(ii.) dχ1/dυ = mz − ny, at the surface of the moving body, but zero over a fixed surface, and at infinity; the same for χ2 and χ3.
For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only,
φ1 = −x, φ2 = −y, φ3 = −z;
(2)
but a rotation will stir up the liquid in the cavity, so that the χ’s depend on the shape of the surface.
The ellipsoid was the shape first worked out, by George Green, in his Research on the Vibration of a Pendulum in a Fluid Medium (1833); the extension to any other surface will form an important step in this subject.
A system of confocal ellipsoids is taken
| x2 | + | y2 | + | z2 | = 1 |
| a2 + λ | b2 + λ | c2 + λ |
(3)
and a velocity function of the form