188 THEORY OF COLLINEATIONS. 
projective transformations. When two groups of transfor- 
mations are so related to each other that there is a one-to-one 
correspondence between the transformations of the two groups, 
the resultant of any two transformations corresponding to 
the resultant of their correspondents, then they break up into 
subgroups in exactly the same way and have exactly the same 
structure. Two such groups are said to be holoedrically 
isomorphic. 
THEOREM 19. The group G; of one-dimensional projective trans- 
formations and G;( A) are holoedrically isomorphic. 
SA) 
5d. Groups of Other Types Defined by 
Linear Relations. 
The groups determined in the last § are all of type I. It 
was assumed that the elements of the matrix M were not 
subject to any of the conditions, Art. 113-117, that cause its 
characteristic equation to have multiple roots or the first mi- 
nors to vanish, 7. e. the collineations were assumed not to be 
of any of the secondary types. We shall now take up each 
type separately and determine the conditions under which all 
the collineations of a given group belong to a given type. 
TYPE II. 
223. No Siz- or Seven-Parameter Groups of Type II. The 
condition that a collineation shall be of type II is, Art. 114, 
that its characteristic equation, 
a—p b Cy 
(oy bo—! Co = 0, 
a bs C3—P 
shall have a double root; 7. e., that its discriminant shall van- 
ish. The vanishing of the discriminant D lays one condition 
on the elements of M. There are therefore ~’ collineations 
of type II. These do not form a seven-parameter group be- 
cause this one condition does not comply with the necessary 
and sufficient conditions laid down in Theorem 11. 
The largest group that could exist is a six-parameter group 
defined by one set of linear relations R, or R,. If we lay 
