262 THEORY OF COLLINEATIONS. 
18. Show that equations (36) represent a collineation of 
type II. 
19. Show that every collineation of type I belongs to one 
and only one two-parameter group G,(AA’A”’). 
20. Show that two groups G,(AA’A”) and G,(AA,'A,’), 
where A’, A”, A,’, A,” are collinear, have a common subgroup 
H,(Al’) of type IV. 
21. Show that G,(A) and G,(A’’) have in common 
G,(AA5); G,(l) and G,(l’) have in common G,(ll’); G,(A) 
and G,(l’’) have in common G,(A,l’’). 
22. Show that the following groups bracketed together are 
dualistic: 
(GA AAQ) on (CG (AA ae Gaal es, 
iG AtAGl), S62 = aGu GU) cee UG (CAd, ee ae 
23. Show that the following groups are self-dualistic : 
G,(AAT),-2, G,(Al),—z. 
24. Show that the two collineations of type I which are 
the resultants of T and 7” in different orders have the same 
invariant cross-ratio, but not the same invariant triangle ; 
show that their invariant triangles have equal areas. 
25. Show that G,’’( Al) is a subgroup of G,(A1l),_.. 
26. Show that G,’’( AlS) is a subgroup of G,( AIS). 
27. ShowthathG,(AA’A”’) contains three and e G,(AA’A”) 
only one involutoric collineation. 
28. Show that the family of path-curves of the group 
eG,(AA’A”’),_, contains both real and pure imaginary conics. 
29. Hither all three cross-ratios along the sides of the in- 
variant triangle of hG,( AA’A”’) are positive or one is posi- 
tive and two negative. 
30. The group G.( AlK’) contains no elliptic collineations of 
type I. 
31. The group eG,(iK) contains only elliptic collineations 
of type I. 
