# 1911 Encyclopædia Britannica/Spheroid

If the generating ellipse has for its equation ${\displaystyle \textstyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$, and revolves about the major axis, i.e. the axis of x, the volume of the solid generated is ${\displaystyle \textstyle {\frac {4}{3}}\pi ab^{2}}$, and its surface is ${\displaystyle \textstyle 2\pi \left\{b^{2}+{\frac {ab}{e}}\sin ^{-1}e\right\}}$, where e denotes the eccentricity. If the curve revolve about the minor axis, the volume is ${\displaystyle \textstyle {\frac {4}{3}}\pi a^{2}b}$, and the surface is ${\displaystyle \textstyle \pi \left\{2a^{2}+{\frac {b^{2}}{e}}\log {\frac {(1+e)}{(1-e)}}\right\}}$. 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 ${\displaystyle \textstyle {\frac {x^{2}}{a^{2}}}+{\frac {(y^{2}+z^{2})}{b^{2}}}=1}$, for the prolate, and ${\displaystyle \textstyle {\frac {(x^{2}+z^{2})}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$, for the oblate, the origin being the centre and the co-ordinate axes the axes of the original ellipse, ${\displaystyle \textstyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$, and the line perpendicular to the plane containing them.