XXXVI. On the comparison of transcendents, with certain applications to the theory of definite integrals

Abstract

1. The following objects are contemplated in this paper:— 1st. The demonstration of a fundamental theorem for the summation of integrals whose limits are determined by the roots of an algebraic equation. 2ndly. The application of that theorem to the problem of the comparison of algebraic transcendents. The immediate object of this application will in each case be the finite expression of the value of the sum of a series of integrals, Ʃ∫X dx , the differential coefficient, X, being an algebraic function, and the values of x at the limits being determined by the roots of an algebraic equation.