SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING NUCLEUS.

ever, this solution applies to a true sphere and to the positive and negative matter filling the space $Σσ_i$, but (iii) is also subject to certain surface forces.

Since points (i) and (ii) together constitute a "bodily force" a force, like gravity, that acts throughout the entire volume of an object, the problem only differs from that of Sir William Thomson in the fact that there are forces acting specifically on the surface of the sphere.

Now, as we are only going to consider small deviations from a perfect sphere original: sphericity, these surface actions will be minor, and an approximation will be permissible.

It is clear that, strictly speaking, there is a tangential action* a "sideways" or shearing force acting along the surface between the layer of matter $Σσ_i$ and the core sphere. However, by far the larger part of the action is "normal" acting perpendicular to the surface, and is simply the weight (either positive or negative) of the matter which lies above or below any point on the surface of the core sphere.

Thus, in order to treat the Earth as a perfect sphere, the appropriate surface action is a normal pull original: traction equal to $-gwΣσ_i$, where $g$ is gravity at the surface and $w$ is the density original: mass per unit volume of the matter making up the Earth.

In order to show what alteration this normal surface pull will make in Sir William Thomson’s solution, I must now give a short account of his method of attacking the problem.

He first shows that, where a "potential function" exists, the solution of the problem may be subdivided. The complete values of the displacements $α, β, γ$ consist of the sum of two parts which are found in different ways.

The first part consists of any values of $α, β, γ$ which satisfy the equations throughout the sphere, without considering the conditions at the surface.

For the second part, the bodily force is treated as non-existent and is replaced by certain surface actions. These are calculated to counteract the surface actions that resulted from the first part of the solution. Thus, the first part satisfies the condition that there is a bodily force, and the second adds the condition that the total forces at the surface are zero.

The first part of the solution is easily found. For the second part, Sir William Thomson discusses the case of an elastic sphere under the action of any surface pulls, but without any bodily force acting on it. In this problem, the component surface pulls parallel to the three axes are expanded in a series of "surface harmonics" mathematical functions used to represent complex shapes or distributions on the surface of a sphere. He shows that the harmonic terms of any "order" have an effect on the displacements that is independent of every other order. Thus, it is only necessary to consider the typical component surface pulls $\text{A}_i, \text{B}_i, \text{C}_i$ of the order $i$.

He proves that (for an incompressible elastic solid, where the resistance to compression $m$ is infinite) this single surface pull $\text{A}_i, \text{B}_i, \text{C}_i$ produces a displacement throughout the sphere given by:

$α = \frac{1}{nai^{i-1}} \left{ \frac{a^2 - r^2}{2(2i^2 + 1)} \frac{dΨ_{i-1}}{dx} + \frac{1}{i-1} \left[ \frac{i+2}{(2i^2 + 1)(2i + 1)} r^{2i+1} \frac{d}{dx} (Ψ_{i-1} r^{-2i+1}) + \frac{1}{2i(2i + 1)} \frac{dΦ_{i+1}}{dx} + \text{A}_i r^i \right] \right} (5)\dagger$

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* I shall consider some of the effects of this tangential action in a future paper, titled "Problems connected with the Tides of a Viscous Spheroid," read before the Royal Society on December 19th, 1878.

$\dagger$ Thomson and Tait’s Natural Philosophy [original: ‘Nat. Phil.’], 1867, § 737, equation (52).