82 THEORY OF COLLINEATIONS. 
98. Invariant Lines and Points. The twoconics K and K’ 
touching the line / will have generally three other common 
tangents designated by a,b,c, Fig. 12. We shall next 
examine the relation of these lines to the transformation de- 
termined by the two conics K and K’. Let us take a point X 
on the line ¢, and draw two tangents from it to the conic K ; 
one of these tangents is the line ¢ which cuts | at C’; the 
other cuts / at some point as Q’. The corresponding point to 
X is found by drawing tangents to K’ from C’ and Q’. One 
of these tangents is again the linec; the tangent from Q’ to 
kK’ intersects ¢ in X’ the corresponding point to X. In like 
manner every point on the line c is transformed into a point 
on the line ¢; in other words, the line ¢ is an invariant line of 
the transformation. Similarly the lines a and 6b are inva- 
riant lines of the transformation. The fixed line / is not an 
invariant line, although a common tangent to the two conics. 
The points A, B, C, which are the intersections of the in- 
variant lines a, b, c, are invariant points of the transformation. 
This is evident from the fact that the two tangents from A 
to K are the lines b and c; the two tangents from C’ and B’ 
to K’ are also 6 and c, which intersect at A, the starting point. 
Thus A is a self-corresponding point of the transformation ; 
the same is true of B and C. 
It is easy to see from the construction that these three 
lines are the only ones left invariant by the transformation 
determined by K and Kk’. In particular cases where the 
conics K and K’ are especially related to one another, e. g., 
touch one another, the invariant figure may be different. 
These special cases will be determined later. 
THEOREM 12. ‘The three common tangents, other than J, to the 
two conies K and K’ are invariant lines, and their three points of 
intersection are invariant points of the transformation. 
99. «* Different Constructions of the Same Collineation. 
There are ~* conics touching the three invariant lines a, b, ¢ 
of the collineation T. From these * conics may be formed 
