ALGEBRAIC FUNCTIONS WITH AUTOMORPHIO FUNCTIONS. 
29 
if w e consider any point in the ^-plane, which is not one of the singularities of the 
group, dtjclT and dTjclt are, in its vicinity, regular functions of t. 
So T is holomorphic at all points except the singularities of the group. 
Now, denoting as before the effect of a substitution of the group by accents, we 
have 
^ ^ (dH'/dr'J 
j dH/dr^ 3 {dHjdT^^f 
^ {dtjdTf ^ {dtldTf _ 
= {yt + Sr T. 
So, T is a function of t of the kind already specified. 
So, the converse of the theorem is true. 
Thus, if we can find any one quasi-uniformising variaMe of an cdgehraic form, we 
can find the totality of all uniformising and quasi-uniformising variables by this 
equation. 
We can now find the functions T. 
If 
* _ at + h 
ct d 
we have 
(If _ _ 1 _ 
dt {ct + dy- ’ 
and so 
{dzjdt'Y = (ct -j- ciy {dzjdty. 
Thus {dz/dty is a Thetafuchsian function of order two; any other Thetafuchsian 
function of order two can be written in the form 
T = fl (2, u). [dzjdty, 
where R [z, u) is an automorphic function of the group, i.e., a rational function of the 
algebraic form. 
If the algebraic form is of genus p, it is known''’ that any function II [z, u) for 
which T is holomorphic is a linear function of ( 3 p — 3 ) special functions. These we 
can write 
Ri (z, u), Ro (z, uj, . . . Rsjj-s (z, u). 
The case q) = 1 is exceptional; here there is one such function, T, namely, a 
constant. 
* Humbert, ‘ Liouville’s Journal,’ (4), vol. 2, p. 239, 1886. 
