EXERCISES. 61 
B. REAL TRANSFORMATIONS. 
I. Analytic Theory. 
(1). Find the transformation which changes the points 
whose coordinates are 2, 8, 9 into the points whose coordi- 
nates are 1/2, 25/44, and 43/75 respectively. 
Ans. (A, A’) = 2x8 -k=31—8Vvi5- 
(2). Show that every real transformation with a negative 
determinant is hyperbolic and its cross-ratio k is negative. 
(3). Show that every transformation with positive deter- 
minant is either elliptic, parabolic, or hyperbolic with positive 
cross-ratio k. 
(4). Show that every transformation with positive deter- 
minant can be generated by the repetition of some real infini- 
tesimal transformation; show also that no transformation 
with negative determinant can be so generated. 
(5). The resultant of two transformations, one with a 
positive and the other a negative determinant, is a hyperbolic 
transformation with negative cross-ratio k. 
(6). Show that the resultant of two hyperbolic transforma- 
tions in G,(A) is (a) parabolic when k, = ,and AY ee AUS (0) 
is identical when k, =, and A, = A’. 
(7). Show that the cross-ratio of four real points on a line 
is always real. 
(8). Show that the cross-ratio of a pair of real and a pair 
of conjugate imaginary points is always of the form e*’. 
(9). Show that the identical and the involutoric trans- 
formations divide the elliptic group eG, into two subdivisions 
each of which contains the transformations inverse to those 
of the other. 
