]> $$\begin{array}{ccc}D\left(\phi ,k\right)& =& {\int }_0^{\phi }\frac{{\sin }^2\theta }{\sqrt{\left(1-k^2{\sin }^2\theta \right)}}d\theta \\ & =& {\int }_0^{\sin \phi }\frac{t^2}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}dt\\ & =& \frac{\left(F\left(\phi ,k\right)-E\left(\phi ,k\right)\right)}{k^2}\\ & =& \frac{1}{3}{R}_D\left(c-\mathrm{1,}c-k^{\mathrm{2,}}c\right);c=\frac{1}{{\sin }^2\phi }\end{array}$$