# What exactly was Lagrange's “grave mistake” with respect to rotating bodies under hydrostatic equilibrium?

A comment below What would be different about satellite orbits if Earth were prolate? Would we have Sun-synchronous and Molniya orbits? got me reading Wikipedia's Jacobi ellipsoid which begins:

Before Jacobi, the Maclaurin spheroid, which was formulated in 1742, was considered to be the only type of ellipsoid which can be in equilibrium. Lagrange in 18114 considered the possibility of a tri-axial ellipsoid being in equilibrium, but concluded that the two equatorial axes of the ellipsoid must be equal, leading back to the solution of Maclaurin spheroid. But Jacobi realized that Lagrange's demonstration is a sufficiency condition, but not necessary. He remarked, "One would make a grave mistake if one supposed that the spheroids of revolution are the only admissible figures of equilibrium even under the restrictive assumption of second degree surfaces" and further adds that "In fact a simple consideration shows that ellipsoids with three unequal axes can very well be figures of equilibrium; and that one can assume an ellipse of arbitrary shape for the equatorial section and determine the third axis (which is also the least of the three axes) and the angular velocity of rotation such that the ellipsoid is a figure of equilibrium.5"

If I understand correctly, Lagrange came to the conclusion that a rotating body under hydrostatic equilibrium (such as a planet) would be of necessity a spheroid and so the cross-section through the equator would be a circle. Jacobi then found that this is not necessarily true, and a rotating triaxial ellipsoid could also be in hydrostatic equilibrium.

Questions:

1. Is my understanding of the situation correct?
2. Did J. L. Lagrange make an error in this case? If so, what was it exactly? Was it intuitive or mathematical in nature?
• – uhoh Jan 19 '20 at 1:28
• Just reading the section of your quote (...sufficien[t] but not necessary..." seems to indicate that the 'grave mistake' was in thinking that failure to meet that sufficiency condition implied you couldn't reach equilibrium. Jacobi shows that Lagrange had rejected possible solutions. – Carl Witthoft Jan 20 '20 at 12:29
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