1439 
if impossible, we should be already in the above mentioned case 
that all electrons are at rest, so that we had to omit all added time 
operations. 
From this example it is evident, that we must try to find out all 
pure time operations that are included in the adding of time opera- 
tions to generative operations of a group G. Evidenty we find them 
all by considering all products, of the generative operations of G 
that are aequivalent to the identity. In most cases these are seen 
on first sight in a sufficient number. In the above example f. 1. 
(U)? = 1 suffices. When however we meet with difficulties we make 
use of the second notation that has been given for each group. In 
this way are obtain the following scheme: 
U Ws} =p, = A, (U), U,} NM, U, Ws} 
Eer Ut = Pe, U, U OPERA Pi LD EO 
RUE IR ME Arse MM, We, Wy, WW} — HM, Me, AU, U} 
SN We WA, Ut = MM, U, U} DM, We, MU, WJM, M,, U, W,} 
ER US A MWE — (DE, Wess UU MM. MU, , We, We, U} IDE, We, Us AU} 
ENH, DOME At == (IDE )?, MM, WE, Me, UW, We, We, U} = (ML, Me)’, 
zel ny > ame ' 5 a1; We NM I, Y 
(MM), (WD APN (De, ME AU (MN, ME y’,} He 
{My Wy, UJ Pt, Wey AU, Ws} 
EN, ML, Ws = EN, Wey, Wy, Ws} De, , WM, MU, Wy} =D, Ms, U, 33 
ED, MA, MUF = (NN, Wy, U} LED, Mes ML, DEU} — DN, MA, Wy 
EN, MMU, U} (MMA, UF | EN, De, MMU, Uj= DN, Me, U, U} 
EN, MA, MM, UJ, We, A, U} | $Me, DM, ADD, U} — (ND, WW} 
FM, DL, ML, ADL, MUD DAA} | $e, Me, We, MA , We, MWe, U LD MAU} 
so that the following 5 groups only remain: 
Ps U} ED, U, U} {OEM A, WL; NM, Wy, A} 
and fi i 
EEM), ORDE A, NM)! Wy 
First we derive all groups, in which no pure time operations are 
added yet to G as generative operations; this simplifies very much 
the derivation of the other groups. Above we saw that everywhere 
MA, must be replaced directly by M and Y%,. As to (NM, M,U,, U} 
we must attend to two things: firstly that (,M,A, == (M, W,)*; 
secondly that (MIM, WU, U', 4)? —= (MMU)? — (M,M,)?, and therefore 
also M M,, are operations of the group, so that the generative operation 
M, M,4, has to be replaced by thegenerative operations M, M, and Y,. 
93 
Proceedings Royal Acad. Amsterdam. Vol. XXIII. 
