This paper contains a demonstration of the theorem given in the fourth volume of the proceedings at p. 419.

The theorem in question was arrived at by the comparison of the well-known formula for the time of descent in a circular arc, with another formula given in the “Edinburgh Philosophical Magazine” for November 1828, by a writer under the signature T. W. L. Each of these series is reached by a long train of transformations, developments, and integrations, which require great familiarity with the most advanced branches of the higher calculus. Yet the theorem which results from their comparison has an aspect of extreme simplicity, and seems as if it could be reached by an easier road.