 1.0,” â Saul | ETE UE FH gy pg. 0 | 
Equations (8) say, that the transformed time depends on the time 
only, and that the transformed space-coordinates are dependent of 
the original ones only. Equations (9) and (10) show, that this last 
transformation is linear orthogonal. By this we have proved the 
proposition stated at the beginning of this §. We need therefore 
only apply equation (5) for values of p‚‚* =O viz. for a pure space- 
transformation and of ¢,*=1 viz. for a pure time-transformation. 
§ 7. Meaning of the cases y,'=0 and yi =1. A RN with 
(mi = 90 is nothing else than a reflection. (Symbol S, symbol of the 
symmetry-plane 8). . 
It might seem interesting to derive all imaginable space-A’s by 
investigating which combinations of G’s when considered as complex 
A, are compatible with the restrictions 1 and 2 of $ 2'). A point 
of consideration could be whether the order of application of the 
reflections in the AA should be chosen arbitrarily or not. In the 
first case’), we find, that each AA may be regarded as a combination 
of those already used by SCHoeNrFLies and predecessors, but we might 
say just as well, that the A’s used by ScroeNrries are but combi- 
nations of ©’s and that there exist combinations, which were not 
treated by him. Proceeding in the indicated way, we find some 
A's that are aequivalent with point- and space-groups of SCHOENELIES. 
After this we might investigate which space-groups can be formed 
from those A’s. 
As however the result of such an investigation has already been 
obtained by ScHoerrLies we shall do better to combine each of his 
1) (Note added during translation). CG. Viota and G. Wurrr partly executed 
such a plan (l.c). 
*) This seems natural by analogy with A's that were known before and is also 
demanded by the principle that around each particle the configuration of the other 
particles is the same. An exception to this last demand is formed by the definition 
of the sense of rotation and translation resp. dilation for a screw resp. time- 
rotation (see further on). 
92 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
