On Poincar\'e's Fourth and Fifth Examples of Limit Cycles at Infinity

Errors are found in example problems from Henri Poincar\'e's paper ``M\'emoire sur les courbes d\'efinies par une \'equation diff\'erentielle.'' Examples four and five from chapter seven and examples one, two, and three from chapter nine do not have the limit cycles at infinity predicted by Poincar\'e. Instead they have fixed points at every point at infinity. In order to understand the errors made by Poincar\'e, examples four and five are studied at length. Replacement equations for the fourth and fifth examples are suggested based on the supposition that terms were omitted from Poincar\'e's equations.