MIXED GROUPS. 281 
THEOREM 7. The mixed group ae ae + contains, besides the 
continuous group ue ay . o* collineations of type I, <*of type 
ILand «* of type IV, which interchange vate and {@l. 
330. Collineations which Permute the Vertices of a Tri- 
angle. Let P, Q, R be the vertices of a triangle; we wish to 
find all collineations which change P intu Q, @ into Rand R 
into P; also, their inverses, viz.: those that change P into R, 
R into Q and Q into P. Let K be any conic circumscribing 
the triangle; with PQR as the triangle of reference, the ho- 
mogeneous equation of K may be written 
$45 +S=0. fo 
Let w’, y’, 2’ be the coordinates of any point A; the polar of 
A with respect to the triangle of reference is 
poe go ee Oe (12) 
The polar of A with respect to the conic K is given by 
a ( bz’ + cy’) + y(az’+ cx’) +2(ay’+ bx’) = 0. (13) 
We wish to determine the point «’, y’, 2’, so that its polars 
with respect to the triangle (PQR) ‘and the conic K coincide. 
Comparing equations (12) and (13), we find w’: y’: 2’ =a: bie. 
When the point A is not on a side of the triangle of refer- 
ence, we find one, and only one, position of A such that its 
polar with respect to the triangle is at the same time its polar 
with respect to the conic K. The converse of this proposition 
is also true; if we choose any point A and take its polar / with 
respect to the triangle of reference, we shall find one, and 
only one, conic K circumscribing the triangle for which A and 
l are pole and polar. 
The line / and conic K, whose equations are respectively 
4 +$+2=0 and 4+—+==0, 
intersect in a pair of points A’and A”. (AA’A”) is the in- 
variant triangle of a two-parameter group G,(AA’A’’); the 
path-curves of its one-parameter subgroup G,(AA’A”), _, 
