1425 
§ 8. The AA’s of two and more W's. Regarding time-A’s as special 
cases of As, we see from the end of § 4, that we can find all 
kinds of time-A’s by only studying AA’s of M's. 
The applicability of the complex symmetry-operation of IN, and 
M, (symbol P, name ‘dilation’, symbol and name of the sy mmetry- 
element p and “period”) to a system of particles, means that when 
at a moment ¢ a particle A is at the point / there are also particles 
B resp. C at P at the moments 2m,—2m,-+ ¢ and 2m,—2m, + ¢ 
respectively. In this case P is every time occupied by a particle 
after a lapse of time, 2(m,—m,). When the number of particles at 
our disposition is not infinite, the same particle A must necessarily 
at the end arrive at P again. Moreover, each particle, when arriving 
at P must have the same velocity and the same direction of motion, 
which will become evident, when we consider the state at moments 
2(m,—m,) +¢-+ At. All particles are thus distributed in unequal 
numbers over differently shaped closed paths in which they circulate 
with phase-differences, that are the same for the different paths and 
also for the different particles in one and the same path. The times 
of revolution in two paths are proportional to the numbers of the 
circulating particles. 
: an 
A AA of a 9 and a rotation through — about a n-al symme- 
n 
try-axis was already used in Comm. n°. 4 Le. We shall call its 
symmetry-element „-al time-axis, the symmetry-operation time-rota- 
tion ') (symbol WW). 
The complex operation of %,, M, and M, (symbol H, name reversal- 
dilation) is a symmetry-operation of a system of particles, when it 
fulfills this condition: When at the moment ¢ the point Pis occupied 
by a particle A, we shall find there particles B, C ete. at the 
moments: j 
— Jm, 2m, 2m —t, 2m,— 2m,+2m,—?¢ and 2m,+2m,—2m,— 1. 
In the first place we have therefore three symmetry moments. 
At those moments all particles must therefore return in their paths. 
As this must happen at more than one moment each particle oscil- 
lates in a different path of arbitrary form, while the moments of 
returning are the same for all paths. It is evident that in each 
path one particle only can circulate now. Secondly there evidently 
exists a period. To find it the following considerations will be of 
use: When to a moment ¢ we apply the order MMM, and to the 
1) The distinction we must make here between the two possible combinations 
of sense of rotation and sense of dilation is analogous to that which Scroenrues 
and his predecessors made between left- and right-handed screws. 
JA 
