260 THEORY OF COLLINEATIONS. 
riant, viz.: hG,(K) and eG,(K). In the first case the conic 
K is real and eG,(K) contains only hyperbolic collineations of 
type I. In the second case the conic is imaginary, having no 
real points, and eG,( K) contains only elliptic collineations of 
type I. 
THEOREM 63. The real groups G,( AA’), G@,(/UV), G@s( A), and 
G@,(AA/’A”) exist in two varieties each, viz.: hyperbolic and elliptic. 
314. Real Collineations of Types II, IT, IV und V. A 
real collineation of type II leaves invariant a figure (AA’/) 
real in all of its parts. There is, therefore, only one variety 
of collineations of this type. Type II appears as the parabolic 
ease between the hyperbolic and elliptic cases of type I. 
A real collineation of type III leaves invariant a real lineal 
element Al and a real pencil of conics S. This type appears 
as the parabolic case between the hyperbolic and elliptic cases 
of type I when the path-curves are conics. 
A real perspective collineation of type IV or V leaves inva- 
riant a real axis, vertex and pencil of lines. 
A list of the real groups of types II, III, IV and V, is iden- 
tical with the list given on page 254, of the present chapter. 
Exercises on Chapter III. 
1. Give several examples of systems of collineations which 
possess the first group property but not the second. 
2. Give examples of systems of collineations that possess 
the second group property but not the first. 
3. Discuss in detail the properties of H,(A,/), the funda- 
mental group of type IV. 
4, Discuss in detail the properties of H,’( Al), the funda- 
mental group of type V. 
5. Show that through a given point P of the plane (not a 
vertex of the triangle AA’A’’) there passes one and only one 
path-curve of the group G,(AA’A”’),. 
