**ELLIPSOID,** a quadric surface whose sections are ellipses. Analytically, it has for its equation *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} + *z*^{2}/*c*^{2} = 1, *a*, *b*, *c* being its axes; the name is also given to the solid contained by this surface (see Geometry: *Analytical*). The solids and surfaces of revolution of the ellipse are sometimes termed ellipsoids, but it is advisable to use the name spheroid (*q.v.*).

The ellipsoid appears in the mathematical investigation of physical properties of media in which the particular property varies in three directions within the media; such properties are the elasticity, giving rise to the strain ellipsoid, thermal expansion, ellipsoid of expansion, thermal conduction, refractive index (see Crystallography), &c. In mechanics, the ellipsoid of gyration or inertia is such that the perpendicular from the centre to a tangent plane is equal to the radius of gyration of the given body about the perpendicular as axis; the “momental ellipsoid,” also termed the “inverse ellipsoid of inertia” or Poinsot’s ellipsoid, has the perpendicular inversely proportional to the radius of gyration; the “equimomental ellipsoid” is such that its moments of inertia about all axes are the same as those of a given body. (See Mechanics.)