.

$$\sin^2 \phi \frac{\partial^2}{\partial \theta^2}$$

$$+\cos^2\phi \cot \theta \frac{\partial}{\partial \theta} + \cot \theta \cos \phi\sin \phi\frac{\partial^2}{\partial \theta\partial \phi}$$

$$-\sin \phi\cos \phi \frac{\partial}{\partial \phi} - \frac{\cos^2... ...\cos \phi \sin \phi}{\sin\theta}\frac{\partial^2}{\partial \phi\partial \theta}$$

$$- \frac{\cos^2 \theta \cos \phi \sin \phi}{\sin^2\theta} \frac{\partial}{\partial \phi}$$

$$+ \frac{\cos^2 \theta \cos^2 \phi}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2}$$

$$+\cos^2\phi \frac{\partial}{\partial \theta}$$

$$+ \cot \theta \sin^2 \phi \frac{\partial}{\partial \theta} - \fra... ...in \phi}{\sin\theta} \cos \phi \frac{\partial^2}{\partial \theta \partial \phi}$$

$$+\cos \phi \sin \phi \frac{\partial}{\partial \phi} +\cos \phi \c... ...-\cos \phi \cot\theta \sin \phi \frac{\partial^2}{\partial \phi\partial \theta}$$

$$+ \frac{\cos^2 \theta \cos \phi\sin\phi}{\sin^2\theta} \frac{\par... ...rac{\cos^2 \theta \sin^2 \phi}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2}$$

$$+ \frac{\partial^2}{\partial \phi^2}$$ (29)

which, gathering terms for clarity becomes

$$\left \{\sin^2 \phi +\cos^2\phi\right \} \frac{\partial^2}{\partial \theta^2}$$

$$+\left \{ \cos^2\phi \cot \theta + \cot \theta \sin^2 \phi\right \} \frac{\partial}{\partial \theta}$$

$$+ \left \{-2\frac{\cos^2 \theta \cos \phi \sin \phi}{\sin^2\theta... ...phi}{\sin^2\theta} -\sin \phi\cos \phi \right \} \frac{\partial}{\partial \phi}$$

$$+ \left \{\cot \theta \cos \phi \sin \phi +\frac{\cos \theta \cos... ... \cot\theta \sin \phi \right \} \frac{\partial^2}{\partial \phi\partial \theta}$$

$$+ \left \{\frac{\cos^2 \theta \cos^2 \phi}{\sin^2\theta} + \frac{... ...theta \sin^2 \phi}{\sin^2\theta} +1 \right \}\frac{\partial^2}{\partial \phi^2}$$ (30)

and finally

$$\begin{eqnarray*}\frac{\partial^2}{\partial \theta^2} \nonumber \\ \cot \theta... ...a}{\sin^2\theta} +K \right \}\frac{\partial^2}{\partial \phi^2} \end{eqnarray*}$$