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

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

3. The shape of the free surface at any time.

If $ρ'$ is the surface value of $ρ$ $ρ$ represents the rate of displacement, then

$ρ' = \frac{i(2i+1)}{2(i-1)[2(i+1)^2+1]} \frac{a^{i+1}}{v} T_i$

Consequently, after a short interval of time $δ t$, the equation for the bounding surface of the spheroid becomes $r = a + σ_i + ρ' δ t$. During this same interval, $σ_i$ has become $\frac{dσ_i}{dt} δ t$, which gives us:

$\frac{dσ_i}{dt} = ρ' = \frac{i(2i+1)}{2(i-1)[2(i+1)^2+1]} \frac{wa^{i+1}}{v} S_i - \frac{i}{2(i+1)^2+1} \frac{gwa}{v} σ_i$

$\frac{dσ_i}{dt} + \frac{i}{2(i+1)^2+1} \frac{gwa}{v} σ_i = \frac{i(2i+1)}{2(i-1)[2(i+1)^2+1]} \frac{wa^{i+1}}{v} S_i$ . . . . (11)

This differential equation shows the way in which the surface changes under the influence of the external potential $r^i S_i$.

If $S_i$ is not a function of time, and if $s_i$ is the value of $σ_i$ when $t=0$, then:

$σ_i = \frac{2i+1}{2(i-1)} \frac{a^i S_i}{g} \left[ 1 - \exp \left( \frac{-gwait}{[2(i+1)^2+1]v} \right) \right] + s_i \exp \left( \frac{-gwait}{[2(i+1)^2+1]v} \right)$ . . (12)original: I write "exp." for "$e$ to the power of."

When $t$ is infinite:

$σ_i = \frac{2i+1}{2(i-1)} \frac{a^i S_i}{g}$ . . . . . . . . . (13)

At this point, the flow stops because the fluid has taken the shape it would have assumed if it were not viscous at all. This result, naturally, agrees with the equilibrium theory of tides The "equilibrium theory" assumes the water or fluid responds instantly to gravity and centrifugal force, staying in perfect balance..

If $S_i$ is zero, the equation shows how the inequalities inequalities original: "inequalities"; here meaning physical irregularities or deviations from a smooth sphere, like mountains or basins. on the surface of a viscous globe would gradually subside under the influence of gravity alone. We can see that this change happens much more slowly if $i$ is large; that is to say, small-scale irregularities disappear much more slowly than wide-spread, large-scale ones. Might this solution provide some insight into the laws governing geological subsidence and upheaval? Darwin is suggesting that the Earth's internal viscosity might explain why certain geological features persist for so long or change slowly over millions of years.

4. Digression on the adjustments of the Earth to a form of equilibrium.

In a previous paper, I discussed certain points regarding the precession of a viscous spheroid and considered the rate at which it adjusts to a new form of equilibrium.