Abel’s Theorem

Abstract

Finally, in order to exhibit a specially remarkable example of the importance of the substitution introduced in Lecture 26, which has already given us the solutions of a series of problems of mechanics, we shall apply it to Abel’s theorem. This theorem likewise concerns a certain system of ordinary differential equations and gives two different systems of integral equations of the same, one of which is expressed through transcendental functions and other though purely algebraic ones. These systems of integral equations, so different in form, are nevertheless completely identical.