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Cut view of a 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

[math]\displaystyle{ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 }[/math]

with semiaxes [math]\displaystyle{ a,b,c }[/math] the inner shell is given for [math]\displaystyle{ 0 \leq m \leq 1 }[/math] by

[math]\displaystyle{ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=m^2 }[/math].

The thin homoeoid is then given by the limit [math]\displaystyle{ m \to 1 }[/math]

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.[4]

See also


  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).
  4. Michel Chasles, Solution nouvelle du problème de l’attraction d’un ellipsoïde hétérogène sur un point exterieur, Jour. Liouville 5, 465–488 (1840)

External links