ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
27 
In all these cases t and t are uniform functions of each other, near the point 
considered. So u and 2 are, near the point, uniform functions of r. This is easily 
seen to be true also of ^ = 00. 
Now, let accented letters denote the effect of opei’ating on t with a substitution 
of the group. 
We have 
at h\ 
ct d) 
d^ 
df^ 
+ TV == 0. 
Now T' 
Then 
{ct + d)^T. Write v = 
dhi' / ^ d \ / . d 
d — + d) — <1 {ct + d) — 
dt 
dt 
Therefore 
or 
(ct + Ciy |f + (ct + rffTf = 0, 
df 
+ = 0 . 
So ^ = Avi + Br2, where A and B are constants, and 
Therefore 
or 
_A-Ti + Bvj 
ct d 
v\ _ Ajt'i + Bi-r., 
V., BoV) 
AjT + Bj 
AjT + B., 
This shows that, when t is transformed by a projective substitution of the group, 
T is transformed by a corresponding projective substitution 
(r . 
\ ’ A.t + B.J 
Thus the theorem is proved, namely, that the dissected Riemann surface can be 
conformally represented on a polygon in the r-plane, and the sides of this polygon 
can be derived from each other in pairs by certain projective substitutions ; in other 
words, T is a quasi-uniformising variable. An infinite number of variables t can be 
got in this way, for T depends linearly on several arbitrary constants. 
E 2 
