THE GENERAL LINEAR GROUP. ole 
G,(Aow’), are logarithmic spirals, 7 = Ce’, where b is the 
cotangent of the constant angle J between the radius vector 
and the tangent to the curve. When b=0, cot 1 =0 and 
Y=7. In this case the spirals degenerate into concentric 
circles about the point A. R, the ratio of linear magnitude, 
in this case is unity. 
The path-curves of this one-parameter group are concentric 
circles; 7. e., they are conics having double contact at w and 
wo’. The group is therefore evidently G,(Aww’),__, or 
G,(HM). There are ~* such one-parameter groups in the 
plane, one for each point A. When the point A falls on the 
line at infinity, the collineation is no longer of type I but de- 
generates, as we shall see, into a collineation of type V. 
376. Collineations of Type Vin G;(EM). The group of 
similarity was found to contain collineations of types 1, IV 
and V (Art. 368). A collineation belonging to the group of 
similarity is a motion when the ratio of expansion is unity. 
We saw in the last article how the ratio of expansion might 
be unity for a collineation of type I. 
The ratio of expansion of a collineation of type IV in the 
group of similarity is R=k; so thatif R=1, thenalsok=1. 
But k=1 is the identical transformation in the group 
H,(Ol.); hence we see that the group of motions contains 
no collineation of type IV. 
The ratio of expansion for a collineation of type V in the 
group of similarity is R = 1; hence all collineations of type 
V in the group of similarity are also to be found in the group 
of motions. 
The two-parameter group of type V, H,’(l.), is therefore 
a subgroup of the group of motions. This group H,’(I..) is 
common to G,(l.), G;(le),, G,(oo’) and the group of mo- 
tions, Giloo)e-=,. 
377. Motion is Hither a Rotation or u Translation. The 
group of motions contains only collineations of type I and 
type V. These must be examined separately. The path- 
curves of a one-parameter group of motions of type I are con- 
