Philosophical Transactions of the Royal Society (1701-)

Also

Now, if we assume the elastic solid is incompressible meaning its volume cannot be reduced by pressure, so that the value m is infinitely large compared to n, then it is clear that the equations of equilibrium for an incompressible elastic solid take exactly the same form as those for the flow of a viscous fluid—with n simply taking the place of v.

Thus, every problem involving the equilibrium of an incompressible elastic solid has a direct counterpart in a problem regarding the flow of an incompressible viscous fluid, provided the effects of inertia are ignored. The solution for one can be applied to the other by simply substituting "velocities" for "displacements" and the coefficient of "viscosity" for the coefficient of "rigidity."

2. A sphere under the influence of internal forces.

Sir William Thomson Sir William Thomson, later Lord Kelvin, was a preeminent physicist who calculated the age of the Earth and studied its internal heat and rigidity has solved the following problem:
To find the displacement of every point within an elastic sphere that is not subject to any surface pull original: traction, but is slightly deformed by a balanced system of forces acting throughout its interior.

If we substitute "velocity" for "displacement" and "viscous" for "elastic," we arrive at the corresponding problem for a viscous fluid; original: "mutatis mutandis" with the necessary changes made, the solution remains the same.

However, we cannot determine the tides of a viscous sphere simply by making the balancing force system equal to the tide-generating influence of the sun or moon. This is because we must assume the substance of the sphere possesses the power of gravitation.

Suppose that at any given time, the equation for the free surface of the Earth (which we may call the "viscous sphere" for brevity) is $r = a + \sum_2^\infty σ_i$, where $σ_i$ is a surface harmonic a mathematical function used to describe shapes on a sphere, such as the bulges and depressions of tides. In this case, the matter—whether positive (a bulge) or negative (a hollow)—filling the space represented by the sum of those harmonics exerts a gravitational pull on every point in the interior. This attraction, combined with the gravity of a perfectly uniform sphere of radius a, must be added to the tide-generating influence to determine the total force within the sphere. Additionally, we are dealing with a spheroid a sphere slightly flattened at the poles, like the Earth rather than a perfect sphere. However, if we mathematically "cut" a perfect sphere of radius a out of the spheroid (leaving out the tidal bulges $\sum σ_i$), then by choosing the right actions to apply to the surface, we can reduce the tidal problem to finding the flow within a perfect sphere under the influence of: (i) the external tide-generating force of the sun or moon, and (ii) the attraction of the