28 
MR. B. T. WHITTAKER OH THE CONHEXIOH OF 
In the above theorem, for the sake of simplicity, we have made a restriction which 
is really unnecessary, namely, we have supposed that i is a uniformising variable. 
t can, hoivever, be any quasi-imiformising variable if we make the corresponding 
extension in the meaning of T. T will now have to be a function of t, which is 
holomorphic in any of the polygons, and which obeys the law 
for substitutions of the group generated from the substitutions which change the 
sides of the ^-polygon into each other. Such functions exist; for, as before, if iv is 
an Abelian integral of the first kind connected with the curve, {divjdty is such a 
function, T is, of course, really a multiform function of t, if ^ is a quasi-uniformising 
variable; but as it is not possible to pass from one of its values to another by any 
paths contained within one of the polygons, we can regard it as uniform within that 
polygon. The proof in this extended case is just as before. Thus we have the more 
general theorem : 
Jft is any uniformising or quasi-uniformising variable of an algebraic form 
f {u, z) = 0, 
and T is any holomorphic Thetafuchsian function of t of order two, then the quotient 
of any t ivo solutions of the di^erential equation 
fv 
dt^ 
Tu = 0 
is another uniformising or quasi-uniformising variable. 
To complete the theorem, we must prove that the converse is also true. Suppose, 
then, that r and t both belong to the set of uniformising and quasi-uniformising 
variables, so that a polygon in the r-plane corresponds to a polygon in the ^-plane. 
point for point, and to each of the substitutions of the group ( t, ) corresponds 
a substitution i t, ~ ^ 
\ ’ c;; -f d 
Now T is the quotient of two integrals of the equation 
if 
d-v 
Tf 
+ 
Tv = 0 
, dH/dr^ ^ (dHIdry- 
2 (dt/dry ^ (dt/dry ' 
Now T has no branch-point, considered as a function of t, and t has no branch¬ 
point, considered as a function of r, except at the limiting points of the groups. So, 
