PERSPECTIVE GROUPS. 235 
Comparing coefficients of x and yin components and result- 
ants we have 
k,=kk,, p.(1—k,) =p,(1—k,) + pk,(1—k), (72) 
ae (Ce eee lle) ale tle key (1K —= i) 
Peer ne mT op tine oP ogy 
l = 
B B B, 
We have here four independent equations involving the three 
quantities k,, B,,p,. Hence the resultant is not of type IV 
and the set of perspective collineations to which S and S, be- 
long does not form a group. Equations (71”) represent a 
collineation of type II leaving invariant the origin, the y-axis, 
the point (0, B,) on the y-axis, and a line l, through the 
origin, but no other point on /,. 
Equations (72) show that no group of the kind H,(A‘l’) 
exists; but if B, = B in (71’) and (71”) they show the exist- 
ence of the group H,(AA’); if p,=p in the same equations 
they show again the existence of the group H,(Il’). 
282. Resultant of Any Two Perspective Collineations. Let 
S and S, be two collineations of type IV whose axes / and 1, 
intersect in O and whose vertices A and A, lie on a line l’. 
Fig. 28(d). The resultant of S and S, leaves O and Il’ invari- 
ant. Along l’ the two one-dimensional transformations due 
to S and S, result generally in a one-dimensional transforma- 
tion with two invariant points A and A’. Hence the result- 
ant of S and S, leaves invariant the triangle (OA A’) and is 
cf typeI. The non-existence of the group property is thus 
proved for the ~* perspective collineations. 
THEOREM 50. The o°% perspective collineations of the plane do 
not form a group; there are five varieties of groups of type LV. viz.: 
Wg (CA) He (1), He (AA), Lg(GV), H,(A,1). 
283. Dualisticand Self-dualistic Groups. From the dual- 
istic character of a collineation, we infer that its fundamental 
invariant figure is self-dualistic. | When there exists a group 
of collineations, leaving a certain figure invariant, there must 
also exist a second group leaving invariant a figure dualistic 
