THE GENERAL LINEAR GROUP. Si. 
H,/(l.) of elations; this group of elations is therefore a sub- 
group of G,(ow’). 
372. One-parameter Groups of Dilations. The path-curves 
of a group H,(A,/.) are the pencil of lines through A. Since 
the characteristic cross-ratio of a collineation S of the group 
H,(A,l.) is constant along all lines through A, it follows 
that S produces a dilation of the whole plane, A being the 
center of dilation. All linear magnitudes are altered in the 
constant ratio k and all areas in the constant ratio k*. For 
negative values of k any figure F' and its corresponding figure 
F, are situated symmetrically on opposite sides of A. 
373. The Mixed Group mG,(oo’). Having pointed out 
all varieties of collineations that leave the circular points « 
and o’ separately invariant, we proceed to consider those col- 
lineations which interchange o and wo’. The mixed group 
MG,(oo') is only a special case of emG,(PQ), which group 
was investigated in $3 of this chapter. We thus see that 
emG,(oo’) contains * collineations of type I, «° collinea- 
tions of type II, and ©* involutoric collineations of type IV. 
Let A A’A” be the invariant triangle of a hyperbolic colline- 
ation of type I which interchanges w and w’. Since A’ and A” 
separate © and w’ harmonically, we see that the angle A’A A” 
isaright angle. Hence such a collineation leaves invariant a 
finite point A and two lines through A at right angles to one 
another. Every angle in the plane is unaltered in magnitude 
but reversed in sense by such a collineation, while all areas 
are altered by a constant ratio, which ratio R is always nega- 
tive. The effect of a collineation of this kind is to transform 
every plane figure into a similar but noncongruent figure. 
With AA’ and AA” for axes the collineation T is given by 
v,=kx, y,=ky with the condition k+k’=0. 
Let AA’l’ be the invariant figure of a collineation T’ of 
type II which interchanges w and w’. Since the points A and 
A’ separate o and wo’ harmonically, we see that lines perpen- 
dicular to /’, the finite invariant line of T’, are transformed 
into lines also perpendicular to /’. A line parallel to /’ is 
