GEOMETRIC METHOD. 167 
variant triangles have a vertex and the opposite side in com- 
mon. Thus we see that G, contains © equivalent four-par- 
ameter sub-groups G,(A, 1’); these groups are self-dualistie. 
203. The Group G,(AA'l’'). Triangles ‘in number can 
be arranged with two of their vertices at A and A’ and their 
third vertex on a line Al’. Each of these triangles is the in- 
variant triangle of a two-parameter group G,(AA’A’’); these 
co! two-parameter groups unite to form a three-parameter 
group G,(AA’L’). 
The group of all collineations in the plane G, contains ©’ 
equivalent subgroups G,(AA/l’); a group of this kind is self- 
dualistic. The group G,(AA’) contains ©‘ subgroups of the 
kind G,(AA/l’), one for each line through A; also one such 
group for each line through A’. The group G,(/l’) contains 
co! subgroups of the kind G,(AA’l’), one for each point on 1’, 
and also one for each point on /. 
204. The Group G,(K). A collineation transforms a conic 
into a conic. Since there are ~’ conics and ©’ collineations 
in the plane, we infer that each conic of the plane may be 
transformed into itself in ©*’ ways. These ©* collineations 
leaving the conic K invariant forma group G,(K). Any col- 
lineation of the group transforms the points on K into points 
on K and tangents to K into tangents to K. Also a lineal 
element of the conic K, consisting of a point on K and the 
tangent to K at the point, is transformed by T into a lineal 
element of K. In particular, if a point on K is invariant the 
tangent to K at A is also invariant. 
A conic K and a point A, not on K, may be simultaneously 
invariant under ~‘ collineations ; for there are only ~’ such 
combinations in the plane. We therefore infer the existence 
of a one-parameter group G,(A,K). In like manner we 
should expect to find a one-parameter group G,(/, K), leaving 
invariant a line / and a conic K. When Ajand K are both 
invariant and A is not on K, the two tangents from A to K 
are both invariant lines; hence their points of contact are 
