ANALYSIS TO THE DYNAMICAL THEORY OP THE TIDES. 
211 
§ 3. Transformation of the Equation of Continuity. 
We proceed now to the transformation of our equations into a form analogous to 
that used by Laplace, dealing first with the equation of continuity. 
Let us refer to a system of orthogonal curvilinear coordinates a, /S, y, and suppose 
that the undisturbed surface of the ocean coincides with one of the surfaces y=const, 
say y = y^. 
On the surface y = yg take a small parallelogram PQLS, bounded by curves of the 
systems a = const, ^ — const. 
.s 
Through the sides of this parallelogram draw the surfaces a = const, yS = const, to 
meet the inner surface of the ocean in the cpiadrilateral P'QTb'S' and the distorted 
free surface in ixirs. 
The surfaces a = const, y8 = const, are, of course, supposed to be in rotation with 
angular velocity w, in common with the axes Ox, Oy, Oz. Let U, Y, W denote the 
relative velocity-components at the point (a, /3, y) parallel to tlie normals to the 
surfaces of reference and in the directions in which a, jS, y respectively increase, and 
let y = y^ where y^ may be regarded as a function of a, yS, be the equation to the 
surface of the solid earth. 
Let hi, h. 2 , be parameters associated with our orthogonal system of coordinates, 
such that the line-element ds is given by 
= da^lhf -f d/Syh^^ + dyfhf. 
Then the volume of liquid which flows in a unit of time across the face a = const 
of an elementary parallelepiped whose adjacent edges are 8a/7q, ^y/ho is 
But if in the above figure we suppose that a, /3, yQ are the coordinates of P, and 
that PQ = PS = 8a//q, the total flow of liquid across the face PP'Q'Q will be 
2 E 2 
