# §19.11(i) General Formulas

 19.11.1 $\mathop{F\/}\nolimits\!\left(\theta,k\right)+\mathop{F\/}\nolimits\!\left(\phi% ,k\right)=\mathop{F\/}\nolimits\!\left(\psi,k\right),$
 19.11.2 $\mathop{E\/}\nolimits\!\left(\theta,k\right)+\mathop{E\/}\nolimits\!\left(\phi% ,k\right)=\mathop{E\/}\nolimits\!\left(\psi,k\right)+k^{2}\mathop{\sin\/}% \nolimits\theta\mathop{\sin\/}\nolimits\phi\mathop{\sin\/}\nolimits\psi.$

Here

 19.11.3 $\displaystyle\mathop{\sin\/}\nolimits\psi$ $\displaystyle=\frac{(\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits% \phi)\Delta(\phi)+(\mathop{\sin\/}\nolimits\phi\mathop{\cos\/}\nolimits\theta)% \Delta(\theta)}{1-k^{2}{\mathop{\sin\/}\nolimits^{2}}\theta{\mathop{\sin\/}% \nolimits^{2}}\phi},$ $\displaystyle\Delta(\theta)$ $\displaystyle=\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{2}}\theta}.$

Also,

 19.11.4 $\displaystyle\mathop{\cos\/}\nolimits\psi$ $\displaystyle=\frac{\mathop{\cos\/}\nolimits\theta\mathop{\cos\/}\nolimits\phi% -(\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi)\Delta(\theta)% \Delta(\phi)}{1-k^{2}{\mathop{\sin\/}\nolimits^{2}}\theta{\mathop{\sin\/}% \nolimits^{2}}\phi},$ $\displaystyle\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\psi\right)$ $\displaystyle=\frac{(\mathop{\sin\/}\nolimits\theta)\Delta(\phi)+(\mathop{\sin% \/}\nolimits\phi)\Delta(\theta)}{\mathop{\cos\/}\nolimits\theta+\mathop{\cos\/% }\nolimits\phi}.$

Lastly,

 19.11.5 $\mathop{\Pi\/}\nolimits\!\left(\theta,\alpha^{2},k\right)+\mathop{\Pi\/}% \nolimits\!\left(\phi,\alpha^{2},k\right)=\mathop{\Pi\/}\nolimits\!\left(\psi,% \alpha^{2},k\right)-\alpha^{2}\mathop{R_{C}\/}\nolimits\!\left(\gamma-\delta,% \gamma\right),$

where

 19.11.6 $\displaystyle\gamma$ $\displaystyle=(({\mathop{\csc\/}\nolimits^{2}}\theta)-\alpha^{2})(({\mathop{% \csc\/}\nolimits^{2}}\phi)-\alpha^{2})(({\mathop{\csc\/}\nolimits^{2}}\psi)-% \alpha^{2}),$ $\displaystyle\delta$ $\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$

# §19.11(ii) Case $\psi=\pi/2$

 19.11.7 $\mathop{F\/}\nolimits\!\left(\phi,k\right)=\mathop{K\/}\nolimits\!\left(k% \right)-\mathop{F\/}\nolimits\!\left(\theta,k\right),$
 19.11.8 $\mathop{E\/}\nolimits\!\left(\phi,k\right)=\mathop{E\/}\nolimits\!\left(k% \right)-\mathop{E\/}\nolimits\!\left(\theta,k\right)+k^{2}\mathop{\sin\/}% \nolimits\theta\mathop{\sin\/}\nolimits\phi,$

where

 19.11.9 $\mathop{\tan\/}\nolimits\theta=1/(k^{\prime}\mathop{\tan\/}\nolimits\phi).$
 19.11.10 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)=\mathop{\Pi\/}% \nolimits\!\left(\alpha^{2},k\right)-\mathop{\Pi\/}\nolimits\!\left(\theta,% \alpha^{2},k\right)-\alpha^{2}\mathop{R_{C}\/}\nolimits\!\left(\gamma-\delta,% \gamma\right),$

where

 19.11.11 $\displaystyle\gamma$ $\displaystyle=(1-\alpha^{2})(({\mathop{\csc\/}\nolimits^{2}}\theta)-\alpha^{2}% )(({\mathop{\csc\/}\nolimits^{2}}\phi)-\alpha^{2}),$ $\displaystyle\delta$ $\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$

# §19.11(iii) Duplication Formulas

If $\phi=\theta$ in §19.11(i) and $\Delta(\theta)$ is again defined by (19.11.3), then

 19.11.12 $\mathop{F\/}\nolimits\!\left(\psi,k\right)=2\!\mathop{F\/}\nolimits\!\left(% \theta,k\right),$
 19.11.13 $\mathop{E\/}\nolimits\!\left(\psi,k\right)=2\!\mathop{E\/}\nolimits\!\left(% \theta,k\right)-k^{2}{\mathop{\sin\/}\nolimits^{2}}\theta\mathop{\sin\/}% \nolimits\psi,$
 19.11.14 $\mathop{\sin\/}\nolimits\psi=(\mathop{\sin\/}\nolimits 2\theta)\Delta(\theta)/% (1-k^{2}{\mathop{\sin\/}\nolimits^{4}}\theta),$
 19.11.15 $\displaystyle\mathop{\cos\/}\nolimits\psi$ $\displaystyle=(\mathop{\cos\/}\nolimits(2\theta)+k^{2}{\mathop{\sin\/}% \nolimits^{4}}\theta)/(1-k^{2}{\mathop{\sin\/}\nolimits^{4}}\theta),$ $\displaystyle\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\psi\right)$ $\displaystyle=(\mathop{\tan\/}\nolimits\theta)\Delta(\theta),$ $\displaystyle\mathop{\sin\/}\nolimits\theta$ $\displaystyle=(\mathop{\sin\/}\nolimits\psi)/\sqrt{(1+\mathop{\cos\/}\nolimits% \psi)(1+\Delta(\psi))},$ $\displaystyle\mathop{\cos\/}\nolimits\theta$ $\displaystyle=\sqrt{\frac{(\mathop{\cos\/}\nolimits\psi)+\Delta(\psi)}{1+% \Delta(\psi)}},$ $\displaystyle\mathop{\tan\/}\nolimits\theta$ $\displaystyle=\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2}\psi\right)\sqrt{% \frac{1+\mathop{\cos\/}\nolimits\psi}{(\mathop{\cos\/}\nolimits\psi)+\Delta(% \psi)}},$
 19.11.16 $\mathop{\Pi\/}\nolimits\!\left(\psi,\alpha^{2},k\right)=2\mathop{\Pi\/}% \nolimits\!\left(\theta,\alpha^{2},k\right)+\alpha^{2}\mathop{R_{C}\/}% \nolimits\!\left(\gamma-\delta,\gamma\right),$
 19.11.17 $\displaystyle\gamma$ $\displaystyle=(({\mathop{\csc\/}\nolimits^{2}}\theta)-\alpha^{2})^{2}(({% \mathop{\csc\/}\nolimits^{2}}\psi)-\alpha^{2}),$ $\displaystyle\delta$ $\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$