Hessian Matrices of Ray R̄_i with Respect to Incoming Ray R̄_i-1 and Boundary Variable Vector X̄_i

Abstract

As discussed in the previous chapter, the study of optical systems basically involves either systems analysis or systems design. Systems design is the reverse problem of systems analysis.

Appendices

Appendix 1

To compute the Hessian matrix $\partial^{2} {\bar{\text{R}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ of a ray with respect to boundary variable vector ${\bar{\text{X}}}_{\text{i}}$ at a spherical boundary surface ${\bar{\text{r}}}_{\text{i}}$, it is first necessary to compute the following terms:

• (1) Determination of $\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}}^{2}$:

$\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}}^{2}$ is a $4 \times 4 \times 9 \times 9$ matrix and can be obtained by further differentiating $\partial ({}^{0}\bar{\text{A}}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}}$ shown in Eqs. (7.64) to (7.72) of Appendix 2 in Chap. 7 with respect to spherical boundary variable vector ${\bar{\text{X}}}_{\text{i}}$. A careful examination of Eqs. (7.64) to Eq. (7.72) of Appendix 2 in Chap. 7 shows that the non-zero components of $\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}}^{2}$ have the forms

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{ix}}} \partial {\upomega}_{{\text{ix}}} }} = \left[ {\begin{array}{*{20}c} 0 & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} + \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} - \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} + \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & { - \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.29)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{iy}}} \partial {\upomega}_{{\text{ix}}} }} = \left[ {\begin{array}{*{20}c} 0 & {\text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & {\text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & { - {\text{S}}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & {\text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.30)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{iz}}} \partial {\upomega}_{{\text{ix}}} }} = \left[ {\begin{array}{*{20}c} 0 & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} + \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & {{\text{S}}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} + \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & {\text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} + \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} + {\text{S}}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.31)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{ix}}} \partial {\upomega}_{{\text{iy}}} }} = \left[ {\begin{array}{*{20}c} 0 & {\text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & {\text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & { - \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & {\text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.32)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{iy}}} \partial {\upomega}_{{\text{iy}}} }} = \left[ {\begin{array}{*{20}c} { - \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ {{\text{S}}{\upomega}_{{\text{iy}}} } & { - \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.33)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{iz}}} \partial {\upomega}_{{\text{iy}}} }} = \left[ {\begin{array}{*{20}c} {{\text{S}}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} } & {\text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & {\text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.34)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{ix}}} \partial {\upomega}_{{\text{iz}}} }} = \left[ {\begin{array}{*{20}c} 0 & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} + \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & {\text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} + \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & {\text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} + \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} + \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.35)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{iy}}} \partial {\upomega}_{{\text{iz}}} }} = \left[ {\begin{array}{*{20}c} {\text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} } & {\text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} } & {\text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], $$

(16.36)

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\upomega}_{{\text{iz}}} \partial {\upomega}_{{\text{iz}}} }} = \left[ {\begin{array}{*{20}c} { - \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} + \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ { - \text{S}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{iy}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{S}{\upomega}_{{\text{ix}}} - \text{C}{\upomega}_{{\text{iz}}} \text{C}{\upomega}_{{\text{ix}}} } & { - \text{S}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{iy}}} \text{C}{\upomega}_{{\text{ix}}} + \text{C}{\upomega}_{{\text{iz}}} \text{S}{\upomega}_{{\text{ix}}} } & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right]. $$

(16.37)

• (2) Determination of $\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$, $\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ and $\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$:

Differentiating Eq. (7.74) of Appendix 2 in Chap. 7 with respect to spherical boundary variable vector ${\bar{\text{X}}}_{\text{i}}$ yields

$$ \frac{{\partial^{2} {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} = \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }}\left[ {\begin{array}{*{20}c} {{\upsigma}_{\text{i}} } \\ {{\uprho}_{\text{i}} } \\ {{\uptau}_{\text{i}} } \\ 1 \\ \end{array} } \right] + \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left[ {\begin{array}{*{20}c} {\partial {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} } \\ {\partial {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} } \\ {\partial {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} } \\ {\bar{0}} \\ \end{array} } \right] + \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\left[ {\begin{array}{*{20}c} {\partial {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} } \\ {\partial {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} } \\ {\partial {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} } \\ {\bar{0}} \\ \end{array} } \right] + {}^{0}\bar{\text{A}}_{\text{i}} \left[ {\begin{array}{*{20}c} {\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2} } \\ {\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2} } \\ {\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2} } \\ {\bar{0}} \\ \end{array} } \right], $$

(16.38)

where $\partial^{2} {\bar{\text{P}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ is given in Eq. (16.16). $\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$, $\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ and $\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ can then be determined from

$$ \left[ {\begin{array}{*{20}c} {\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2} } \\ {\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2} } \\ {\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2} } \\ {\bar{0}} \\ \end{array} } \right] = \left( {{}^{0}\bar{\text{A}}_{\text{i}} } \right)^{ - 1} \left( {\frac{{\partial^{2} {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} - \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }}\left[ {\begin{array}{*{20}c} {{\upsigma}_{\text{i}} } \\ {{\uprho}_{\text{i}} } \\ {{\uptau}_{\text{i}} } \\ 1 \\ \end{array} } \right] - \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left[ {\begin{array}{*{20}c} {\partial {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} } \\ {\partial {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} } \\ {\partial {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} } \\ 0 \\ \end{array} } \right] - \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\left[ {\begin{array}{*{20}c} {\partial {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} } \\ {\partial {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} } \\ {\partial {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} } \\ 0 \\ \end{array} } \right]} \right). $$

(16.39)

• (3) Determination of $\partial^{2} {\upalpha}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ and $\partial^{2} {\upbeta}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$:

$\partial^{2} {\upalpha}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ and $\partial^{2} {\upbeta}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ can be obtained by differentiating Eqs. (7.76) and (7.77) of Appendix 2 in Chap. 7, respectively, with respect to spherical boundary variable vector ${\bar{\text{X}}}_{\text{i}}$, yielding

$$ \begin{aligned} \frac{{\partial^{2} {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} & = \frac{1}{{{\upsigma}_{\text{i}}^{ 2} + {\uprho}_{\text{i}}^{ 2} }}\left[ {{\upsigma}_{\text{i}} \frac{{\partial^{2} {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} + \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} - {\uprho}_{\text{i}} \frac{{\partial^{2} {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} - \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right] \\ & \quad - \frac{2}{{\left( {{\upsigma}_{\text{i}}^{ 2} + {\uprho}_{\text{i}}^{ 2} } \right)^{2} }}\left[ {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right]\left( {{\upsigma}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} - {\uprho}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right), \\ \end{aligned} $$

(16.40)

$$ \begin{aligned} \frac{{\partial^{2} {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} & = \frac{{\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )}}\frac{{\partial^{2} {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} - \frac{{2\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )^{2} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uptau}_{\text{i}} \frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right)\frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} \\ & \quad + \frac{1}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right)\frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} \\ & \quad - \frac{{{\uptau}_{\text{i}} }}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\upsigma}_{\text{i}} \frac{{\partial^{2} {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} + \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial^{2} {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }}} \right) \\ & \quad - \frac{1}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right) \\ & \quad + \frac{{2{\uptau}_{\text{i}} }}{{\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} } \right)^{2} \sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uptau}_{\text{i}} \frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right)\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right) \\ & \quad + \frac{{{\uptau}_{\text{i}} }}{{\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} } \right)\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)^{{{3 / 2}}} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right)\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right). \\ \end{aligned} $$

(16.41)

• (4) Determination of $\partial^{2} {\bar{\text{n}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$:

One approach to compute $\partial^{2} {\bar{\text{n}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ is to differentiate Eq. (7.78) of Appendix 2 in Chap. 7 with respect to spherical boundary variable vector ${\bar{\text{X}}}_{\text{i}}$, to give

$$ \frac{{\partial^{2} {\bar{\text{n}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} = \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }}{}^{\text{i}}{\bar{\text{n}}}_{\text{i}} + \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}\frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {}^{0}\bar{\text{A}}_{\text{i}} \frac{{\partial^{2} ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }}, $$

(16.42)

where

$$ \begin{aligned} \frac{{\partial^{2} ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} & = {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{S}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ {{\text{C}}{\upbeta}_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{\partial^{2} {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} + {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{C}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ {{\text{C}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial^{2} {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} \\ & \quad + {\text{s}}_{\text{i}} \left( {\left[ {\begin{array}{*{20}c} { - {\text{C}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ { - {\text{C}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{\partial {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + \left[ {\begin{array}{*{20}c} {{\text{S}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right)\frac{{\partial {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} \\ & \quad + {\text{s}}_{\text{i}} \left( {\left[ {\begin{array}{*{20}c} {{\text{S}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }} + \left[ {\begin{array}{*{20}c} { - {\text{C}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ { - {\text{C}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}\underline{{_{\text{i}} }} }}} \right)\frac{{\partial {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}. \\ \end{aligned} $$

(16.43)

• (5) Determination of $\partial^{2} ({\text{C}}{\uptheta}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}}^{2}$:

The term $\partial^{2} ({\text{C}}{\uptheta}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}}^{2}$ can be computed from Eq. (7.80) of Appendix 2 in Chap. 7 as

$$ \frac{{\partial^{2} ({\text{C}}{\uptheta}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} = - \left( {{\ell}_{{\text{i} - 1{\text{x}}}} \frac{{\partial^{2} {\text{n}}_{{\text{ix}}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} + {\ell}_{{\text{i} - 1{\text{y}}}} \frac{{\partial^{2} {\text{n}}_{{\text{iy}}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }} + {\ell}_{{\text{i} - 1{\text{z}}}} \frac{{\partial^{2} {\text{n}}_{{\text{iz}}} }}{{\partial {\bar{\text{X}}}_{\text{i}}^{2} }}} \right), $$

(16.44)

where the three components of $\partial^{2} {\bar{\text{n}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}}^{2}$ are given in Eq. (16.42) of this appendix.

Appendix 2

To compute the ray Hessian matrix $\partial^{2} {\bar{\text{R}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ for a spherical boundary surface ${\bar{\text{r}}}_{\text{i}}$, it is first necessary to determine the following terms:

• (1) Determination of $\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}} {\bar{\text{R}}}_{{{\text{i}} - 1}}$, a matrix with dimensions $4 \times 4 \times 9 \times 6$:

${}^{0}\bar{\text{A}}_{\text{i}}$ is not a function of the incoming ray ${\bar{\text{R}}}_{{{\text{i}} - 1}}$. Therefore, the following equation is obtained:

$$ \frac{{\partial^{2} ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = \left[ {\begin{array}{*{20}c} {\bar{0}} & {\bar{0}} & {\bar{0}} & {\bar{0}} \\ {\bar{0}} & {\bar{0}} & {\bar{0}} & {\bar{0}} \\ {\bar{0}} & {\bar{0}} & {\bar{0}} & {\bar{0}} \\ {\bar{0}} & {\bar{0}} & {\bar{0}} & {\bar{0}} \\ \end{array} } \right]_{4 \times 4 \times 9 \times 6} . $$

(16.45)

• (2) Determination of $\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$, $\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ and $\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$:

Differentiating Eq. (7.57) of Appendix 1 in Chap. 7 with respect to spherical boundary variable vector ${\bar{\text{X}}}_{\text{i}}$ yields

$$ \frac{{\partial^{2} {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left[ {\begin{array}{*{20}c} {\partial {\upsigma}_{\text{i}} /\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial {\uprho}_{\text{i}} /\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial {\uptau}_{\text{i}} /\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\bar{0}} \\ \end{array} } \right] + {}^{0}\bar{\text{A}}_{\text{i}} \left[ {\begin{array}{*{20}c} {\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\bar{0}} \\ \end{array} } \right], $$

(16.46)

where $\partial^{2} {\bar{\text{P}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ is given in Eq. (16.23). $\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$, $\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ and $\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ can then be determined from

$$ \left[ {\begin{array}{*{20}c} {\partial^{2} {\upsigma}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial^{2} {\uprho}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial^{2} {\uptau}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\bar{0}} \\ \end{array} } \right] = \left( {{}^{0}\bar{\text{A}}_{\text{i}} } \right)^{ - 1} \left( {\frac{{\partial^{2} {\bar{\text{P}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} - \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\left[ {\begin{array}{*{20}c} {\partial {\upsigma}_{\text{i}} /\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial {\uprho}_{\text{i}} /\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial {\uptau}_{\text{i}} /\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\bar{0}} \\ \end{array} } \right]} \right). $$

(16.47)

• (3) Determination of $\partial^{2} {\upalpha}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ and $\partial^{2} {\upbeta}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$:

$\partial^{2} {\upalpha}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ and $\partial^{2} {\upbeta}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ can be obtained by differentiating Eqs. (7.59) and (7.60) of Appendix 1 in Chap. 7, respectively, with respect to boundary variable vector ${\bar{\text{X}}}_{\text{i}}$, to give

$$ \begin{aligned} \frac{{\partial^{2} {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} & = \frac{1}{{{\upsigma}_{\text{i}}^{ 2} + {\uprho}_{\text{i}}^{ 2} }}\left[ {\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}\frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\upsigma}_{\text{i}} \frac{{\partial^{2} {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} - \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} - {\uprho}_{\text{i}} \frac{{\partial^{2} {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right] \\ & \quad - \frac{2}{{\left( {{\upsigma}_{\text{i}}^{ 2} + {\uprho}_{\text{i}}^{ 2} } \right)^{2} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\left( {{\upsigma}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} - {\uprho}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right), \\ \end{aligned} $$

(16.48)

$$ \begin{aligned} \frac{{\partial^{2} {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} & = \frac{{\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )}}\frac{{\partial^{2} {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + \frac{1}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}\\ & \quad - \frac{{2\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}{{\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} } \right)^{2} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uptau}_{\text{i}} \frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} \\ & \quad - \frac{{{\uptau}_{\text{i}} }}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}\frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\upsigma}_{\text{i}} \frac{{\partial^{2} {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}\frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\uprho}_{\text{i}} \frac{{\partial^{2} {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right) \\ & \quad - \frac{1}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {\frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)^{\text{T}} \left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right) \\ & \quad + \frac{{2{\uptau}_{\text{i}} }}{{\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} } \right)^{2} \sqrt {\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uptau}_{\text{i}} \frac{{\partial {\uptau}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right) \\ & \quad + \frac{{{\uptau}_{\text{i}} }}{{({\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} + {\uptau}_{\text{i}}^{2} )\left( {{\upsigma}_{\text{i}}^{\text{2}} + {\uprho}_{\text{i}}^{2} } \right)^{{{3 / 2}}} }}\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\left( {{\upsigma}_{\text{i}} \frac{{\partial {\upsigma}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\uprho}_{\text{i}} \frac{{\partial {\uprho}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right). \\ \end{aligned} $$

(16.49)

• (4) Determination of $\partial^{2} {\bar{\text{n}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$:

$\partial^{2} {\bar{\text{n}}}_{\text{i}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ can be obtained by differentiating Eq. (7.61) of Appendix 1 in Chap. 7 with respect to ${\bar{\text{X}}}_{\text{i}}$, to give

$$ \frac{{\partial^{2} {\bar{\text{n}}}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} = \left[ {\begin{array}{*{20}c} {\partial^{2} {\text{n}}_{{\text{ix}}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial^{2} {\text{n}}_{{\text{iy}}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\partial^{2} {\text{n}}_{{\text{iz}}} /\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} } \\ {\bar{0}} \\ \end{array} } \right] = \frac{{\partial ({}^{0}\bar{\text{A}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} }}\frac{{\partial ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {}^{0}\bar{\text{A}}_{\text{i}} \frac{{\partial^{2} ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}, $$

(16.50)

where $\partial^{2} ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ is obtained by differentiating Eq. (7.62) of Appendix 1 in Chap. 7, to give

$$ \begin{aligned} \frac{{\partial^{2} ({}^{\text{i}}{\bar{\text{n}}}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} & = {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{S}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ {{\text{C}}{\upbeta}_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{\partial^{2} {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\text{s}}_{\text{i}} \left[ {\begin{array}{*{20}c} { - {\text{C}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ {{\text{C}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial^{2} {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} \\ & \quad + {\text{s}}_{\text{i}} \left( {\left[ {\begin{array}{*{20}c} { - {\text{C}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ { - {\text{C}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} } \\ 0 \\ \end{array} } \right]\frac{{\partial {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + \left[ {\begin{array}{*{20}c} {{\text{S}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\frac{{\partial {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} \\ & \quad + {\text{s}}_{\text{i}} \left( {\left[ {\begin{array}{*{20}c} {{\text{S}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ { - {\text{S}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial {\upbeta}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + \left[ {\begin{array}{*{20}c} { - {\text{C}}{\upbeta}_{\text{i}} {\text{C}}{\upalpha}_{\text{i}} } \\ { - {\text{C}}{\upbeta}_{\text{i}} {\text{S}}{\upalpha}_{\text{i}} } \\ 0 \\ 0 \\ \end{array} } \right]\frac{{\partial {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right)\frac{{\partial {\upalpha}_{\text{i}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}. \\ \end{aligned} $$

(16.51)

• (5) Determination of $\partial^{2} ({\text{C}}{\uptheta}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$:

$\partial^{2} ({\text{C}}{\uptheta}_{\text{i}} )/\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ can be computed from Eq. (7.63) of Appendix 1 in Chap. 7 as

$$ \begin{aligned} \frac{{\partial^{2} ({\text{C}}{\uptheta}_{\text{i}} )}}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} & = - \left( {{\ell}_{{\text{i} - 1{\text{x}}}} \frac{{\partial^{2} {\text{n}}_{{\text{ix}}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\ell}_{{\text{i} - 1{\text{y}}}} \frac{{\partial^{2} {\text{n}}_{{\text{iy}}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }} + {\ell}_{{\text{i} - 1{\text{z}}}} \frac{{\partial^{2} {\text{n}}_{{\text{iz}}} }}{{\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}} \right) \\ & \quad - \left( {\frac{{\partial {\ell}_{{\text{i - 1x}}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}\frac{{\partial {\text{n}}_{{\text{ix}}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + \frac{{\partial {\ell}_{{\text{i} - 1{\text{y}}}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}\frac{{\partial {\text{n}}_{{\text{iy}}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }} + \frac{{\partial {\ell}_{{\text{i} - 1{\text{z}}}} }}{{\partial {\bar{\text{R}}}_{{{\text{i}} - 1}} }}\frac{{\partial {\text{n}}_{{\text{iz}}} }}{{\partial {\bar{\text{X}}}_{\text{i}} }}} \right), \\ \end{aligned} $$

(16.52)

where the three components of $\partial {\bar{\text{n}}}_{\text{i}}/\partial {\bar{\text{X}}}_{\text{i}}$ in Eq. (16.52) are given in Eq. (7.78) of Appendix 2 in Chap. 7, while the three components of $\partial^{2} {\bar{\text{n}}}_{\text{i}}/\partial {\bar{\text{X}}}_{\text{i}} \partial {\bar{\text{R}}}_{{{\text{i}} - 1}}$ are given in Eq. (16.50) of this appendix.