ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
209 
water is small in comparison with the radius of the solid globe on which it resides^ a 
hypothesis which will certainly be applicable in the case of the earth. 
§ 2. llie Boundary-conditions. 
Before proceeding with the approximations which we propose to employ hereafter, 
let us examine the boundary-conditions to which the functions u, v, iv, ip are subject 
in the general case. 
Suppose the fluid resides on the surface of a solid nucleus which is constrained to 
rotate with uniform angular velocity co about the axis of 2 . We introduce this con¬ 
straint so as to avoid the complications resulting from the reactions of the fluid 
motion on that of the nucleus. Since in the case of the earth the mass of the ocean 
is exceedingly small compared with that of the solid parts, such reactions would be 
very minute, while for most of the more important types of oscillation they would not 
exist at all. In such types the problem is not affected by the introduction of the 
constraints. The boundaries of the ocean where it is in contact with the solid 
nucleus may then be regarded as fixed relatively to the moving axes Ox, Oy, Oz, and 
the condition to be satisfied at these boundaries is that there is no flow of fluid across 
them. Denoting by I, m, n the direction-cosines of the normal to the surface, this 
condition is expressed analytically by the equation 
\lu -f mv -h nid\ =0.(4). 
Next, let us examine the boundary-conditions at the free surface. Let I, m, n 
denote the direction-cosines of the normal to this surface in its undisturbed position, 
and let ^ be the distance between the displaced surface and the mean surface 
measured along the normal to the latter. Then we may equate the velocity at the 
mean surface in the direction of this normal to the rate at which ^ increases ; thus at 
the undisturbed surface we have 
\Ju -j- mv -f- mv\ — .(5), 
Lastly, we must express the condition that the pressure at the actual free surface 
is zero (or constant). Now if dn' denote an element of the normal to the undisturbed 
surface, and «, denote the values at this surface of the pressure and its rate of 
0)1 
increase along the normal, the pressure at the actual surface is 
v-\- i 
4 
7)1 
and this we have seen is to be equated to a constant. 
2 E 
MDCCCXCVII.-A. 
