Abel's Curve Theorem

The sum of the values of an Integral of the ``first'' or ``second'' sort

$\begin{displaymath} \int_{x_0,y_0}^{x_1,y_1} {P\,dx\over Q}+\ldots+\int_{x_0,y_0}^{x_N,y_N} {P\,dx\over Q}=F(z) \end{displaymath}$

and

$\begin{displaymath} {P(x_1,y_1)\over Q(x_1,y_1)} {dx_1\over dz}+\ldots+{P(x_N,y_N)\over Q(x_N,y_N)}{dx_N\over dz}={dF\over dz}, \end{displaymath}$

from a Fixed Point to the points of intersection with a curve depending rationally upon any number of parameters is a Rational Function of those parameters.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 277, 1959.