192 THEORY OF COLLINEATIONS. 
A, A’,l,l’, where A and / are respectively the point and 
line corresponding to the single root of A(p) =0 and A’ and 
those corresponding to the double root of A(p)=0. The in- 
variant lineal element of matrix (1) is Al’, of (2) is A’l, of 
(3) is Al’, of (4) is A’l’. Since (1) and (8) are the same we 
have three distinct cases and hence three distinct four-par- 
ameter groups, one for each distinct lineal element in the fig- 
ure AA’ll’. These groups are designated respectively by 
G/(Al), G/(A’)), G,(A). 
Since the vanishing of either 6, or a, is both necessary and 
sufficient to establish the first group property we have proved 
the following important theorem : 
THEOREM 21. A necessary and sufficient condition for the ex- 
istence of a group of collineations of type II is that the invariant 
figures of all the collineations in the {sy stem have in common the 
same lineal element. 
226. Other Groups of Type II. Each of the above four- 
parameter groups in its unreduced form is defined by the fol- 
lowing sets of relations: R,, R,, SAl and D. In precisely the 
same manner we can show the existence of groups of type II 
as follows; R., R,’ and D: i, ke,’ and Wc h,, Wa, fu, oA 
and D, where the relation S’./ exists between the coordinates 
of the single line and double point. These groups are desig- 
nated respectively by G,/(AA), G,’(ll’), G,(AA'l’). We have 
therefore six varieties of groups of type II defined by linear 
relations on the elements of M. 
In order to make the discussion complete we should exam- 
ine for the group property the set of »* collineations defined 
by R, and D. But since the group defined by R, alone is irre- 
ducible, the application of the above process would be very 
tedious. Later we shall attack the problem by an indirect 
method and reach a negative result. 
THEOREM 22. There are six varieties of groups of type II de- 
fined by linear relations on the elements of M, viz.: 
G,/(Al'), GAGAW): GCA). G,'(AA’), Gly), G,/(AA'l’). 
