388 Sir William Thomson on the General Integration 



is isochronous in its motion with the water in V. The vessel V 

 is shown out of proportion small. When the right length of E 

 is obtained, the disk H moves up and down with the water with- 

 out a ripple for twenty minutes or more, showing how truly pen- 

 dular is the wave-motion. It follows that, for small disturbances, 

 the rate of vertical motion at the normal level must be closely 

 proportional to the altitude of the wave or amplitude of undulation. 



XLIV. General Integration of Laplace's Differential Equation of 

 the Tides. By Sir William Thomson, F.R.S.* 



1. " T APLACE considers the ocean as a rotating mass of fric- 

 J-^ tionless incompressible liquid covering a rotating rigid 

 spheroid to .a depth everywhere infinitely small in proportion to 

 the radius, and investigates its oscillations under the influence 

 of periodic disturbing forces, with the limitation that the rise and 

 fall is nowhere more than an infinitely small fraction of the 

 depth, the condition that the mean angular velocity of every 

 part of the liquid is the same as that of the solid, and the 

 assumption that the distance from summit to summit of the dis- 

 turbed water-surface is nowhere less than a large multiple of the 

 depth. This last assumption is, though not explicitly stated by 

 Laplace, implied in, and is virtually equivalent to, his assump- 

 tions (Mecanique Celeste, Livre I. No. 36) that the vertical mo- 

 tion of the water is small in comparison with its horizontal 

 motion, and that the horizontal motion is sensibly the same for 

 all depths. 



2. Let now h be the elevation of the water-surface above mean 

 level, and f and tj sin 6 the southward and eastward horizontal 

 component displacements of the water at time t, and at the place 



whose north latitude is ^ — 6 (or north-polar distance 6) and 



east longitude ^ . The " equation of continuity " [Mec. Cel. 

 Liv. I. No. 36, or Airy, " Tides and Waves " {Encyclopedia 

 Metropolitana) , art. (72)], is 



<%?) , Tfcosj? d(rn) , hn ,,\ 



~W" + ~sTnT~ + ~d^ +/l =°> • ' ' C 1 ) 

 or 



sm0d0- + -dir- +h -°> ' ' ' ' 0)** 



where y denotes the ratio of the depth of the sea to the earth's 

 radius. And the dynamical equations [Mec. Cel. Liv. I. No. 



* Communicated by the Author, being an extension of a paper read in 

 Section A of the British Association at its recent Meeting in Bristol. 



