  The Biot-Savart law is used in aerodynamic theory to calculate the velocity
induced by curved vortex lines. Explicit formulas are developed, using
multivariate Appell hypergeometric functions, for the velocity induced by a
general parabolic vortex segment. The formulas are derived by constructing a
particular pencil of elliptic curves whose period integrals provide the
solution to the induced velocity. We use numerical integration and a
perturbation expansion to evaluate the validity of our formulas.
