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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[edit]

If the outer shell is given by

with semiaxes the inner shell is given for by


The thin homoeoid is then given by the limit

Physical meaning[edit]

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.


  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).

See also[edit]