230 THEORY OF COLLINEATIONS. 
) Bititm 
litte i 
Sie: = and Saree 
(66") 
The resultant will be of type I in general; it will reduce to 
type III, if B=0; and to type V if both A=0 and B,=0, 
thus proving the existence of the group H,’(A). 
THEOREM 49. The o/elations of the plane donot form a group: 
there are three varieties of groups of elations, viz.: //,/(Al). //./ (/) 
and A,/( A). 
Ba GROURSTOR Mi YErnmslive 
277. Resultant of Sand S,. We shall now take up the 
problem of determining the character of the resultant of any 
two collineations of type IV, and all continuous groups of 
such collineations. We shall begin with the analytic deter- 
mination of the resultant of two perspective collineations 
having the same axis. 
Let S and S, be two perspective collineations having the 
same axis but not the same vertex. Fig. 28(a). Let us 
choose the common axis of S and S, as the z-axis, and let the 
vertices of S and S, be any two points in the plane. Starting 
with the normal form of type I we put k=1, A=0, B=0O, 
B’' =0 in equations (12), chap. II. We thus get, after ex- 
panding and reducing, 
zt ey (I = ty ey 
eo Emer mee eae Ne Re eg 
1+ (- a) 1+ ( ae 
The vertex is the point whose coordinates are (A’’ B’’) and 
k’ is the characteristic cross-ratio of the perspective collinea- 
tion S. Dropping primes and double primes, since they are 
no longer necessary, we can put (67) in the form, 
v1 x A {k-1 1 1 t fk—1) 
eae ty k ), Wi == k ). (67) 
Writing S, in the same form with different vertex (A,B,) and 
different cross-ratio k,, we have 
a A en i re (22 (67’) 
Y2 ky Y\ By k | Y2 ky, B\ k 
