1911 Encyclopædia Britannica/Spheroid

SPHEROID (Gr. σφαῖρα-ειδής, like a sphere), a solid resembling, but not identical with, a sphere in shape. In geometry; the word is confined to the figures generated by an ellipse revolving about a diameter. If the axis of revolution be the major axis of the ellipse, the spheroid is “prolate”; if the minor axis, “oblate”; if any other, “universal.”

If the generating ellipse has for its equation $\textstyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$, and revolves about the major axis, i.e. the axis of x, the volume of the solid generated is $\textstyle {\frac {4}{3}}\pi ab^{2}$, and its surface is $\textstyle 2\pi \left\{b^{2}+{\frac {ab}{e}}\sin ^{-1}e\right\}$, where e denotes the eccentricity. If the curve revolve about the minor axis, the volume is $\textstyle {\frac {4}{3}}\pi a^{2}b$, and the surface is $\textstyle \pi \left\{2a^{2}+{\frac {b^{2}}{e}}\log {\frac {(1+e)}{(1-e)}}\right\}$. The figure of the earth is frequently referred to as an oblate spheroid; this, however, is hardly correct, for the geoid has three unequal axes. The Cartesian equation to a spheroid assumes the forms $\textstyle {\frac {x^{2}}{a^{2}}}+{\frac {(y^{2}+z^{2})}{b^{2}}}=1$, for the prolate, and $\textstyle {\frac {(x^{2}+z^{2})}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$, for the oblate, the origin being the centre and the co-ordinate axes the axes of the original ellipse, $\textstyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$, and the line perpendicular to the plane containing them.

In physics, the term “spheroidal state” is given to the following phenomenon. If drops of a liquid be placed on a highly heated surface, for example, the top of a stove, the liquid forms a number of tremulous globules which continually circulate internally. There is no visible boiling, although the globule diminishes slowly in size. The theory of the experiment is that the liquid is surrounded by an elastic envelope of its vapour which acts, as it were, as a cushion preventing actual contact of the drop with the plate. On the formation of a similar protective cushion of vapour depends the immunity of such experiments as plunging a hand into a bath of molten metal.