In regard to Mark Sylvester's case (i): the stake through the spheres
contributes a force not present in the gravitational problem. I would
expect this force to be important as it has the same magnitude as the
gravitational attraction. Why is this case relevant to the discussion?
I strongly disagree with the proposition that the effects of oceans,
rotations, and revolutions can be added later. They can be meaningfully
added only if the stake is simultaneously removed. In other words, only
if we solve the problem at hand.
His case (ii) of the spheres falling freely toward each other is just
the case we are discussing. The Earth and the Moon *are* falling freely
toward one another, and in this case (where they revolve about one
another in circular orbits) it is possible to simplify the problem by
moving to a rotating frame of reference after the scheme of that clever
Frenchman, Joseph Lagrange.
If I change my model to an elastic sphere it will not greatly affect
the outcome. I see no reason to stake the sphere at all. It will stay
put quite nicely in the rotating frame without any restraints applied.
Once it has settled down (Jell-O, as it is called in North America, is
quite elastic) I'm willing to stake it at any or all points. So long
as the stakes exert no forces I see no role for them, however. With
the elastic sphere the figure will still approach the equipotential;
there will be a bulge on each side, and the circumference in the plane
perpendicular to the axis will be reduced. In my original model (with
a rigid sphere) I can stake it anywhere and the stake will exert no
force.