310 THEORY OF COLLINEATIONS, 
the angle between the radius vector and the tangent at its 
extremity is constant is the logarithmic spiral. Hence the path- 
curves of the one-parameter group G,(Aww’), are logarithmic 
spirals about the point A,; these spirals cut all the lines 
through A at a constant angle. 
It may also be shown from the equations to the path-curves 
of G,( AA’A”), that, when A’ and A” are the circular points 
» and w’, these path-curves become spirals. The equation of 
the path-curves of G,(AA’A”), is, Art. (251), 
CaO cme 
Putting z = 7 the side A’ A” of the invariant triangle is shifted 
to infinity. To make A’ and A” the circular points at infinity, 
we write « —iy for x and «+ 7y for y (or vice versa). The 
above equation then becomes 
etiy=C(“%— wy)’. 
Setting , =a-+ 7b we have 
SCY Cems OO Ceeke Uae 
Also by changing the sign of 7 
x — a Cebdiere , 
Multiplying we get 
a? + y? = C7e*?, 
Setting «+ y’ = 4° we have 
s= Cees, 
which is the polar equation of a family of logarithmic spirals. 
THEOREM 28. The path-curves of a one-parameter group 
G,(ee') whose invariant triangle has two vertices at the circular 
points at infinity are a family of logarithmic spirals about 4. 
371. Collineations of Types IV and V in G,(ow’). The 
group G,(oo’) contains the three-parameter group H,(1.); 
for the ~* perspective collineations S(/.) form a group and 
in the invariant figure of this group is every point on the line 
at infinity, including therefore the two circular points » and 
w’, Within this group H,(/,) is the two-parameter group 
