where is an elliptic integral of the first kind and is an elliptic integral of the second kind,
with the amplitude being
and the moduli
and
where the anisotropy parameters, and , are
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_b = \left( \frac{b}{a} \right)^2 -1,}
and
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_c = \left( \frac{c}{a} \right)^2 -1.}
The volume of the ellipsoid is given by the well known
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = \frac{4 \pi}{3}abc.}
![{\displaystyle R={\frac {a}{2}}\left[{\sqrt {\frac {1+\epsilon _{b}}{1+\epsilon _{c}}}}+{\sqrt {\epsilon }}_{c}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{1})+E(\varphi ,k_{1})\right\}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e881e50476a138c1b82d0eff3fa835b1212160d)
![{\displaystyle S=2\pi a^{2}\left[1+{\sqrt {\epsilon _{c}(1+\epsilon _{b})}}\left\{{\frac {1}{\epsilon _{c}}}F(\varphi ,k_{2})+E(\varphi ,k_{2})\right\}\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0370210651aa0d817cf6bd9f11fc9898f4a23b)
