242 THEORY OF COLLINEATIONS. 
292. The Group G,( AlS). Let us select from the sys- 
tem of conics touching / at A any pencil S such that all conics 
of S have contact of the third order with each other at A. 
One property of this pencil of conics is that the polars of a 
point on / with respect to the pencil S coincide, and this com- 
mon polar passes through A. A pencil of conics having con- 
tact of the third order is projectively transformed into a 
pencil of the same kind. We can construct * triangles hav- 
ing one side AA’ along /, one vertex at A, and one vertex at 
A” on one of these conics, so that A’A’’ isa tangent to the 
curve at A’, and AA” is the chord of contact. Belonging to 
each of these triangles isa one-parameter group G,(AA’A’’),_, 
such that one conic of the pencil S is included in its family of 
path-curves. Every collineation in such a one-parameter 
group leaves invariant one of the conics of S and interchanges 
the other conics of S; hence, it leaves invariant the pencil S 
as a whole and the lineal element Al. The aggregate of all 
collineations, leaving S invariant, forms a three-parameter 
group G,(AlS). 
Since the group G,(A/S) leaves invariant the lineal element 
Al, the pencil of conics S having contact of the third 
order at A, we see that G,(AJS) can be built up out of ~’ 
two-parameter groups G,(Al/K), one for each conic in S. 
Again, since the polars of any point A’ on / with respect to 
each conic of S coincide inl’, a line through A, we see that 
G,( AIS) is composed of ~’ subgroups G,(A’Al’),_.. 
THEOREM 54. All collineations leaving invariant the configu- 
ration consisting of a pencil of conics S having contact of the third 
order with each other, and their common point 4 and line /, forma 
three-parameter group G;( A/S). 
293. Analytic Determination of G,( AlS) and G,( AIK). 
The group G,(Al) is reducible, hence the point A may be 
taken as one vertex of the triangle of reference, as (0, 0, 1), 
and the line / as an adjacent side, as y= 0 of the triangle of 
reference. We may therefore put A= B= B’=0 in the 
normal form of 7, and thus get the reduced normal form 
