SPHEROID

(Yunanca köken — orijinale bakınız.) In Geo metry, a solid generated by the revolution of an ellipse about one of its axes. If the generating ellipse revolves about its major axis, the spheroid is prolate, or oblong; if about its minor axis, the spheroid is oblate. Let 2 a be the axis of revolution, and 2 b the diameter of the generating ellipse perpendicular to the axis; then the origin of the co-ordinates being at the centre, and x being taken on the semi-axis a, the equation of the surface of the spheroid is = 1. x2, yl -4-;2 _ H Let k2 denote the ratio of the difference of the squares of a and b to the square of a; that is, tnako>n2 A-2= ai — b'~, and put w = 3-14159; then the whole surface S of the spheroid is expre ed by the following series, in which the upper signs are to be used if the spheroid is oblong (that is, if a is greater than b), and the under signs if the spheroid is ob late: viz.,+: The solid content of any spheroid, whether oblate or ob long, is equal to two thirds of its circumscribing cylinder, and is therefore equal to | jr a 62. And since the content of a sphere, of which the radius is equal to a, is A tt a\ it follows that the content of the spheroid is to the content of a sphere whose diameter is equal to the axis of revolution as 62: o2. The oblate spheroid being the figure a umed by the earth and the other planets, its properties are of great importance In astronomy and geodesy. Newton demonstrated that the attraction of a sphere on any exterior body is th0 same as if all the matter in the sphere were collected into a point at the centre; but this property does not hold true of spheroids, and the calculation of the effects of attraction is thereby rendered greatly more difficult. Sec Gravitation. In geodetical operations, it is nece ary to have regard to the curvature of the spheroidal surface of the earth. Sup posing the elements of the spheroid (that is, the polar and equatorial axes) to be known, the cirvature of the surface [s. 1160]