LINEAR RELATIONS. TyPE II. 189 
upon the elements of M simultaneously the conditions R, and 
D, or R,and D, we have only ~’ collineations. There can 
not be, therefore, a six-parameter group of type II. 
There may or may not be a five-parameter group of type 
II, so far as we yet know. We must examine the system of 
«* collineations defined by say R, and D. If the resultant of 
any two collineations of this system is also of type II, then 
this system has the first group property. 
224. No Five-Parameter Group of Type Il. The group of 
collineations defined by R, alone is reducible ; its matrix and 
its characteristic equation may therefore be written in the 
reduced forms : 
{a, b O ja,—-p by 0 
M= |e %& 0|, and ANG) = 2 bo—? 0 =o. (38 ) 
a3 b; 1 as b, i/o 
One root of A(p) is given by the factor p—1=0; the other 
two roots of A(p) are the roots of the quadratic 
p?—(a,+b,)p+(a,b,—a.b,)=0. (39) 
Since A(p)=0 is to have a double root, two cases arise ; 
either 7 is one root of (39 ) or (39) has equal roots. We must 
consider these cases separately. 
(1). Let us suppose first that 1 is one root of (39); then 
we have for T the condition 
1—(a,+ 6.) + (a,b, —a,b,) =0. (1) 
The same condition holds also for T,, the elements of whose 
matrix are a,, 3,, etc.; thus: 
1—(4,+.)+4,3,—a,3,=0. (11) 
We wish to see if the same condition holds also for T,, where 
TT,=T,, 1. €., is it true that the vanishing of (i) and (11) 
makes the function 
(Aye 8e) ct Ay Be Ae (iii) 
also vanish? 
Since the determinant of 7, is equal to the product of the 
determinants of 7 and T,, we have 
ob, = (a,b, — a,,) (a1/32— 42/3;), 
Ache 8s =(a,+0,—1) (a,+,—1). 
