(11)
with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary, cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of a line doublet along each axis.
47. The extension of Green’s solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity function
φ = xyχ
(1)
for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface λ may be in rotation about Oz without disturbing the motion if
| R = − | [ 1/ (a2 + λ) + 1/ (b2 + λ) ] χ + 2 dx/dλ | , |
| 1 / (b2 + λ) − 1 / (a2 + λ) |
(2)
and that the continuity of the liquid is secured if
| (a2 + λ)3/2 (b2 + λ)3/2 (c2 + λ) ½ | dχ | = constant, |
| dλ |