SOLUTION

$\overrightarrow{OS}|_{\mathcal{R}_L}=\mathcal{Q}^{-1}\cdot\mathcal{P}^{-1}\cdot\overrightarrow{OS}|_{\mathcal{R}_{SL}}=\left\lbrack \begin{array}{ccc} \cos A & -\sin A & 0 \\ \sin A & \cos A & 0 \\ 0 & 0 & 1 \end{array} \right\rbrack \cdot \left\lbrack \begin{array}{ccc}\cos h & 0 & -\sin h \\ 0 & 1 & 0 \\ \sin h & 0 & \cos h \end{array}\right \rbrack \cdot \left(\begin{array}{c}1 \\ 0 \\ 0 \end{array}\right)=\left(\begin{array}{c}\cos A \cos h \\ \sin A \cos h \\ \sin h \end{array}\right)$ .

$\overrightarrow{OS}|_{\mathcal{R}_H}=\mathcal{T}\cdot\overrightarrow{OS}|_{\mathcal{R}_{L}}=\left\lbrack \begin{array}{ccc} \cos \bar{\varphi} & 0 & \sin \bar{\varphi} \\ 0 & 1 & 0 \\ -\sin \bar{\varphi} & 0 & \cos \bar{\varphi} \end{array} \right\rbrack \cdot \left(\begin{array}{c}\cos A \cos h \\ \sin A \cos h \\ \sin h \end{array}\right) = \left(\begin{array}{c}\cos \bar{\varphi} \cos A \cos h + \sin \bar{\varphi} \sin h \\ \sin A \cos h \\ -\sin \bar{\varphi} \cos A \cos h + \cos \bar{\varphi} \sin h \end{array}\right )= \left(\begin{array}{c}\sin \varphi \cos A \cos h + \cos \varphi \sin h \\ \sin A \cos h \\ -\cos \varphi \cos A \cos h + \sin \varphi \sin h \end{array}\right )$ .

Ainsi on a $\begin{array}{rcl}\cos H \cos \delta &=& \sin \varphi \cos A \cos h + \cos \varphi \sin h \\ \sin H \cos \delta &=& \sin A \cos H \\ \sin \delta &=& -\cos \varphi \cos A \cos h+\sin \varphi \sin h \end{array}$ .