Appendix B

From Fig. (1.5) it follows that

$$\hat{r}_{i+1}=\sin \vartheta \cos \varphi _{i}\hat{e}^{x}(i)+... ...sin \varphi _{i}\hat{e}^{y}(i)+\cos \vartheta \hat{e} ^{z}(i).$$ (1.50)

Using this in

$$\hat{e}^{x}(i+1) -\frac{1}{\sin \vartheta }\hat{r}_{i}+\frac{\cos \vartheta }{\sin \vartheta }\hat{r}_{i+1}$$ = (1.51)

$$\hat{e}^{y}(i+1) \frac{1}{\sin \vartheta }\hat{r}_{i}\times \hat{r} _{i+1}$$ = (1.52)

$$\hat{e}^{z}(i+1) \hat{r}_{i+1}$$ = (1.53)

one easily derives

$$\hat{e}^{x}(i+1) \cos \vartheta \cos \varphi _{i}\hat{e} ^{x}(i)+\cos \vartheta \sin \varphi _{i}\hat{e}^{y}(i)-\sin \vartheta \hat{e}^{z}(i)$$ = (1.54)

$$\hat{e}^{y}(i+1) -\sin \varphi _{i}\hat{e}^{x}(i)+\cos \varphi _{i}\hat{e}^{y}(i)$$ = (1.55)

$$\hat{e}^{z}(i+1) \sin \vartheta \cos \varphi _{i}\hat{e} ^{x}(i)+\sin \vartheta \sin \varphi _{i}\hat{e}^{y}(i)+\cos \vartheta \hat{e}^{z}(i)$$ = (1.56)

from which the matrix $\underline{M}(\varphi _{i})$ may be read off.