Homoeoid and focaloid

If the outer shell is given by

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1}$

with semiaxes ${\displaystyle a,b,c}$, the inner shell of a homoeoid is given for ${\displaystyle 0\leq m\leq 1}$ by

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=m^{2},\quad {\displaystyle {\frac {x^{2}}{a^{2}+\lambda }}+{\frac {y^{2}}{b^{2}+\lambda }}+{\frac {z^{2}}{c^{2}+\lambda }}=1.}}$

and a focaloid is defined for ${\displaystyle \lambda \geq 0}$ by

${\displaystyle {\frac {x^{2}}{a^{2}+\lambda }}+{\frac {y^{2}}{b^{2}+\lambda }}+{\frac {z^{2}}{c^{2}+\lambda }}=1.}$

The thin homoeoid is then given by the limit ${\displaystyle m\to 1}$, and the thin focaloid is the limit ${\displaystyle \lambda \to 0}$.^[3]

Thin focaloids and homoeoids can be used as elements of an ellipsoidal matter or charge distribution that generalize the shell theorem for spherical shells. The gravitational or electromagnetic potential of a homoeoid homogeneously filled with matter or charge is constant inside the shell, so there is no force on a test particle inside of it.^[5] Meanwhile, two uniform, concentric focaloids with the same mass or charge exert the same potential on a test particle outside of both.^[4][1]