236 Mr. S. 8. Hough. On the Application of Harmonic 



44 On the Application of Harmonic Analysis to the Dynamical 

 Theory of the Tides. Part I. On Laplace's ' Oscillations 

 of the First Species/ and on the Dynamics of Ocean 

 Currents." By S. 8. Ho UGH, 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 March 12,— "Read April 8, 1897. 



(Abstract.) 



1. By a transformation of the differential equations of Poincare* 

 for the oscillations of a rotating mass of liquid, an equation for the 

 tidal oscillations of the ocean is obtained in a form similar to that 

 employed by Laplace. f From this equation it is deduced that if the 

 surface-value of the disturbing potential which gives rise to the 

 oscillations is expressible as a series of zonal harmonics in the 

 form 



27«P W (/*), 



then, provided the depth be a function of the latitude alone, the 

 height of the surface-waves will also be expressible as a series of 

 zonal harmonics ; if this series take the form 



the following relation connecting successive C's is shewn to hold 

 good when the depth is uniform : — 



c H . a Cn r vyw-i 2_ 



(2n— l)(2n— 3) * I n (> + 1) (2 n— l) (2 rc-f-3) 



_JH_/ 1 3p \\ C n+ 2 _Mn_ / x 



4wV\ (2rc + l)ff/J (2w + 3)(2rc + 5) 4wV 



where A. denotes the "speed" of the oscillation dealt with, w the 

 angular velocity of rotation, p, a the density of the water and the 

 mean density of the earth respectively, h the depth, and a the earth's 

 radius. 



A similar relation connecting three successive C's is also shewn to 

 hold good if the depth be a function of the latitude given by the 

 formula 



where sin -1 yu, denotes the latitude. 



2. If we put all the 7's zero in the formula (a), we may eliminate 



* 1 Acta Math.,' vol. 7, p. 356. 



r k Mec. C41.,' Part I, book 4, cap. i. 



