206] LAPLACE'S PROBLEM. 345 



where M* is the gravitation-potential due to the earth's attraction, 

 whilst II denotes the potential of the extraneous forces. 



So far the only approximation consists in the omission of terms 

 of the second order in u, v, w. In the present application, the 

 depth of the sea being small compared with the dimensions of the 

 globe, we may replace R + z by R. We will further assume that 

 the effect of the relative vertical acceleration on the pressure may 

 be neglected, and that the vertical velocity is small compared with 

 the horizontal velocity. The last of the equations (5) then re- 

 duces to 



az \p 



Let us integrate this between the limits z and f, where f 

 denotes the elevation of the disturbed surface above the surface 

 of reference. At the surface of reference (z 0) we have 



w 2 w 2 = const., 



by hypothesis, and therefore at the free surface (z = f) 

 -i?i 2 OT 2 = const. 



provided ?-;r(-i'** a ) .................. (?) 



\_az Jz=o 



Here g denotes the value of apparent gravity at the surface of 

 reference; it is of course, in general, a function of 6. The 

 integration in question then gives 



const. +#+fl ............ (8), 



the variation of H with z being neglected. Substituting from (8) 

 in the first two of equations (5), we obtain, with the approxima- 

 tions above indicated, 



, 



- 



dt 



where ?=-fl/0 ........................... (10). 



These equations are independent of z> so that the horizontal 

 motion may be assumed to be sensibly the same for all particles 

 in the same vertical line. 



