Chasles's Contact Theorem

If a one-parameter family of curves has index $N$ and class $M$, the number tangent to a curve of order $n_1$ and class $m_1$ in general position is

$$m_1N+n_1M.$$

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 436, 1959.