344 TIDAL WAVES. [CHAP. VIII 



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

 of reference, and denote by 6 and a> 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, a>, 2, the kinetic energy of 

 unit mass is given by 



2T = (R + zj- 2 4- -C7 2 (n + o>) 2 + z* (1), 



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

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

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

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

 geometrical considerations that 



d^l(R + z) dd = cos 6, d&ldz = 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 



I/ j j ?n 3 1 

 I Ci (LI. CL 



^ + 



zr A dta 



1 /d dT dT\ 

 -^- T T--r-= 

 w \dtdA dcoj 



dd v ' dz~" "' (8) ' 



d dT dT_.. 



dt~dz~dz- = ""'" dz 



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

 particle, viz. 



-(+*) 4, w = r, w = z (4), 



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

 the forms 



C ^+Znucos0 + 2nwsin0= - - 

 dt vf da> \p 



dt 



'. (5) 



