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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. 



By S. S. HOUGH, M.A., Fellow of St. John's College and Isaac Newton Student in 



the University of Cambridge. 



Communicated by Professor G. H. DARWIN, F.R.S. 



Received October 27, Read December 9, 1897. 



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 

 AlBYt 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. 



I was originally led to attack the problem by a totally different method from that 

 of LAPLACE based on the work of POINCABEJ and BRYAN, and the principal 

 analytical results, both in this paper and in the preceding, were at first obtained by 



* " On the General Integration of LAPIACE'S Differential Equation of the Tides." ' Phil. Mag.,' 1875. 

 t ' Encyc. Metropolitana.' Art, " Tides and Waves," Section III., 116. 



% " Snr 1'equilibre d'une masse fluide animee d'un mouvement de rotation." ' Acta Math.,' vol. 7, 

 p. 355, et seq. 



" The Waves on a Rotating Liquid Spheroid of Finite Ellipticity." ' Phil. Trans.,' A, 1889. 



T 2 24.5.98. 



