MacCullagh ellipsoid

The MacCullagh ellipsoid is defined by the equation:

${\displaystyle {\frac {x^{2}}{A}}+{\frac {y^{2}}{B}}+{\frac {z^{2}}{C}}=2E,}$

where ${\displaystyle E}$ is the energy and ${\displaystyle x,y,z}$ are the components of the angular momentum, given in body's principal reference frame, with corresponding principal moments of inertia ${\displaystyle A,B,C}$. The construction of such ellipsoid was conceived by James MacCullagh.^[1]