SPHEROID

, a solid body approaching to the figure of a sphere, though not exactly round, but having one of its diameters longer than the other.

This solid is usually considered as generated by the rotation of an oval plane figure about one of its axes. If that be the longer or transverse axis, the solid so generated is called an oblong Spheroid, and sometimes prolate, which resembles an egg, or a lemon; but if the oval revolve about its shorter axis, the solid will be an oblate Spheroid, which resembles an orange, and in this shape also is the figure of the earth, and the other planets.

The axis about which the oval revolves, is called the fixed axis, as AB; and the other CD is the revolving axis: whichever of them happens to be the longer.

When the revolving oval is a perfect ellipse, the so- lid generated by the revolution is properly called an ellipsoid, as distinguished from the Spheroid, which is generated from the revolution of any oval whatever, whether it be an ellipse or not. But generally speaking, in common acceptation, the term Spheroid is used for an ellipsoid; and therefore, in what follows, they are considered as one and the same thing.

Any section of a Spheroid, by a plane, is an ellipse (except the sections perpendicular to the fixed axe, which are circles); and all parallel sections are similar ellipses, or having their transverse and conjugate axes in the same constant ratio; and the sections parallel to the fixed axe are similar to the ellipse from which the solid was generated. See my Mensuration pa. 267 &c, 2d edit.

For the Surface of a Spheroid, whether it be oblong or oblate. Let f denote the fixed axe, r the revolving axe, ; then will the surface s be expressed by the following series, using the upper signs for the oblong spheroid, and the under signs for the oblate one; viz, &c; where the signs of the terms, after the first, are all negative for the oblong Spheroid, but alternately positive and negative for the oblate one.

Hence, because the factor 4arf is equal to 4 times the area of the generating ellipse, it appears that the surface of the oblong Spheroid is less than 4 times the generating ellipse, but the surface of the oblate Spheroid is greater than 4 times the same: while the surface of the sphere falls in between the two, being just equal to 4 times its generating circle.

Huygens, in his Horolog. Oscillat. prop. 9, has given two elegant constructions for describing a circle equal to the superficies of an oblong and an oblate Spheroid, which he says he found out towards the latter end of the year 1657. As he gave no demonstrations of these, I have demonstrated them, and also rendered them more general, by extending and adapting them to the surface of any segment or zone of the Spheroid. See my Mensuration, pa. 308 &c, 2d ed. where also are several other rules and constructions for the surfaces of Spheroids, besides those of their segments, and frustums.

Of the Solidity of a Spheroid. Every Spheroid, whether oblong or oblate, is, like a sphere, exactly equal to two-thirds of its circumscribing cylinder. So that, if f denote the fixed axe, r the revolving axe, and ; then 2/3 afr² denotes the solid content of either Spheroid. Or, which comes to the same thing, if t denote the transverse, and c the conjugate axe of the generating ellipse; then (2/3)ac²t is the content of the oblong Spheroid, and (2/3)act² is the content of the oblate Spheroid. Consequently, the proportion of the former solid to the latter, is as c to t, or as the less axis to the greater.

Farther, if about the two axes of an ellipse be ge- | nerated two spheres and two spheroids, the four solids will be continued proportionals, and the common ratio will be that of the two axes of the ellipse; that is, as the greater sphere, or the sphere upon the greater axe, is to the oblate Spheroid, so is the oblate Spheroid to the oblong Spheroid, and so is the oblong Spheroid to the less sphere, and so is the transverse axis to the conjugate. See my Mensuration, pa. 327 &c, 2d ed. where may be seen many other rules for the solid contents of Spheroids, and their various parts. See also Archimedes on Spheroids and Conoids.

Dr. Halley has demonstrated, that in a sphere, Mercator's nautical meridian line is a scale of logarithmic tangents of the half complements of the latitudes. But as it has been found that the shape of the earth is spheroidal, this figure will make some alteration in the numbers resulting from Dr. Halley's theorem. Maclaurin has therefore given a rule, by which the meridional parts to any Spheroid may be found with the same exactness as in a sphere. There is also an ingenious tract by Mr. Murdoch on the same subject. See Philos. Trans. No. 219. Mr. Cotes has also demonstrated the same proposition, Harm, Mens. pa. 20, 21. See Meridional Parts.

Universal Spheroid, a name given to the solid generated by the rotation of an ellipse about some other diameter, which is neither the transverse nor conjugate axis. This produces a figure resembling a heart. See my Mensuration, pa. 352, 2d ed.