282 THEORY OF COLLINEATIONS. 
are conics and K is one of these path-curves. Consequently, 
all collineations of the group G,(AA’A”),__, transform K 
into itself. 
The coordinates of A are (a,b,c) of A’ (a, 0b, wc), of A” 
(a, wb, o¢), Where w?=1. The lines AA’, AA”, AP, AQ, 
AR are given by the following equations respectively: 
x y ray ua & y ahd 
SF SET Sarr OW ETT TE Sane 
we 
a ee 
From these equations we readily find that the cross-ratios of 
the three pencils A (A’A”PQ), A(A’A” QR), A(A’A’RP) are 
equal to each other and each equal to o, 7. e., 6 aa. Conse- 
quently, the collineations of the group G,(AA’A”),__,, 
U 
for which k = e a changes P into Q, Q into R, and R into P. 
The inverse of this collineation, for which k =e = , changes 
Pinto R, R into Q, and Q into P. 
There are ~’ conics circumscribing the triangle PGR; for 
each of these conics there is a point A and a line / which are 
pole and polar with respect to both triangle and conic. Con- 
sequently, there are ~’* two-parameter groups G,(AA’A”), 
each of which contains a pair of inverse collineations that 
permute the vertices of the triangle (PQR). Each of these 
collineations is of order 3. 
THEOREM 8. There are <? collineations of type Land period 3 
that permute the vertices of a triangle. 
331. The Mixed Group mG,(AA’'A”). The mixed group 
m G,(AA’'A’’) is composed of all collineations which leave the 
triangle (A A’A”’) invariant. These consist of the ~? colline- 
ations belonging to the continuous group G,(AA’A”’), the 
o* eollineations of type I which leave one vertex fixed and 
interchange the other two, the ~” collineations of type I and 
period 3 which permute the vertices of the triangle, and of 
the -/ collineations of type IV which leave one vertex inva- 
riant and interchange the other two. 
