V. On the application of harmonic analysis to the dynamical theory of the tides.—Part II. On the general integration of Laplace’s dynamical equations

Abstract

In the former paper on this subject I have dealt with the formation of Laplace’s dynamical equation for the tides, and the integration of it, subject to the limitation that the solutions obtained should be symmetrical with respect to the axis of rotation. In the present paper I propose to extend the method of solution so as to free it from this restriction. The difficulties experienced by Laplace in his attempts to integrate the equation in question were so great that he abandoned all efforts to obtain a general solution, and confined his discussion to a few of the special cases which present the greatest interest from a practical point of view; even in these simple cases however his original attempts to express the solutions by means of the coefficients associated with his name were discarded in favour of series proceeding according to powers of a certain variable used to define the position of a point on the Earth’s surface. These power-series have been further employed by Lord Kelvin to obtain a more general solution of the problem, but the results obtained, though of considerable analytical interest, do not lend themselves well to a numerical discussion. Both Airy and Kelvin condemn the employment of the surface-harmonic functions as inappropriate, but a profound conviction that the efforts of Laplace, though unsuccessful, were well directed, has led me to take up the problem again from his point of view; with what success will be seen hereafter.