LINEAR RELATIONS. TYPE IIL. 193 
TYPE III. 
We shall now investigate the conditions that must be satis- 
fied by collineations of type III, if they are to form a group. 
The necessary and sufficient condition that a collineation be 
of type III is that its characteristic equation A(p) =0 shall 
have three equal roots, for which its first minors do not all 
vanish. ‘This is equivalent to two conditions on the elements 
of M, call them D and D’, and shows that there are ~° col- 
lineations of type III. For the same reasons that hold in the 
case of type I, we see that there can be no five- or six-par- 
ameter groups of type III. 
227. The Group G,’(Al). We shall first investigate the 
existence of a four-parameter group of type III, defined by 
R,, Dand D’. We may write the matrix M and the charac- 
teristic equation \(~) in the same form as in Art. 224. The 
factor p—1 gives one root of A(p)=0 which must be a 
triple root. It follows that the quadratic equation (39) must 
be satisfied by 7 and also have equal roots. This is equiva- 
lent to saying that the two conditions, which gaye us cases 
(1) and (2) of Art. 224, hold simultaneously. Since these two 
conditions separately lead to the same result, when combined 
they give us that same result. | Hence we infer that there is 
no four-parameter group of type III defined by R,, D and D’, 
or by R,, Dand D’. It also follows that a necessary and 
sufficient condition for a group of type III is the vanishing of 
either 6, or a, in the reduced form of the matrix M. 
The matrix M of this group may therefore be written in 
either form 
109 S040! 
te OT 0 
(1) ||. z olf or(2)\o x ol. 
|@3 63 7 las by 1 
The group leaves invariant the lineal element A/, 7. e., the 
collineations of the group all have the same invariant figure ; 
it is designated by G,/’(Al). We have thus proved that if 
a system of collineations of type III form a group, it is neces- 
sary and sufficient that each collineation of the system leave 
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