ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
25 
To this corresponds a Rieraann surface of two sheets, which can be resolved 
by two cuts into a simply-connected surface. 
Let P be the Weierstarssian elliptic tnnction associated with this curve ; 
and 'iDi, IV2, its periods. 
Consider u and 2 as functions of t, where 
u = P' (log r), 2 = P (log t). 
In the T-plane, form a curvilinear parallelogram ABCD, of which the side CB 
is derived from AD by the projective substitution 
(t, e-r), 
and the side CD is derived from AB by the projective substitution 
( t , e^-^r). 
Then wdthin this parallelogram ABCL), the dissected Biemann surface corre¬ 
sponding to the curve 
= 4 z^ — C/2Z — ^3 
is conformally represented ; the sides AD, CB of the parallelogram correspond to 
the two edges of one cross-cut, and the sides AB, CD to the other ; and, as we have 
seen, the opposite sides of the parallelogram are derived from each other by projective 
substitutions. But in spite of this, u and 2 are not uniform functions of t. The 
reason is, that t is only a quasi-uniformising variable ; when w^e derive all possible 
polygons from ABCD by applying the group of substitutions generated from 
(r, and (r, 
the polygons so derived cover the plane more than once. 
The connexion between the uniformising and quasi-uniformising variables for any 
algebraic form is given by the following theorems. 
I/t is a uniformising variable of an equation 
f{u, 2) = 0, 
and T is any holomorpMc Thetafuchsian function of t of order two, then the quotient 
of any tivo solutions of the differential equation 
d'v 
dd 
+ 
Tv = 0 
( 1 ) 
IS a quasi-uniformising variable. 
The term “ holomorphic Thetafuchsian function ol‘ order two ” may require some 
explanation. 
VOL. cxcii.— A. 
E 
