94 THEORY OF COLLINEATIONS. 
This kind of a collineation may be considered as a special 
ease of the last, when two sides of the invariant triangle co- 
incide. This case gives us type II. 
120. Type III. Instead of simple contact, as in the last 
case, the two conics K and K’ may have contact of the second 
order at a point A. When the conics have a contact of the 
second order, they have one and only one other common tan- 
gent. In this case the common tangent, a, to the two conics 
at A is the only invariant line of the transformation and the 
point A is the only invariant point on the invariant line, 
Fig 15). 
In this case the invariant figure consists of a line a anda 
point A on this line a. This combination of line and point is 
a lineal element. A collineation of this special kind is of type 
III and leaves invariant a lineal element. 
121. Type IV. Again, the two conics may both touch the 
line / at the same point, the contact being of the first order. 
In this case the two conics K and K’ have two other common 
tangents, 6 and a, which intersect at some point C (Fig. 15c). 
It is at once evident from the figure that the transformation 
determined by K and K’ leaves the lines b and a and the point 
Cinvariant. A little further consideration of the construc- 
tion shows that the points A, B, and L on the line! are inva- 
riant points. Consider the point 5A on b infinitesimally near 
to A. From $A the two tangents to K are b and a line infin- 
itesimally near to 1, meeting /at dL. From $Z and A the 
tangents to K’ intersect at 6’A. So that SA is transformed 
to \‘A. AsdA approaches A, 3/A also approaches A. In 
the limit A is an invariant point. Similar constructions hold 
for B and L. 
But if a collineation leaves more than two points of a line in- 
variant it leaves all points on the line invariant (art. 8). There- 
fore, every point on the line/ is an invariant point of the trans- 
formation. Any line g drawn through C intersects / in some 
point as G. Therefore, the line g, having two points G and 
