286 THEORY OF COLLINEATIONS. 
respond to infinitesimal transformations, one in each half of 
the spiral and adjacent to the unit point. These are given 
by kK=et (1 +*)6? andk=e-(a+%)6@, Kivery finite trans- 
formation corresponding to a point on either half of the 
spiral can be generated by the repetition of its corresponding 
infinitesimal transformation. 
Different values of ¢ give us different spirals. c varies 
continuously through all real values from — © to +, so 
that these spirals lie infinitely close to one another. They all 
pass through the unit point. As ¢ approaches zero, the cor- 
responding spiral approaches as a limit the circle of unit 
radius about the origin; as ¢ approaches infinity, the corre- 
sponding spiral approaches as a limit the straight line which 
is the axis of reals. 
Two problems now present themselves for solution: Can 
every finite transformation in the group G,( AA’) be gener- 
ated by the repetition of an infinitesimal transformation of the 
group? Cana given finite transformation T of the group be 
generated by more than one infinitesimal transformation of 
the group? To answer these questions we proceed as follows: 
Let P be the point on the Argand diagram corresponding to 
the given transformation T;, and let the coordinates of P be p, 
and 6,+22n(” any integer). Since 
k, =" et(h+2nz) — g(e+72) (h+2nz7) i then 
log k, = loge, +71(0,+ 2n2)=¢(0,+2nx)+7(0,+ 2n7z) ; 
whence log p, =¢(#,+2n2) or ¢= ; 29 . Since » is any 
integer, there are an unlimited number of values of ¢ which 
satisfy the equation. Thus there are an infinite number of 
spirals of the family p = e (* +7)" through the point P. When 
n=0,1,2,3, .... the corresponding spiral starting from 
the unit point, makes 0, 1, 2, 3, . . . . turns about the origin 
before passing through P. Hence, every point in the plane 
of the Argand diagram, not on the unit circle, lies on an 
infinite number of discrete spirals, from which we infer that 
every transformation of the group G,( AA’), whose corre- 
