60 EXERCISES. 
(9). Find the equation of all transformations in the group 
G, that interchange two points A and A’; show that they are 
all involutoric transformations. 
(10). Prove that the system of ©* involutoric transforma- 
tions in G, does not form a group; also the system contains 
no infinitesimal transformation. 
(11). Ina parabolic transformation show that t= eS 5 
(12). Prove that the ~* parabolic transformation in G, do 
not form a group. 
(13). Show that two transformations are commutative 
when, and only when, they belong to the same one-parameter 
group. 
Let T transform S into S’ according to the formula 
S’=T'tST. Then— 
(14). If S is of type I, so is S’; and k =k’, where k and k’ 
are the cross-ratios of S and S’ respectively. 
(15). If S is of type II, so is S’; find the relation between t 
and ¢’, the characteristic constants of S and S’ respectively. 
(16). If S and T have one invariant point in common, S’ 
has the same invariant point; in this case if S is of type II 
and T of type I, then t/= kt, where k is the cross-ratio of T. 
(17). If S and T belong to the same one-parameter group, 
S’=S. 
(18). Show from equations I to V, art. 21, that = = 
1—In) (1— kn) Lt nth ; 
Conn = += where 1 =(A,AA'A,’, the eross-ratio of 
CH] 
the four invariant points of T and T,. 
(19). Deduce equation (31a) from equation (26). 
(20). Deduce the canonical forms, v,=kv+ A(1—k), 
“, = kx, and «,=x-+t, from equation (22). 
