MR. DARWIN ON THE BODILY TIDES OF VISCOUS AND SEMI-ELASTIC The title continues: "...Spheroids, and on the Ocean Tides upon a Yielding Nucleus."

$$α'' = \frac{1}{v} \left[ \left{ \frac{i(i+2)}{2(i-1)[2(i+1)^2+1]} a^2 - \frac{(i+1)(2i+3)}{2(2i+1)[2(i+1)^2+1]} r^2 \right} \frac{dW_i}{dx} \right.$$
$$\left. - \frac{i}{(2i+1)[2(i+1)^2+1]} r^{2i+3} \frac{d}{dx} (r^{-2i-1} W_i) \right] . \text{ . . . . . . . (8)*}$$

With symmetrical expressions for $β''$ and $γ''$.

I will first consider (ii); original: i.e. that is, the material of the earth is now supposed to possess the power of gravitation.

The gravitational potential the energy per unit mass at a point due to a gravitational field of the spheroid $r = a + σ_i$ (taking only a typical term of $σ$) at a point in the interior, estimated per unit volume, is

$$\frac{gw}{2a} (3a^2 - r^2) + \frac{3gw}{2i+1} \left( \frac{r}{a} \right)^i σ_i$$

according to the usual formula in the theory of the potential.

Now the first term, being symmetrical around the center of the sphere, can clearly cause no flow in the incompressible viscous sphere. We are therefore left with $\frac{3gw}{2i+1} \left( \frac{r}{a} \right)^i σ_i$.

Now if $\frac{3gw}{2i+1} \left( \frac{r}{a} \right)^i σ_i$ is substituted for $W_i$ in equation (8), and if the resulting expression is compared with equation (7) when $-gwσ_i$ is written for $S_i$, it will be seen that $-α'' = \frac{3}{2i+1} α'$.

Thus
$$α' + α'' = α'' \left( 1 - \frac{2i+1}{3} \right)^\dagger = -\frac{2}{3}(i-1)α''.$$

And if $V_i = \frac{3gw}{2i+1} \left( \frac{r}{a} \right)^i σ_i$,

$$α' + α'' = -\frac{1}{v} \left[ \left{ \frac{i(i+2)}{2(i-1)[2(i+1)^2+1]} a^2 - \frac{(i+1)(2i+3)}{2(2i+1)[2(i+1)^2+1]} r^2 \right} \frac{d}{dx} \left( \frac{2}{3}(i-1)V_i \right) \right.$$
$$\left. - \frac{i}{(2i+1)[2(i+1)^2+1]} r^{2i+3} \frac{d}{dx} \left( r^{-2i-1} \frac{2}{3}(i-1)V_i \right) \right] . \text{ . . . . . (9)}$$

with symmetrical expressions for $β' + β''$ and $γ' + γ''$.

Equation (9) then represents the solution as far as it depends on (ii) and (iii). And since (9) is the same as (8) when $-\frac{2}{3}(i-1)V_i$ is substituted for $W_i$, we may include all the effects of mutual gravitation the gravitational attraction between the different parts of the earth itself in producing a state of flow in the viscous sphere. We do this by adopting Thomson's solution (8) and using it instead of the true potential of the layer.

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* See 'Natural Philosophy', section 834, equation (8) where $m$ is infinite compared with $n$, $i-1$ is substituted for $i$, and $v$ replaces $n$. Referring to Thomson and Tait's 'Treatise on Natural Philosophy' (1867).

$\dagger$ The case in section 815 of Thomson and Tait's 'Natural Philosophy' is a special instance of this.