MIXED GROUPS. 279 
eight only three may be the invariant figures of mixed groups, 
viz.: (AA’), (ll’) and (AA’A”), for these are the only fig- 
ures in which points or lines may be interchanged without 
changing the figures as a whole. 
328. The Mixed Group mG.(APQ). Let A, P,Q be any 
three points forming a triangle. Any collineation that inter- 
changes P and Q must leave the line PQ invariant. Let 
(AA’A”’) be the invariant triangle of a collineation of type 
I, which interchanges P and @; then two of the vertices, as 
A’ and A”, must be on the line PQ and so situated that the 
cross-ratio (A’A’’PQ) = — 1. All collineations in the group 
G,(AA’A’’), for which the cross-ratio along the side A’A” is 
—1, leave A invariant and interchange P and Q; they be- 
long, therefore, to the mixed group mG,(APQ). 
Let & and k’ be the two independent parameters of the con- 
tinuous group G,(AA’A’’). Since the product of the three 
cross-ratios in the same order around the triangle must be unity, 
we have (k) (— 1)(1/k’) =1; thus, k+k’=0. Hence, out 
of the ~’ collineations in G,(AA’A”), where (A’A”PQ)=—1, 
there are ~/ that satisfy the condition k + k’ = 0 and inter- 
change Pand Q. The pair of points A’A” can be chosen in 
co! different ways, so that (A’A’” PQ) = — 1; if out of each 
of these groups G,(AA’A’’) we select the collineations that 
satisfy the relation k + k’ = 0, we obtain ~* collineations of 
type I that leave A invariant and interchange P and Q. 
Among these ~* collineations of type I interchanging P and 
Q, there are ~‘of type IV. In every group G,(AA’A”) there 
are two collineations, viz.: k = 1, k’ = — 1and k= — 1, k’/=1 
which satisfy the condition k +k’ = 0 and are not of type I. 
They are involutoric perspective collineations with the vertex 
always on the line PQ. 
Since no collineation of type III or V is involutorie along 
an invariant line, it follows that the mixed group m G,(A PQ) 
contains no collineations of these types. A collineation of 
type II, whose invariant figure is (AA‘/’), may be involutoric 
along AA’, but cannot belong to the mixed group m G.(A PQ), 
