INFINITESIMAL COLLINEATIONS. 285 
THEOREM 9. The group G/ (A) contains «</infinitesimal trans- 
formations; every finite transformation in the group ean be gen- 
erated from one, and only one, infinitesimal transformation of the 
group. 
335. The One-dimensional Loxodromic Group G,(AA’). 
It was shown in Art. 26 that the variable parameter of the 
loxodromic group G,(A A’) is k, a complex number, and that 
the law of combination of parameters in this group is ex- 
pressed by the equation k, =kk,. In this group as in G,/(A) 
the transformations are commutative and k assumes in turn 
all complex values. 
This group contains the identical transformation corre- 
sponding to the value k=7. When the values of k are 
represented on the Argand diagram, the unit point, k = 1, 
corresponds to the identical transformation. Each point on 
the circle about the unit point with radius 5, an infinitesimal, 
corresponds to an infinitesimal transformation of the group. 
Hence the group G,(A A’) contains -! infinitesimal transfor- 
mations. 
Let us set k= re’ and r = ee’; whencek = e(¢+*)", where 
¢isareal number. When 6=0, k= for all finite values 
ofc. When #=34¢, an infinitesimal, we have an infinitesi- 
mal transformation corresponding to each finite value of c. 
Let us choose one of these infinitesimal transformations, say 
that corresponding to the fixed value, c,, and designate it by 
I,, If J, be repeated n times we have k,=e(a+%n59, By 
choosing 7 sufficiently large we may thus generate from J, cer- 
tain finite transformations of the group. The locus of the 
point k,=e(+%"", as n varies, is a logarithmic spiral 
about the origin, passing through the unit point and making 
an angle ¥ with the axis of reals such that cot’ =c. 
Such a spiral makes an infinite number of turns about the 
origin and the unit point divides it into two distinct portions, 
which we shall call the two halves of the spiral. One of these 
halves lies entirely within the unit circle and the other en- 
tirely without it. This spiral contains two points which cor- 
