ALGEBRAIC FUNCTIONS WITH AUTOMORPHIO FUNCTIONS. 
9 
be a self-inverse substitution with real coeflBcients a, h, c. Then the substitution 
transforms real values of t into other real values, so the real axis in the i-plane is 
unaffected by the substitution. If (cr-j- he) is negative, it is easily seen that the part 
of the ^-plane above the real axis transforms into itself; if be) is positive, the 
part of the ^-plane above the real axis transforms into the part below the real axis. 
We shall suppose that our groups are generated only from the former kind of substi¬ 
tutions, so we need only consider the half of the ^-plane above the real axis. 
Assuming then throughout that {a^-\-he) is negative for the substitution considered’ 
it is obvious that the double points of the substitution are conjugate complex 
quantities; for the double points are the roots of the equation 
et"^ — 2 at — 6 = 0. 
Now draw any circle through the double points of the substitution. This circle cuts 
the real axis orthogonally. 
Then the substitution transforms the parts of the t-plane outside and inside this 
eirele into eaeh other. 
For, let the double points be 
^ = y + ih, and t — y — ih, 
and let t' be the point into which any point t is transformed. Then the substitution 
may be written 
t' — y + ih t — y -\- ih 
t' — y — ih t' — y — ih 
This shows that the angle subtended by t at the double ])oints is changed into its 
supplement by the transformation ; and therefore the circumferences of all circles 
through the double points transform into themselves, the part on one side of the 
double points transforming into the part on the other side of them. By considering 
the whole plane as made up of the circumferences of circles through the double 
points, we obtain the theorem. 
Now consider the infinite group generated from a number [n + 2) of these self¬ 
inverse substitutions. 
Cyt — 
which satisfy the relation 
b»2 — [t, 
+ b.i' 
c 4 — «„ 
s,. 
!^ ~t ^L+2 
V Cj+2^ 
n+'l. 
S1S2S3. . . S„,,2= 1. 
If n — 1 , we find that it is inqjossible to satisfy this relation by self-inverse 
substitutions with conjugate complex double points ; and if u = 2, it will be seen later 
VOL. CXCII.—A. C 
