MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC

with symmetrical expressions for $β$ and $γ$; where $Ψ$ and $Φ$ are auxiliary functions helper functions used to simplify complex equations defined by:

$$\left. \begin{aligned} Ψ_{i-1} &= \frac{d}{dx}(A_i r^i) + \frac{d}{dy}(B_i r^i) + \frac{d}{dz}(C_i r^i) \ Φ_{i+1} &= r^{2i+3} \left{ \frac{d}{dx}(A_i r^{-i-1}) + \frac{d}{dy}(B_i r^{-i-1}) + \frac{d}{dz}(C_i r^{-i-1}) \right} \end{aligned} \right} \dots \text{(6)}$$

In the case considered by Sir W. THOMSON Sir William Thomson, later Lord Kelvin, a preeminent physicist of the 19th century of an elastic sphere deformed by internal stress original: bodily stress and subject to no surface action, we must substitute into equations (5) and (6) only those surface actions that are equal and opposite to the surface forces corresponding to the first part of the solution;* but in the case we now wish to consider, we must add to these latter forces the components of the normal traction a pulling force acting perpendicular to the surface $-gwΣσ_i$. Furthermore, we must include in the internal force both the external disturbing force and the gravitational attraction of the spheroid's matter upon itself.

Now, from the forms of equations (5) and (6), it is obvious that the tractions corresponding to the first part of the solution and the traction $-gwΣσ_i$ produce entirely independent effects. Therefore, we only need to add the terms arising from the normal traction $-gwΣσ_i$ to the complete solution of Sir W. THOMSON’s problem of the elastic sphere. Finally, we must transition from the elastic problem to the viscous one by substituting viscosity ($v$) for the shear modulus ($n$), and velocities for displacements.

I will now proceed to find the state of internal flow in the viscous sphere that results from a normal traction at every point of the sphere's surface, as given by the surface harmonic a mathematical function used to describe patterns on the surface of a sphere $S_i$.

In order to use formulas (5) and (6), it is first necessary to express the component tractions $\frac{x}{a} S_i, \frac{y}{a} S_i, \frac{z}{a} S_i$ as surface harmonics.

If $V_i$ is a solid harmonic, then:

$$\frac{d}{dx}(r^{-2i-1}V_i) = -(2i+1)r^{-(2i+3)}xV_i + r^{-(2i+1)}\frac{dV_i}{dx}$$

So that:

$$xV_i = \frac{1}{2i+1} \left{ r^2 \frac{dV_i}{dx} - r^{2i+3} \frac{d}{dx}(r^{-2i-1}V_i) \right}$$

Therefore:

$$\frac{x}{a} S_i = \frac{1}{2i+1} \left{ \left[ r^{-i+1} \frac{d}{dx}(r^i S_i) \right] - \left[ r^{i+2} \frac{d}{dx}(r^{-i-1} S_i) \right] \right}$$

The quantities within the brackets [ ] are independent of $r$ and are surface harmonics of orders $i-1$ and $i+1$ respectively. We now have $\frac{x}{a} S_i$ expressed as the sum of two surface harmonics $A_{i-1}, A_{i+1}$, where

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* Where the solid is incompressible cannot be squeezed into a smaller volume, this surface traction is perpendicular original: normal to the sphere at every point, provided that the potential of the internal force can be expressed as a series of solid harmonics.