EXERCISES. 63 
(8). Show that the group G,(A) contains only hyperbolic 
and parabolic transformations. 
(9). Show that the two groups G,(A) and G,(A’) have in 
common all the transformations of the group hG,(AA’). 
(10). Show that the »* conics touching / and /’ determine 
o* transformations which form the group G,. 
(11). Show that the »’ parabolas touching / and /’ deter- 
mine a two-parameter group; find its invariant point. 
(12). Show that the system of parabolas having O~ for 
their common axis and touching / and l’ determines a one- 
parameter parabolic group ; find its invariant point. 
(13). Let 7 be a transformation whose invariant points 
are A and ~ ; and let P and P, be a pair of corresponding 
points of 7’; show that the characteristic cross-ratio of T is 
k= AP/AP,. 
(14). If the transformation T(A ~ ) transforms the seg- 
ment PQ into P,Q,, show that the length of the segment PQ 
is k times the segment P,Q,, 7. e., PQ = k( P,Q,). 
This is identical with the mechanical effect of stretching a rubber cord with one 
end fixed at A. Such a transformation is called a Dilation, and the group G2(A) is 
called the group of dilations. 
(15). Ifa tangent be drawn to one of the parabolas K of 
example (12) cutting / and l’ at Pand P’, show that the dif- 
ference of the segments OP and OP’ is constant for all tan- 
gents to K; hence show that the transformation determined 
by K transforms segments into equal segments. 
Such a transformation is called a Translation of the line into itself, and the group 
pGi() is called the group of translations. 
(16). Show that the characteristic cross-ratio of a per- 
spective transformation whose center is at @ (Fig. 9), is 
k= A'Q/ AQ. 
