318 THEORY OF COLLINEATIONS. 
tion in hG,( AA’), can be generated from complex infinitesi- 
mal transformations in G,(AA’). 
20. Show that a real collineation hT(AA’A’’) of type I, 
for which k and k’ are both positive, belongs to one and only 
one one-parameter subgroup of hG,(AA’A”). 
21. Find the determinant of the normal form of a collinea- 
tion of type I in G,(l..), and hence show analytically that R, 
the ratio of areas, is kk’. 
22. Find ina similar manner the value of FR for collinea- 
tions of types II, III, IV and V, inG,(l.). 
23. Show analytically that the group of invariant areas is 
composed of collineations of types I, III and V. 
24. Deduce from the equations of the normal form of type 
I the following equations of the Group of Similarity, G,( ow’): 
@=F(k+h)at+t(k—k)ytA—S(k+k)+F(k—k), 
y= —3(k—K e+ i (k+h )y+B-S(k—k’) —F(b+k’). 
25. Deduce from the equations of problem 24 the equations 
of the group H,(/..) and show that this group is a subgroup 
of Gi(@O") ° 
26. Show that H,’(/.) is also a subgroup of G,(o’) and 
deduce its equations from those of problem 24. 
27. Deduce from the equations of problem 24 the following 
equations of a rotation about the point (AB) through an 
angle ): 
v,=«xcosd—ysind+ A—Acos#+Bsine, 
y,=«xsino+ycos#+ B—Bcos#—Asiné. 
28. Prove that the resultant of two rotations, T and T,, 
through angles # and ¢, respectively, about points (AB) and 
(A,B,) respectively, is a rotation about a third point, (A,B,), 
through an angle #, = 6-+¢,, and find the coordinates (A,B,) 
of the new centre of rotation. 
