EXERCISES. 261 
6. Show that one and only one path-curve of the group 
G,(AA’A”), touches a given line / (not a side of the triangle 
AA’A”’) of the plane, 
7, Show that the constant 7 in the one-parameter group 
G,(AA’A”), is the cross-ratio of the four points (TLL’/L’’) on 
any line g of the plane, where T is the point of the path-curve 
touching g and L, L’, L’’, are the points where g cuts the sides 
of the invariant triangle (AA’A”’). 
8. Show that an involutorie collineation in its plane is 
necessarily of type IV. 
9. From the self-dualistic character of a collineation show 
that, when the path-curves consist of a pencil of conics, they 
. must have double contact with each other and with the inva- 
riant triangle. 
10. What is the geometrical meaning of a in the equation 
B=tGOM yeu 
11. Develop the whole theory of perspective collineations 
by the methods suggested in Article 276. 
12. Show that 2a is the common radius of curvature at the 
origin of all conics of the system 
a+ 2hay+ 4aCy=4ay. 
13. Show that each collineation of type III in G,’’( Al) can 
be resolved into two elations, one belonging to the group 
H,/(A) and the other to H,’(/). 
14. Show that the group G,’’( AlN) contains as a subgroup 
the group of elations H,’( Al). 
15. Verify equations (24) of Article 217; prove that the 
determinant of a complete family of automorphic forms of 
degree n is equal to A”. 
16. Verify equation (35;) of Article 217 and factor the 
general determinant \,(1)=0. 
17. Deduce the equations and give an analytic proof of the 
existence of the groups H,(ll’) and H,( AA’). 
