1440 
In the case of {M,M, A, MM, A’,} the operations (MN, M,A,)? = (M,M,)*, 
and in the same way (2,,)*, but also (MMA, (MN,m,W',)—)? = 
= (MM, MMU jr AN, ws, MM) = MM)? belong to the 
group. Now these three dilations must give together a G.C.M. of 
their periods, by which the new dilation WM, is determined. Then 
we must have (IN, M,)? = (MWE, (MM)? — (MM) and (MM) — 
= (MN, M,)", (4, Ll and m integers). This involves Mt, IM, = (MW,)4/3, 
MM, = (MM, )!/F and MM, (DM)? so that MMM, MN, = 
= (MM, )//F+-m/2 — MD, = (MMS or 1/3 + m/2— k/3 and l=k+3n 
(n an integer). Taking the newly found dilation as a generative opera- 
tion the group becomes therefore {DM M,, MN, M,) 4/3 AU, (MIM, )" +43 A’. 
In the factors of A, and U, we may of course omit powers of 
M,M, with integer exponents, so that ” vanishes and we can choose 
for k 1 and 2 only. Here (IN,M,)?/5 A, means nothing but AN W,)—/3 A, 
the time rotation with period opposite to that of QN,I",)1/3U,. When 
besides in order to avoid fractions we substitute (M, M,) for M,M,, 
we obtain {M, M,)*, (MN, Me, )LtA,, WI, WM, ji W441, where +--+ means 
that in the exponents only the + signs or only the — signs may 
be combined. 
Other time operations that must be attended to will not easily 
be found beside those above mentioned. Here we have a case, that 
we consider the figure using at the same time the new form for 
the symbol of the group. Then it is evident, that a simple dilation . 
is allowed. 
In the investigation of the following groups the new symbols for the 
groups obtained until now can be used. (Example: SON, M,, MM, W,, W’,}). 
Not only we attend then to the groups that have been discussed 
already and to which a generative time operation is joined, but 
also to the groups treated before that are obtained by joining that 
time operation to each of the generative operations of the first group. 
Finally we may draw the attention to a way of proceeding, which 
gives us easily a new generative time operation when we are joining 
a generative time operation. When f.i. M, is joined to {M,2,, M, U}, 
so that we obtain {M,,Dt,%,, 7, W',}, then according to the above we 
consider also {M,, M, Mt, A,, Wt, A}. However not only M,W,A,, but 
also MMU, and therefore (MMA) — UMM, is an opera- 
tion of the group, and so M,M,U, AM, M, — (MM)? too. In the 
same way of course (9, M,). Further we then proceed in a similar 
way as has been indicated above for {,M,U, Me, Me, W’,}. 
$ 9. Final remarks. With the aid of the preceding we can find 
in a rather simple way the 167 kinds of groups of space-time 
