656 Wisconsin Academy of Sciences, Arts, and Letters. 
face. In order to do this we must classify surfaces according 
to their deficiencies and treat each class separately. 
In the case p = o there are no cuts on the surface and no in¬ 
tegrals of the first kind. Integrals of the second kind exist on 
all surfaces. Let us select one of these, w, which has a single 
algebraic infinity. Since there are no period paths on the sur¬ 
face, w assumes only one value for each point of F n and since w 
has the one infinity, and only one, it is a function of “ weight ” 
one belonging to F n where the weight of a function is defined 
as the number of times which it assumes the value q© , hence 
the number of times which it assumes any assigned value, for 
points on Fn. The function w being of weight one assumes one 
and only one value corresponding to each point of the n-leaved 
surface 
F{x, y) = o. 
These values, real and complex, may be represented by the 
points in a plane by the ordinary representation of complex 
numbers. The given w-leaved surface can then be conformally 
represented upon a plane by means of the real and complex 
values assumed by w. 
I f p = 1 two cuts are required to make the surface simply 
connected. We know that on any surface of deficiency p, there 
exist p linearly independent integrals of the first kind. Here 
then there exists only one which we will call u. Let the 
periods across the cuts be Wi and w- 2 . If u has the value u 0 at a 
given point, for all the region around containing no branch 
point the u will vary continuously, and since u can be no¬ 
where infinite, and since its values may be represented by 
points of a plane just as any complex number is represented, 
these points must all be within a parallelogram whose bound¬ 
aries are determined by the limits of the values of the real and 
imaginary parts of u as it varies over the surface, never cross¬ 
ing a boundary. 
If we seek then to represent our entire surface by means of 
the integral u which has an infinite number of values we get 
corresponding to a given point of F n an infinite number of 
homologous points in similar parallelograms. We will form 
the doubly periodic functions 
P (u | w lf w 2 ) and P' (u | iv L , w 2 ) 
We know that all doubly periodic functions of u, wi, and^ may 
be expressed rationally in terms of these two, 
