218 THEORY OF COLLINEATIONS. 
Two groups G,’/(AA’l), and G,’(AA‘l), have in common 
the identical and the pseudo-collineations corresponding to 
t=0 and t=. We wish to ascertain if they have any 
other collineation incommon. If a collineation whose con- 
stants are k and t belongs to both the above groups, then k 
and ¢ must satisfy both equations k=atandk=a,'. This is 
possible only when a=a,. Hence the two groups have no 
collineation in common, save the identical and the pseudo- 
collineations. 
D. FUNDAMENTAL GROUP OF TYPE III AND ITS SUBGROUPS. 
262. Fundamental Group of Type II. The canonical 
form of a collineation T’ of type III has been found, Art. 150, 
to be 
Loar x 
—_—=—-+2at, 
ap 4 eh y : / 
fe ; (57') 
1 1 x 
—=—+t—+ (a#+ht) 
Y. y ) 
the origin is the only invariant point and the a-axis the only 
invariant line. 7’ depends upon three constants, a, h, t, and 
hence there is a set of ©’ collineations of type III having the 
same fundamental invariant figure. Let 7,’ be a second col- 
lineation of the same set given by the equations 
P Y2 Yi 
(AES : ov’ 
t 1 1 xy ( v) 
—=—+t4—+a 47+ hth 
Y2 Yi Yi 
The resultant of 7” and T,” is found by the elimination of 
x, and y, to be 
Hip) x 
— = — 4+ 2aots 
Y2 y 
/} /} 
iE. 1 1 x (57 ) 
== — th talk, 
Y2 y ¥y 
where 
t=t+t ’ 
d2t,=at+aqyiti, (58 ) 
Aote+ hot, =at+ 2att, +a,t, +ht+h, ty 
