62 EXERCISES. 
(10). The group eG, contains no pseudo-transformations. 
(11). In an involutoric transformation the middle point of 
the segment AA’ is transformed into the point at infinity. 
(12). If Q is transformed to infinity and the point at in- 
finity to Q’ by a parabolic transformation T whose invariant 
point is A, then the points Q and Q’ are equally distant from 
A on opposite sides. 
Il. Geometric Theory. 
(1). The range of conics inscribed in the quadrilateral 
ABB’'A’, Fig. 3, contains three degenerate conics, viz.: the 
diagonals of the quadrilateral Om, AB’, A’B. Show (a) 
that the transformation determined by O-@ is the identical 
transformation of the group hG,(AB); and (6) that the 
transformations determined by AB’ and A’B are the pseudo- 
transformations of the group. 
(2). If the conic K’ is the reflection of K on the line OX, 
show that the transformations T, and JT, form an inverse 
pair. 
(3). Show that the transformation determined by the conic 
which has the line OX for one of its axes is involutoric. 
(4). What two conics determine the two infinitesimal 
transformations of the group hG,(AB)? 
(5). What conics of the range inscribed in ABB’A’ deter- 
mine transformations belonging to subdivisions I, II, III, 
respectively? 
(6). In the elliptic group eG,(AB), what conic determines 
(a) the identical, and (b) the involutoric transformation of 
the group? 
(7). In the parabolic group pG,(A) what conic determines 
(a) the identical and (b) the pseudo-transformation of the 
group? 
