178 Mr. D. D. Heath on the Dynamical Theory of 



We have now also a horizontal force arising from the attrac- 

 tion of the wave on particles situated in it, which cannot he 

 treated as insensible. When we deal with the problem by assu- 

 ming a wave-form with indeterminate coefficients, Laplace's func- 

 tions enable us to express this attraction. In the particular 

 case which I propose to consider, where y is expressed in terms 

 each proportional to one of the terms of P 2 , this attraction is 

 taken account of by reducing g, in the expressions for the pres- 

 sure-forces, by a constant quantity —■ (where p is the mean den- 

 sity of the earth compared to water). I will continue to write g 

 for this. 



Collecting these forces and joining with them the moon's force, 

 which we will represent by the differentiations of HP 2 , H being a 

 positive constant (see 8)., we have, after transpositions, and using 

 the equations (F), 



H-— — q ~ = 2?isin 0cos 6v— m-r-i 

 , dd * dd dco 



1 fUrfPfl dy\ . a dv a 



< H -r^ —g -j- >=z—msmd~ 7 2ra cosera. 



sin 6 



In forming the equation of continuity, we will suppose the 

 depth of the sea uniform in the same latitude, as seems abso- 

 lutely necessary in order to have a permanent wave-form ; but 

 it may vary from one latitude to another, or k may be a func- 

 tion of d. Conceive, then, a vertical column denned by two me- 

 ridians dco apart, and by two great circles running east and west 

 dd apart. Then the area of the northern face is k x sin @ dco, 

 and, the velocity southward being w, the whole influx through it 

 is proportional to k sin du dw. On the eastern face similarly the 

 influx is proportional to v sin d /cdd. The effluxes on the oppo- 

 site faces are these quantities increased by their differentials in 

 dd and dco respectively ; and to obtain the rate of depression, or 



— -— ( —m~- J, we must, as in (3), divide the balance of efflux 



by the base sin 6 dco dd. Whence, remembering that k is inde- 

 pendent of coj we get 



dy _ 1 d . k sin du dv 

 dco sin d dd dco 



d/c fdu cos 6 av7\ 



(H) 



„• 10. We must now, as before, attempt the solution of these 

 equations by taking each term of P 2 separately as if it wer^e the 



