GROUPS*OF TYPE I. 239 
k,’=k'k, and which therefore break up into groups of the 
second class, viz.: G,( Al), G,(AA’), G,(ll’), G,( AA’) and 
G,(AA’A’’). We shall designate these groups of the second 
class by G,(Al),, G,(AA’),, G(U’),, G(AA),, G(AA'A”),. 
It is easy to show that the remaining subgroups of G, of 
the first class do not break up into subgroups in the above 
manner. The groups G,(A), G,(l) and G,(A,/) have the 
single k-relation k,k,’=kk’k,k/; if we put k’=k" and 
k,/ =k, we do not find k,’=k, for all values of r. Hence 
these groups do not break up into subgroups of the second 
class. There is, however, one exceptional value of 7 for which 
those three groups have one subgroup each; this exceptional 
case is considered later, Art. 289. 
The six critical values of 7, viz.: r=1,0, ©, — 1, 2, 4, are 
to be excepted for each of these groups of the second class. 
For r = 1,0, © the collineations are all perspective collinea- 
tions and the groups corresponding to these values of 7 are 
groups of perspective collineations. For r= — 1,2,4 the 
path-curves are conics and the resulting groups belong to the 
third class to be considered below. 
THEOREM 51. There are five, and only five, varieties of groups 
of the second class. viz.: G,;(4ABC),, G.(ABI),, G3(AB),, Gs (W),, 
G,(Al),. 
288. Groups Whose Path-curves are Conics. We come 
now to the consideration of the groups of collineations of 
type I which are made up of one-parameter groups whose 
path-curves are conics. These groups are in many instances 
only special cases of groups of the second class when 
r= -—1, 2, 4; but we shall also find many groups which 
are not special cases of the above. _ These latter cases are 
usually of great interest. 
There is only one variety of one-parameter group of this 
kind; in this case the two common tangents to the conic and 
the common chord of contact form the invariant triangle. 
