SPHEROIDS, AND ON THE OCEAN TIDES UPON A YIELDING CORE original: nucleus; here referring to the Earth's central core.

2 B

$$\left. \begin{aligned} -\frac{dp}{dx} + v\nabla^2α + \text{X} = 0 \ -\frac{dp}{dy} + v\nabla^2β + \text{Y} = 0 \ -\frac{dp}{dz} + v\nabla^2γ + \text{Z} = 0 \end{aligned} \right} \text{ . . . . . . . . . . (1)}$$

In these equations, $x, y, z$ represent the rectangular coordinates of a point within the fluid; $α, β, γ$ the Greek letters alpha, beta, and gamma are the component velocities moving parallel to those axes; $p$ is the average of the three pressures measured across planes perpendicular to the three axes respectively; X, Y, and Z are the component forces acting on the fluid, calculated per unit of volume; $v$ is the coefficient of viscosity a measure of a fluid's resistance to flow; essentially its thickness or "stickiness"; and $\nabla^2$ represents the Laplacian operation a mathematical tool used to describe how a quantity spreads or changes through space $\frac{d^2}{dx^2} + \frac{d^2}{dy^2} + \frac{d^2}{dz^2}$.

In addition to these, we have the equation of continuity a mathematical rule stating that mass is neither created nor destroyed as fluid flows:
$$\frac{dα}{dx} + \frac{dβ}{dy} + \frac{dγ}{dz} = 0$$

Also, if P, Q, R, S, T, and U are the normal and tangential stresses forces that act parallel to a surface, like friction, rather than pushing directly against it calculated in the standard way across three planes perpendicular to the axes:

$$\left. \begin{aligned} \text{P} = -p + 2v\frac{dα}{dx}, \quad \text{Q} = -p + 2v\frac{dβ}{dy}, \quad \text{R} = -p + 2v\frac{dγ}{dz} \ \text{S} = v\left(\frac{dβ}{dz} + \frac{dγ}{dy}\right), \quad \text{T} = v\left(\frac{dγ}{dx} + \frac{dα}{dz}\right), \quad \text{U} = v\left(\frac{dα}{dy} + \frac{dβ}{dx}\right) \end{aligned} \right} \text{ . . . . . (2)}$$

Now, in an elastic solid a material that returns to its original shape after being deformed, if $α, β, γ$ are the displacements (the distance a point has moved), $m - \frac{1}{3}n$ is the coefficient of dilatation a measure of how much the volume of the material expands or contracts, and $n$ is the coefficient of rigidity the material's resistance to changing its shape; and if $δ$ the Greek letter delta represents the change in volume ($δ = \frac{dα}{dx} + \frac{dβ}{dy} + \frac{dγ}{dz}$), then the equations of equilibrium equations describing a state where all forces are balanced and the object is at rest are:

$$\left. \begin{aligned} m\frac{dδ}{dx} + n\nabla^2α + \text{X} = 0 \ m\frac{dδ}{dy} + n\nabla^2β + \text{Y} = 0 \ m\frac{dδ}{dz} + n\nabla^2γ + \text{Z} = 0 \end{aligned} \right} \text{ . . . . . . . . . . (3)*}$$

Also:

$$\text{P} = (m-n)δ + 2n\frac{dα}{dx}, \quad \text{Q} = (m-n)δ + 2n\frac{dβ}{dy}, \quad \text{R} = (m-n)δ + 2n\frac{dγ}{dz} \text{ . . . (4)}$$

The values for S, T, and U take the same forms as seen in equation (2), but with $n$ written instead of $v$.

Therefore, if we define $-p$ as one-third of the sum of P, Q, and R ($-p = \frac{1}{3}(\text{P} + \text{Q} + \text{R})$), we find that $p = -(m - \frac{n}{3})δ$. Consequently, equation (3) can be rewritten as:

See Thomson and Tait’s Natural Philosophy*, section 698, equations 7 and 8.