238 THEORY OF COLLINEATIONS. 
of one-parameter groups whose path-curves are conics. 
Strictly speaking the third class is only a special case of the 
second, but we shall see that the special case is so important 
that these groups should be classified by themselves. 
286. Groups of the First Class. One list of groups of the 
first class is already complete. It consists of G, and the sub- 
groups of G, defined by one or more sets of linear relations 
on the elements of M@. These are by Art. (209) G,(A), G,(J), 
G,(Al), G,(AA’), GWU’), G,(A, 1), G,(AA'l), G,(AA’A”). 
We include here the general projective group G,, since it can 
be built up of the »’ two-parameter groups G,(AA’A’’) of 
the plane and cannot be included in any of the other classes 
of type I. 
287. Groups of the Second Class. We now proceed to 
the determination of groups of another variety, viz.: those in 
which every collineation is characterized by the same value of 
r. Ithas already been shown, Art. 249, that the two-para- 
meter group G.(AA’A’’) breaks up into «’ one-parameter 
subgroups, each subgroup being characterized by a constant 
value of 7. If we take all types of groups of the first class 
and in these set k’ = ky and keep r constant and let the other 
parameters vary, we will sometimes find subgroups by this 
process and sometimes not. The groups G,; G,(/), G.(A); 
G,(Al); G,(AMW), G,AA’),-G,W); G,(AA), must -each 
be examined separately. 
Every group of the first class which has the double k-rela- 
tions k, = kk, and k,’ = k’k,’ may be broken up into subgroups 
of the second class. To show this let k’ =k" and k, =k,, 
where r is any constant. Thenk,!=k’k/ =k ky; (kk,) =k7. 
Hence the two conditions k, = kk, and k,’=k’k,' reduce to a 
single one and all the collineations in the group of the first 
class satisfying the relations k’ = k” form a subgroup. There 
will be one such subgroup for every value of 7. 
It was shown in Art. 234 that there are five groups of the 
first class which have the double k-relations k,=kk, and 
