CHAPTER III. 
CONTINUOUS GROUPS OF COLLINEATIONS. 
Theory of Continuous Groups of Collineations. 
Resultant of Two Collineations; Gs. 
Analytic Conditions for a Sub-group of Gs. 
Groups of Type I Defined by Linear and Quadratic Relations. 
Groups of Other Types Defined by Linear Relations. 
Normal Form of Groups of Type I; k-Relations. 
Fundamental Groups; One-Parameter Groups and Path-Curves. 
Groups of Perspective Collineations. 
Groups of Types I, II, and III; Table of Groups. 
Groups of Real Collineations. 
Exercises. 
Un UR UR LR LR UP U2 SH SR Gr 
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159. The present chapter is devoted to the theory of con- 
tinuous groups of plane collineations and the determination 
of all essentially distinct varieties of such groups. We shall 
also investigate the chief properties of those groups and 
classify them according to their characteristic properties. In 
$1 there is developed the fundamental group concept and a 
general method of handling groups of collineations. In $2 
we find in three different ways the resultant of two collinea- 
tions and establish the existence of the general projective group 
G,. The analytic conditions which are necessary and suffi- 
cient to define a sub-group of G, are developed in $3. In $4 
we determine all sub-groups of G, for which the defining re- 
lations are limited to linear and quadratic relations among the 
elements of the matrix M of G,. In $5 we determine all va- 
rieties of sub-groups of G, that are defined by these same 
linear and quadratic relations and the additional relations 
among the elements of M that cause a collineation to degen- 
erate into one of the lower types. The normal form of T is 
used in $6 to further develop the theory of these groups and 
to uncover the fundamentally important k-relations. In $7 
we investigate by means of the normal form the fundamental 
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