EXERCISES. 59 
similar to that of a pencil of lines that nothing further than 
a brief statement is required on this point. The transforma- 
tions are hyperbolic, elliptic, and parabolic; these groups are 
the same as for the other one-dimensional forms, the range 
of points and the pencil of lines. 
Exercises on Chapter Il. 
A. GENERAL ANALYTIC THEORY. 
(1). Show that the determinants of a pair of inverse trans- 
formations have the same value. 
(2). Show that a pair of inverse transformations are of the 
same type, have the same invariant points and, if of type I, 
reciprocal cross-ratios. 
(3). Show that the resultant of a pair of inverse transfor- 
mations is always the identical transformation. 
(4). Show that the determinants of a pair of conjugate 
transformations always have the same value. 
(5). Show that two conjugate transformations have the 
same cross-ratio but not the same invariant points. 
(6). Let the invariant points of a pair of conjugate trans- 
formations be (A A’) and (BB’); show that the ,segment 
(AA’) is equal to the segment (BB’). 
(7). Show that the transformation, x, = a , 
transforms 
the point «= =< to infinity and the point «= == to the 
origin. Into what points are the origin and the point at in- 
finity transformed? 
(8). Find the invariant points of the following transforma- 
Hons) t— On 20> i), — aa il) ty— 2-1 Ol 
