1886.] Dynamical Theory of the Tides of Long Period. 337 



IV. " On the Dynamical Theory of the Tides of Long Period." 

 By Gr. JEL Darwin, LL.D., F.R.S., Fellow of Trinity College 

 and Plnmian Professor in the University of Cambridge. 

 Received November 5, 1886. 



In the following note an objection is raised against Laplace's method 

 of treating these tides, and a dynamical solution of the problem, 

 founded on a paper by Sir William Thomson, is offered. 



Let 0, be the colatifcude and longitude of a point in the ocean, let £ 

 and rj sin be the displacements from its mean position of the water 

 occupying that point at the time t, let |y be the height of the tide, 

 and let z be the height of the tide according to the equilibrium theory ; 

 let n be the angular velocity of the earth's rotation, g gravity, a the 

 earth's radius, and 7 the depth of the ocean at the point 0, 0. 



Then Laplace's equations of motion for tidal oscillations are — 



d^B . „ dt] q d 



-~—2n sin cos 0- 1 — ~ 



at* dt 



sin 0^ + 2n cos 0% = 



g d 



dt asin0d^ ^ J 

 And the equation of continuity is — 



Id, . dri 

 - — ; sin 0) + 7^= 0. 



sin d0 y 



(2) 



The only case which will be considered here is where the depth of 

 the ocean is constant, and we shall only treat the oscillations of long 

 period in which the displacements are not functions of the longitude. 



As the motion to be considered only involves steady oscillation, we 

 assume — 



% = ecos (2nft + a) 



Ij = h cos (2nft + a) 

 £ — x cos (2nft + x) )> 



— y sin (2nft-\- a) 

 u = h—e J 

 Hence, by substitution in (1), we have 



xf 2, + yf sin cos 0= — — 



. JJ 4>md0 



(3) 



?// 2 sin + xfcos 0=0 



