96 THEORY OF COLLINEATIONS. 
124. Elations. When the point Q is any point in the plane 
bisecting the external angle of 2 and 7’, the perpendicular 
from Q on the internal bisecting plane of a and a’ meets that 
plane in a point on /. In this case the invariant point Cis a 
point on J and the invariant pencil of lines through C has 
its vertex on 1. But this is the invariant figure of type V. 
Such a collineation is called an Elation.* All collineations 
of type V are elations ; elation is a special case of perspective 
collineation. 
125. Second Construction of the Five Types. When the two 
conics K and K, of the construction of art. 101 intersect in four 
points S, ABC, Fig. 18, we have a collineation of typeI. If 
the two conics have contact of the first order, as for example 
when A and C coincide, the collineation is of type II and the 
invariant figure is the degenerate triangle AB/, Fig. 16(a). 
If the two conics intersect at S, and have contact of the sec- 
ond order at a point A, the invariant figure is a lineal element 
Al and the collineation is of type I, Fig. 16(b). If 
the two conics K and K, have contact of the first order at S,, 
Fig. 16(c), then the common tangent to K and K, at S, is an 
invariant line, also the lines joining S, to the other two points 
of intersection are invariant lines. Thus we have three in- 
variant lines through S, and one invariant line not through 
S, The collineation is of type IV with vertex at S, and axis 
through the other two points of intersection of K and K,. If 
the two conics have contact of the second or third order at 
S,, Fig. 16(d) or 16(e) respectively, the collineation is of 
type V, as may readily be seen. 
*Lie: Vorlesungen iiber Continuierliche Gruppen, p. 262. 
