344 TIDAL WAVES. [CHAP. VIII 



We adopt this equilibrium-form of the free surface as a surface 

 of reference, and denote by 6 and o&amp;gt; the co-latitude (i.e. the angle 

 which the normal makes with the polar axis) and the longitude, 

 respectively, of any point upon it. We shall further denote by 

 z the altitude, measured outwards along a normal, of any point 

 above this surface. 



The relative position of any particle of the fluid being specified 

 by the three orthogonal coordinates 0, o&amp;gt;, z, the kinetic energy of 

 unit mass is given by 



where R is the radius of curvature of the meridian-section of the 

 surface of reference, and tzr is the distance of the particle from 

 the polar axis. It is to be noticed that .R is a function of 6 only, 

 whilst -or is a function of both 6 and z ; and it easily follows from 

 geometrical considerations that 



d^l(R -f z) dO = cos 0, dvr/dz = sin 6 (2). 



The component accelerations are obtained at once from (1) by 

 Lagrange s formula. Omitting terms of the second order, on 

 account of the restriction to infinitely small motions, we have 



1 / d dT dT\ v * 1 



...(3). 



I (ddT dT\ ( d f&amp;gt; 



-y -TT 3 = OTft) + 2ll -ja V + 



GT \ctt da) d(oj \du 



ddT dT .. , 2 LO ., dsr 

 -J--J. -j- = z (nt + 2n) & y- 

 dt dz dz ^ J dz 



Hence, if we write u, v, w for the component relative velocities of a 

 particle, viz. 



u=(R + z}Q ) v = &a) ) w = z ............ (4), 



and make use of (2), the hydrodynamical equations may be put in 

 the forms 



at 



C ^-+2nucos0 + 2nwsm0= - - 

 dt vr dco \p 



dt dz \p 



........................... (5), 



