# Homoeoid

Homoeoid in 3D

A homoeoid is a shell (a bounded region) bounded by two concentric, similar ellipses (in 2D) or ellipsoids (in 3D). [1][2] When the thickness of the shell becomes negligible, it is called a thin homoeoid. The name homoeoid was coined by Lord Kelvin and Peter Tait[3].

## Mathematical definition

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 is given for ${\displaystyle 0\leq m\leq 1}$ by

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

The thin homoeoid is then given by the limit ${\displaystyle m\to 1}$

## Physical meaning

A homoeoid can be used as a construction element of a matter or charge distribution. The gravitational or electromagnetic potential of a homoeoid homogeneously filled with matter or charge is constant inside the shell. This means that a test mass or charge will not feel any force inside the shell.

## References

1. ^ Chandrasekhar, S.: Ellipsoidal Figures of Equilibrium, Yale Univ. Press. London (1969)
2. ^ Routh, E. J.: A Treatise on Analytical Statics, Vol II, Cambridge University Press, Cambridge (1882)
3. ^ Harry Bateman. "Partial differential equations of mathematical physics.", Cambridge, UK: Cambridge University Press, 1932 (1932).