ON THE CONFORMAL REPRESENTATION OF PLANE 
CURVES, PARTICULARLY FOR THE CASES 
p = 4, 5, and 6. 
CHARLOTTE E. PENGRA. 
A given non-homogeneous function of x and y 
n m 
F {x, y)~o 
of degree n in x and m in y may, we know, be regarded either 
as a plane curve or as a Riemann’s surface. Klein reaches a 
number of important results here briefly outlined by consider¬ 
ations based on the latter view. 
Let 
n m 
F (x, y) = o 
be an irreducible algebraic equation defining the surface F n 
which is an n-leaved surface spread over the y plane. The de¬ 
ficiency of the surface, p, is fixed by the number of cuts, 2 p, 
which is necessary to reduce F n to a simply connected sur¬ 
face. The deficiency so arrived at is numerically the same as 
the deficiency of the plane curve 
F(x , y)~o 
which is precisely the number representing the number of 
double points which the curve lacks of having the maximum. 
An algebraic function or an integral of the first or second 
kind belongs to a surface when it has but one value for each 
point on the surface, and when it has only a finite number of 
infinities and these only algebraic infinities of finite integral 
order.* Klein proves that upon F n exist integrals of the first 
and second kinds. With this work as a basis the surface F n 
may be conformally represented by another much simpler sur- 
* See Klein Theorie der Elliptischen Modulfucctionen, Vol. I., p. 499. 
