viii PREFACE. 
and practical bearings, are projective transformations or col- 
lineations. 
Lie’s work on the theory of collineations was both syntheti- 
cal and analytical; synthetical in its earliest conception and 
announcement, analytical in its final form as presented to the 
mathematical world in the books published in his later years. 
Lie throughout kept his eye fixed on the properties of the col- 
lineation itself rather than on the effect of the collineation on 
certain configurations of space But it is evident that his 
chief interest in projective transformations was in their group 
properties, and not in those more fundamental properties 
which form the natural basis for a classification both of col- 
lineations and their groups. 
But after all is said the most important and most interest- 
ing properties of collineations are their group properties ; and 
no discussion of the theory of collineations is full and sym- 
metrical which fails to lay the major stress on the considera- 
tion of the collineation groups. ‘The group of projective trans- 
formations, or collineations, is by far the most important of 
the continuous groups discovered by Lie and developed by him 
in his ‘‘ Theorie der Transformationsgruppen.’’ This group lies 
at the very heart and core of his theory for the reason that all 
finite continuous groups ean, by a suitable transformation of 
variables, be shown to be similar in structure to some projec- 
tive group. Therefore every contribution to our knowledge of 
collineations and their groups reacts upon the wider theory of 
all continuous groups. <A transformation of the elements of 
a space is defined as an operation which interchanges among 
themselves the elements of a space, but leaves the space, con- 
sidered as the aggregate of all its elements, unchanged as ¢ 
whole. The operation may be produced by means of a me- 
chanical device, an analytical formula, a geometrical construc- 
tion, or in any other way. Sometimes there are several 
different methods of producing one and the same transforma- 
tion; but the effect is the same no matter by what method 
produced. A collineation is defined as one that transforms 
