(15)
with similar equations for β and γ.
If we can make
(4πρA + α) x2 = (4πρB + β) b2 = (4πρC + γ) c2,
(16)
the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided α, β, γ remain constant.
This is so when the axis of revolution is a principal axis, say Oz; when
Ω1 = 0, Ω2 = 0, ξ = 0, η = 0.
(17)
If Ω3 = 0 or θ3 = ζ in addition, we obtain the solution of Jacobi’s ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin’s solution is obtained of the rotating spheroid.