WITH THE TIDES OF A VISCOUS SPHEROID. 
541 
tangential to tlie surface of the sphere, which are exercised by the layer cr on the 
sphere. 
Let 9, (f> be the colatitude and longitude of a point in the layer. Then consider a 
prismatic element bounded by lire two cones 9, 9-\-h9, and by the two planes </>, <£+§</>. 
The radial faces of this prism are acted on by the pressures and tangential stresses 
communicated by the four contiguous prisms. But the tangential stresses on these 
faces only arise from the fact that contiguous prisms are solicited by slightly different 
forces, and therefore the action of the four prisms, surrounding the prism in question, 
must be principally pressure. I therefore propose to consider that the prism resists 
the tendency of the impressed forces to move tangentially along the surface of the 
sphere, by means of hydrostatic pressures on its four radial faces, and by a tangential 
stress across its base. 
This approximation by which the whole of the tangential stress is thrown on to the 
base, is clearly such as slightly to accentuate, as it were, the distribution of the 
tangential stresses on the surface of the sphere, by which the equilibrium of the layer 
or is maintained. For consider the following special case :—Suppose cr to be a surface of 
revolution, and Y to be such that only a single small circle of latitude is solicited by a 
tangential force everywhere perpendicular to the meridian. Then it is obvious that, 
strictly speaking, the elements lying a short way north and south of the small circle 
would tend to be carried with it, and the tangential stress on the sphere would be a 
maximum along the small circle, and would gradually die away to the north and 
south. In the approximate method, however, which it is proposed to use, such an 
application of external force would be deemed to cause no tangential stress to the 
surface of the sphere to the north and south of the small circle acted on. This special 
case is clearly a great exaggeration of what holds in our problem, because it postulates 
a Unite difference of disturbing force between elements infinitely near to one another. 
We will first find what are the hydrostatic pressures transmitted by the four 
prisms contiguous to the one we are considering. 
Let p be the hydrostatic pressure at the point r, 9, <£ of the layer cr. Then if we 
neglect the variations of gravity due to the layer cr and to Y, p is entirely due to the 
attraction of the mean sphere of radius a. 
The mean pressure on the radial faces at the point in question is ; where cr 
is negative the pressures are of course tractions. 
We will first resolve along the meridian. 
The excess of the pressure acting on the face 9 +8# over that on the face 9 (whose 
area is era sin 9S(f>) is 
f ]JA9 Wcr - aa s ^ u 98(f)]S9, or |ycca ^(cr ' sin 9)S9S(f>, 
and it acts towards the pole. 
The resolved part of the pressures on the faces <£+§<£ and (j> (whose area is <raS9) 
along the meridian is 
