TYPES OF COLLINEATIONS. 95 
C invariant, is an invariant line. Thus we see that every line 
through C is an invariant line. Hence we conclude that the 
collineation determined by the two conics K and K’ touching 
the line / at the same point leaves the point C, all points of 
the line J, and all lines through C invariant. If the two re- 
maining points of intersection of K and Kk’ are coincident, 
i. e., if the two conics have double contact, the resulting col- 
lineation is still of the same character and the invariant figure 
the same. This case is type IV. 
122. Type V. When the two conics have a contact of the 
second order at the point LZ on the line /, the invariant figure 
takes still another form. In this case only one other common 
tangent, a, can be drawn to the two conics. This common 
tangent intersects / at A (Fig. 15d). The collineation deter- 
mined by the two conics in this position leaves invariant all 
points on the line / and all lines through A. If the two conics 
have contact of the third order at L, then / is the only com- 
mon tangent they have (Fig. 15e). Sucha collineation leaves 
invariant every point of the line / and every line through L. 
The invariant figure is the same as before. This constitutes 
type V. 
123. Perspective Collineations. Types IV and V constitute 
what are known as perspective collineations. In article 87 
we discussed two projective planes in perspective position. 
When the plane z’ is revolved about the line / until it coin- 
cides with a, the resulting collineation in ~ is called a 
perspective collineation. Evidently all points on / are self- 
corresponding or invariant points of the collineation. The 
perpendicular ray from Q on the plane bisecting the angle be- 
tween x and za’ cuts a and a’ in C and C’ respectively. Revo- 
lution about / brings C’ to C, and thus C is an invariant point 
of the collineation. All lines through C are necessarily inva- 
riant lines, for they each pass through two invariant points, 
viz.: C and an invariant point on/. Therefore, the invariant 
figure of a perspective collineation is the same as that of type 
IV. All collineations of type IV are perspective collineations. 
