1438 
may be represented as well by {%,, W,}*) as by {%,, U, }. Adding 
to the operations M, and M,M, in every combination, we find for 
the second notation 9, for the first one 6 groups, as evidently 
Er, AU, U} and {A,, MW}, and also {MM A, Wy} and (U, MM, AW} 
and also {Mt,W,, Wt, MW’, } and {MM A,, MW} are aequivalent groups. 
Therefore the first notation is preferable. 
Generally both desiderata on the choice of the generative opera- 
tions cannot be fulfilled at the same time. Then it is desirable to 
satisfy both separately and to consider the two notations compara- 
tively. : 
All point groups (between which therefore the 32 classes in ques- 
tion of ScHorneiirs) are then included in the preceding scheme of 
14 kinds of groups for some of which two notations have been 
given (see table p. 1437). 
§ 8. Investigation of the aequivalence of groups that are found. 
Here we ean only give some indications, which together with the 
drawing of a figure or the writing out of the non- aequivalent opera- 
tions of the group (the two last ways of proceeding are most times 
superfluous) are at all events sufficient. 
Ry way of illustration we shall directly apply, these indications 
to one of the groups G for which we shall choose {%,, U}. 
We must try to reduce to one all notations, by which a 
group can appear. When therefore a M,M, that is added to one 
of the generative operations, can be reduced to ® or even to the 
identity, we shall do it. Example: {¥t,2,, M,M,W'}.. In the first 
place we may replace MM, by M,M, de loss of zh 
(let this proceeding never De omitted). Further, (MN, U,)? = M, is 
an operation of the group. When therefore we consider M, As U, 
separately as generative operations of the group we have already 
followed the precept partly. We can however follow it still more 
completely. As namely ®, has now become a generative operation, 
we may substitute Dt, U, for M,M,U',. As then however (MW, W',)* = M, 
is also an operation of the group, we must take as generative ope- 
rations again M', and U, instead of MA. The group {M,A,, 
MMA} is thanefere aequivalent with ie group SM, My, A, Wf. 
When the newly found generative (pure time-) operations had not 
happened to appear in one of the above given forms MW, MM, or 
MM, we should have reduced them to one of those forms or, 
; ul vate 4 
1) In the following Y, means a rotation through En 360°, S a reflection in a 
plane, %, a rotatory reflection. 
