ELLIPSOID, a closed surface of the second degree. Its plane sections are ellipses, its central (rectangular) equation being -2 =1, ly it may be formed after the analogy of the ellipse (q.v.) by aline transformation (see AFFINE GEOMETRY) the sphere by compressing uniformly, in the ratios b/a and c/a, all y and z coordinates, two sets of mutually lar chords, also perpendicular to OX. Less vividly it may be imagined as the path of a varying ellipse always parallel to a fixed plane (X Y), its vertices moving on two ellipses in two perpendicular planes, YZ and ZX (fig. 1) . For any equality among a, b, c, the surface becomes a roid. (See CONOID.) The elliptic plane sections become circular for two directions. For suppose a > b > c, then the plane XY cuts out the ellipse having the major and minor axes 2a, 2b. Turn the plane about Y or 2b, the minor axis 2b of the ellipse remains unchanged while the major axis decreases from 2a to 2C, for on turning through 90° the ellipse becomes At some stage 2b must have ceased to be minor and become major; at that stage the axes of the ellipse were equal, both being 2b, and the ellipse was a circle, and so were all parallel sections. The sections shrink into cyclic points as the planes become tangent. If 0 be the slope of the plane to the greatest axis 2a the opposite slope -9 would yield a like result; hence there two sets of cyclic sections and four cyclic points. (See fig. 2.) The volume of the ellipsoid is that of the major sphere reduced in the ratio In MECHANICS (q.v.), the ellipsoid plays an important role in the study of moments and inertia. (See SURFACE and MATHEMATICAL MODELS.)