EXERCISES. 317 
G,(AIUS) are the resultants of a system of involutoric per- 
spective collineations with the elations of the subgroup 
HSGAL): 
12. Show that the system of collineations selected from the 
group G,(AA’A”), so as to satisfy the condition k+k'’=0, 
form a continuous system but not a continuous group. 
13. Show that the system of parabolic transformations 
within G,’(A), which correspond on the Argand diagram to 
all points ona straight line through the origin, has both 
group properties and hence forms a subgroup of G,/(A). 
14. Show that there is one such subgroup of G,/(A) for 
each line through the origin; hence show that the real group 
pG,(A) is a subgroup of G,/(A), when A is real. 
15. Show that the system of loxodromic transformations 
within G,(AA’), which correspond on the Argand diagram 
to all points ona logarithmic spiral r=e(¢+*)’ about the 
origin and through the unit point, has both group properties 
and hence forms a subgroup of G,(AA’). 
16. Show that G,(AA’) contains one such subgroup for 
each value of ¢; hence show that eG,(AA’) is a subgroup of 
G,(AA’) when A and A’ are conjugate imaginary points; 
show also that hG,(AA’) is a subgroup of G,(A A’) when A 
and A’ are a pair of real points. 
17. Show that each of the plane collineation groups, 
H, (Al), G,’(AlS), and G,(AA’l),, in which the para- 
meter t assumes in turn all complex values, breaks up into 
oo! continuous subgroups, one for each value of 6 in t=+re"’. 
18. Show that each of the plane collineation groups, H’(A1) 
and G,(AA’A’’),, in which k assumes in turn all complex val- 
ues, breaks up into continuous subgroups, one for each value 
of cin k =e (e+7)6, ; 
19. Show that those hyperbolic transformations in 
hG,(AA’), for which k is negative and which therefore can 
not be generated from either real infinitesimal transforma- 
