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

$$A_{i-1} = \frac{1}{2i+1} r^{-i+1} \frac{d}{dx} (r^i S_i), A_{i+1} = - \frac{1}{2i+1} r^{i+2} \frac{d}{dx} (r^{-i-1} S_i)$$

Similarly, $\frac{y}{a} S_i$ and $\frac{z}{a} S_i$ may be expressed as $B_{i-1} + B_{i+1}$ and $C_{i-1} + C_{i+1}$, where the $B$ and $C$ values only differ from the $A$ values by having $y$ and $z$ written instead of $x$.

We now have to form the auxiliary functions helper functions used to simplify complex equations $Ψ_{i-2}$ and $Φ_i$ corresponding to $A_{i-1}, B_{i-1}, C_{i-1}$, and $Ψ_i$ and $Φ_{i+2}$ corresponding to $A_{i+1}, B_{i+1}, C_{i+1}$.

Then, by the formulas original: formulæ (6):

$(2i+1) Ψ_{i-2} = \left( \frac{d^2}{dx^2} + \frac{d^2}{dy^2} + \frac{d^2}{dz^2} \right) r^i S_i = 0$

$\frac{2i+1}{r^{2i+1}} Φ_i = \frac{d}{dx} \left[ r^{-2i+1} \frac{d}{dx} (r^i S_i) \right] + \frac{d}{dy} \left[ \quad \right] + \frac{d}{dz} \left[ \quad \right] = - \frac{i(2i-1)}{r^{2i+1}} r^i S_i$

$-(2i+1) Ψ_i = \frac{d}{dx} \left[ r^{2i+3} \frac{d}{dx} (r^{-i-1} S_i) \right] + \frac{d}{dy} \left[ \quad \right] + \frac{d}{dz} \left[ \quad \right] = - (i+1)(2i+3) r^i S_i$

$- \frac{2i+1}{r^{2i+5}} Φ_{i+2} = \left( \frac{d^2}{dx^2} + \frac{d^2}{dy^2} + \frac{d^2}{dz^2} \right) r^{-i-1} S_i = 0$

Thus:

$Ψ_{i-2} = 0, Φ_i = - \frac{i(2i-1)}{2i+1} r^i S_i, Ψ_i = \frac{(i+1)(2i+3)}{2i+1} r^i S_i, Φ_{i+2} = 0$

Then, using equation (5), we form $α$ corresponding to $A_{i-1}, B_{i-1}, C_{i-1}$, as well as to $A_{i+1}, B_{i+1}, C_{i+1}$, and add them together. The final result is that a normal traction a pulling or pushing force acting perpendicular to the surface $S_i$ gives:

$α' = \frac{1}{v a^i} \left[ \left\{ \frac{i(i+2)}{2(i-1)[2(i+1)^2+1]} a^2 - \frac{(i+1)(2i+3)}{2(2i+1)[2(i+1)^2+1]} r^2 \right\} \frac{d}{dx} (r^i S_i) \right.$

$\left. - \frac{i}{(2i+1)[2(i+1)^2+1]} r^{2i+3} \frac{d}{dx} (r^{-i-1} S_i) \right] \dots \dots \dots (7)$

There are also symmetrical expressions for $β'$ and $γ'$.

In this context, $α', β',$ and $γ'$ are used instead of $α, β,$ and $γ$ to show that this is only a partial solution. Furthermore, $v$ is written instead of $n$ to show that it corresponds to the viscous problem the study of how thick, fluid-like materials flow under stress. If we now set $S_i = - g w σ_i$, we obtain the state of flow of the fluid caused by the transmitted pressure from the deficiencies and excesses of matter below and above the true spherical surface. This constitutes the solution as far as it depends on part (iii).

There remain the parts dependent on (i) and (ii), which may for now be classified together. For this part, Sir W. THOMSON's Sir William Thomson, later Lord Kelvin (1824–1907), a pioneer in thermodynamics and planetary physics solution is directly applicable. The state of internal strain of an elastic sphere—which is subject to no surface action but is under the influence of a bodily force a force, like gravity, that acts on the entire mass of an object rather than just its surface with a potential a mathematical function describing the intensity of a force field of $W_i$—may be immediately adapted to give the state of flow of a viscous sphere under similar conditions. The solution is—