<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\batchmode


\documentclass[a4j,twoside]{jreport}
\RequirePackage{ifthen}


\usepackage{Dennou6}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{html}	

%
\providecommand{\Dmodelbase}{dcpam}   % モデル名 (バージョンなし)%
\providecommand{\Dmodelver}{4}        % モデルバージョン%
\providecommand{\Dmodel}{dcpam4}  % モデル名 (バージョンあり)

%
\providecommand{\titleblank}{\\[70mm]}   % タイトルと著者間の空白%
\providecommand{\titleblank}{\\[5mm]} 
\Dtitle[dcpam4支配方程式系とその離散化]
         {dcpam4\\[5mm]
          支配方程式系とその離散化 \\[5mm]}
  \Dauthor[地球流体電脳倶楽部]{地球流体電脳倶楽部}
  %%
\providecommand{\Dmodelverprint}{20080626}%
\providecommand{\Dmodeldateprint}{2008/06/26}%
\providecommand{\DmodeldateprintJ}{2008年06月26日} 
  \Ddate[2008/06/26]{2008年06月26日(dcpam4-20080626)}
  \Dparskip

%
\providecommand{\Drireki}[1]{
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\footnotetext[0]{#1}
\renewcommand{\thefootnote}{\arabic{footnote}}
} 


\makeindex




\usepackage[dvips]{color}


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\lthtmlinlinemathA{tex2html_wrap_indisplay9949}%
$\displaystyle =  \delta_{nn'},$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay9950}%
$\displaystyle \sum_{i=1}^{I} \exp(im \lambda_i) \exp(-im' \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay9951}%
$\displaystyle =  I \delta_{mm'}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9953}%
$ w_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9955}%
$ {\displaystyle w_j \equiv
\frac{(2J-1)(1-\sin^2 \varphi_j)}
{\left\{J P_{J-1}(\sin \varphi_j)\right\}^2 }    }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9958}%
$ A$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9960}%
$ (\lambda_i,\varphi_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9962}%
$ i=1,2,\cdots,I. \quad j=1,2,\cdots,J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9964}%
$ A_{ij}=A(\lambda_i,\varphi_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9968}%
$ m=-M,\cdots,M. \quad n=|m|,\cdots,N(m)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9970}%
$ \tilde{A}_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay9973}%
$\displaystyle A_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay9974}%
$\displaystyle \equiv              \sum_{m=-M}^{M} \sum_{n=|m|}^{N}               \tilde{A}_n^m               Y_n^m (\lambda_i,\varphi_j),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay9975}%
$\displaystyle \tilde{A}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay9976}%
$\displaystyle =              \frac{1}{I}              \sum_{i=1}^{I} \sum_{j=1}^{J}                A_{ij} Y_n^{m*} (\lambda_i,  \varphi_j) w_j .$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9980}%
$ {\displaystyle 
\left\{\tilde{A}^m_n \exp(im\lambda) \right\}^*
= \tilde{A}^{-m}_n \exp(-im\lambda) }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9982}%
$ m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9988}%
$ n-m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9990}%
$ \varphi=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9996}%
$ A_{ij}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9998}%
$ A_{ij}^{even}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10000}%
$ A_{ij}^{odd}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10002}%
$ A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10004}%
$ A_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10006}%
$ A_{i,j}+A_{i,J-j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10008}%
$ A_{i,j}-A_{i,J-j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10010}%
$ j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10012}%
$ J/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10014}%
$ *$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10018}%
$\displaystyle =  \sum_{m=0}^{M} \sum_{n=m}^{N}             \Re \tilde{A}_n^m Y_n^m(\lambda_i, \varphi_j),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10020}%
$\displaystyle =   \left\{        \begin{array}{ll}      {\displaystyle \frac{1}{I}                   \sum_{i=1}^{I} \sum_{j=1}^{J}  }                  A_{ij} Y_n^{m*}(\lambda_i, \varphi_j) w_j ,      &amp;   \  \ \  m=0, \quad m \le n \le N, \\{\displaystyle \frac{2}{I}                   \sum_{i=1}^{I} \sum_{j=1}^{J}  }                  A_{ij} Y_n^{m*}(\lambda_i,\varphi_j) w_j ,      &amp;   \  \ \  1 \le m \le M, \quad m \le n \le N.   \end{array}       \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10025}%
$ A(\lambda,\varphi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10032}%
$\displaystyle A(\lambda,\varphi)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10033}%
$\displaystyle \equiv \sum_{m=-M}^{M} \sum_{n=|m|}^{N}                 \tilde{A}_n^m Y_n^m (\lambda, \varphi)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10037}%
$\displaystyle \left( \DP{f}{\lambda} \right)_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10038}%
$\displaystyle \equiv \sum_{m=-M}^M                      \sum_{n=|m|}^N im \tilde{f}_n^m                  Y_n^m (\lambda_i,\varphi_j) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10039}%
$\displaystyle \widetilde{ \left( \DP{f}{\lambda} \right) }_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10040}%
$\displaystyle =   \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J                     im f_{ij} Y_n^{m*} (\lambda_i, \varphi_j) w_j  .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10043}%
$\displaystyle \left(\DP{f}{\varphi}\right)_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10044}%
$\displaystyle \equiv  \sum_{m=-M}^M \sum_{n=|m|}^N \tilde{f}_n^m                    \left. \DD{P_n^m}{\varphi} \right|_j                     \exp(im \lambda_i) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10045}%
$\displaystyle \widetilde{\left(\DP{f}{\varphi}\right)}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10046}%
$\displaystyle = - \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J f_{ij}                     \left. \DD{P_n^m}{\varphi}\right|_j                     \exp(-im \lambda_i) w_j .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10053}%
$\displaystyle \DP{\pi}{t}   + \frac{u}{a \cos \varphi} \DP{\pi}{\lambda}   + \frac{v}{a} \DP{\pi}{\varphi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10054}%
$\displaystyle =   - D   - \DP{\dot{\sigma}}{\sigma}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10056}%
$\displaystyle \DP{\Phi}{\sigma} = - \frac{RT_v}{\sigma}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10058}%
$\displaystyle \DP{\zeta}{t}       \ $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10059}%
$\displaystyle = \  \Dinv{a \cos \varphi}            \left\{ \DP{v_A}{\lambda} - \DP{(u_A \cos \varphi)}{\varphi} \right\}           + {\cal D}(\zeta),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10060}%
$\displaystyle \DP{D}{t}       \ $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10061}%
$\displaystyle = \  \Dinv{a \cos \varphi}            \left\{ \DP{u_A}{\lambda} + \DP{(v_A \cos \varphi)}{\varphi} \right\}           - \nabla^{2}_{\sigma}            ( \Phi + R \overline{T} \pi +$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10062}%
$\displaystyle )            + {\cal D}(D).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar10064}%
\begin{align*}\begin{split}   \DP{T}{t}     \  &amp;= \            - \Dinv{a \cos \varphi}              \left\{  \DP{(u T^{\prime})}{\lambda}                     + \DP{(v T^{\prime} \cos \varphi)}{\varphi}              \right\}           + T^{\prime} D \\\  &amp; \  \quad        - \dot{\sigma}                \DP{T}{\sigma}        + \kappa T_v            \left(                    \DP{\pi}{t}                  + \frac{u}{a \cos \varphi} \DP{\pi}{\lambda}                  + \frac{v}{a} \DP{\pi}{\varphi}                  + \frac{ \dot{\sigma} }{ \sigma }             \right)        + \frac{Q}{C_p}        + {\cal D}(T)         + {\cal D}^{\prime}(\Dvect{v}).  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar10066}%
\begin{align*}\begin{split}  \DP{q}{t}     \  &amp;= \            - \Dinv{a \cos \varphi}              \left\{  \DP{(u q)}{\lambda}                     + \DP{(v q \cos \varphi)}{\varphi}              \right\}           + q D  \\\  &amp; \quad           - \dot{\sigma} \DP{q}{\sigma}           + S_{q}           + {\cal D}(q).  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10067}%
$\displaystyle \varphi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10068}%
$\displaystyle : \quad 緯度 [\mathrm{deg.}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10069}%
$\displaystyle \lambda$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10070}%
$\displaystyle : \quad 経度 [\mathrm{deg.}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10071}%
$\displaystyle \sigma$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10072}%
$\displaystyle \equiv p/p_s,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10073}%
$\displaystyle t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10074}%
$\displaystyle : \quad 時間 [\mathrm{s}].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10079}%
$\displaystyle \pi\  (\varphi, \lambda)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10080}%
$\displaystyle \equiv \ln p_s,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10081}%
$\displaystyle T\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10082}%
$\displaystyle : \quad 気温 [\mathrm{K}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10083}%
$\displaystyle q\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10084}%
$\displaystyle : \quad 比湿 [\mathrm{kg}\  \mathrm{kg}^{-1}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10085}%
$\displaystyle \zeta\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10086}%
$\displaystyle \equiv \Dinv{a \cos \varphi}                         \left\{   \DP{v}{\lambda}                                 - \DP{(u \cos \varphi)}{\varphi} \right\}                       : \quad 渦度 [\mathrm{s}^{-1}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10087}%
$\displaystyle D\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10088}%
$\displaystyle \equiv  \Dinv{a \cos \varphi}                         \left\{   \DP{u}{\lambda}                                 + \DP{(v \cos \varphi)}{\varphi} \right\}                       : \quad 発散 [\mathrm{s}^{-1}].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10090}%
$ u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10092}%
$ v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10094}%
$ (\varphi, \lambda, \sigma)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10095}%
$\displaystyle \Phi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10096}%
$\displaystyle \equiv gz                       : \quad ジオポテンシャル高度                               [\mathrm{m}^{2}\   \mathrm{s}^{-2}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10097}%
$\displaystyle \dot{\sigma}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10098}%
$\displaystyle \equiv \DD{\sigma}{t}                       \  \equiv \                            \DP{\sigma}{t}                        + \frac{u}{a \cos \varphi} \DP{\sigma}{\lambda}                        + \frac{v}{a} \DP{\sigma}{\varphi}                        + \DP{\sigma}{\sigma},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10099}%
$\displaystyle \overline{T}\  (\sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10100}%
$\displaystyle : \quad 水平平均気温 [\mathrm{K}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10101}%
$\displaystyle T^{\prime}\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10102}%
$\displaystyle \equiv T - \overline{T},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10103}%
$\displaystyle T_v\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10104}%
$\displaystyle \equiv T                           \left\{                             1 + \left(\epsilon_v^{-1} - 1\right) q                           \right\},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10105}%
$\displaystyle T_v^{\prime}\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10106}%
$\displaystyle \equiv T_v - \overline{T},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10107}%
$\displaystyle u_A\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10108}%
$\displaystyle \equiv                           ( \zeta + f ) v                          - \dot{\sigma} \DP{u}{\sigma}                          - \frac{R T_v^{\prime}}{a \cos \varphi} \DP{\pi}{\lambda}                          + {\cal F}_{\lambda},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10109}%
$\displaystyle v_A\  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10110}%
$\displaystyle \equiv                         - ( \zeta + f ) u                          - \dot{\sigma} \DP{v}{\sigma}                          - \frac{R T_v^{\prime}}{a} \DP{\pi}{\varphi}                          + {\cal F}_{\varphi},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10111}%
$\displaystyle \nabla^{2}_{\sigma}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10112}%
$\displaystyle \equiv                 \frac{1}{a^{2} \cos^2 \varphi} \DP[2]{}{\lambda}               + \frac{1}{a^{2} \cos \varphi}                 \DP{}{\varphi}                   \left( \cos \varphi \DP{}{\varphi} \right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10113}%
$\displaystyle \  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10114}%
$\displaystyle \equiv  \frac{u^{2}+v^{2}}{2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10115}%
$\displaystyle {\cal D}(\zeta)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10116}%
$\displaystyle : \qquad 渦度水平拡散,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10117}%
$\displaystyle {\cal D}(D)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10118}%
$\displaystyle : \qquad 発散水平拡散,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10119}%
$\displaystyle {\cal D}(T)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10120}%
$\displaystyle : \qquad 温度水平拡散,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10121}%
$\displaystyle {\cal D}(q)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10122}%
$\displaystyle : \qquad 比湿水平拡散,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10123}%
$\displaystyle {\cal F}_\lambda \  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10124}%
$\displaystyle : \qquad 小規模運動過程 (経度方向),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10125}%
$\displaystyle {\cal F}_\varphi \  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10126}%
$\displaystyle : \qquad 小規模運動過程 (緯度方向),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10127}%
$\displaystyle Q \  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10128}%
$\displaystyle : \qquad 放射, 凝結, 小規模運動過程等による加熱・温度変化,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10129}%
$\displaystyle S_q \  (\varphi, \lambda, \sigma)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10130}%
$\displaystyle : \qquad 凝結, 小規模運動過程等による水蒸気ソース,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10131}%
$\displaystyle {\cal D}' \  (\Dvect{v})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10132}%
$\displaystyle : \qquad 摩擦熱.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10133}%
$\displaystyle a$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10134}%
$\displaystyle : \quad 惑星半径 [\mathrm{m}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10135}%
$\displaystyle R$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10136}%
$\displaystyle : \quad 乾燥大気の気体定数 [\mathrm{J\   kg}^{-1}\   \mathrm{K}^{-1}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10137}%
$\displaystyle C_p$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10138}%
$\displaystyle : \quad 乾燥大気の大気定圧比熱 [\mathrm{J\  kg}^{-1}\  \mathrm{K}^{-1}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10139}%
$\displaystyle f$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10140}%
$\displaystyle : \quad コリオリパラメータ [\mathrm{s}^{-1}],$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10141}%
$\displaystyle \kappa$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10142}%
$\displaystyle \equiv R/C_p,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10143}%
$\displaystyle \epsilon_v$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10144}%
$\displaystyle : \quad 水蒸気分子量比.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10146}%
$\displaystyle \dot{\sigma} = 0  \  \ \  at \  \ \sigma = 0 , \  1 .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10150}%
$ \dot{\sigma}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10151}%
$\displaystyle \DP{\pi}{t}    = - \int_{0}^{1} \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma      - \int_{0}^{1} D  d \sigma ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10152}%
$\displaystyle \dot{\sigma}     = - \sigma       \DP{\pi}{t}      - \int_{0}^{\sigma} D d \sigma      - \int_{0}^{\sigma}           \Dvect{v}_{H} \cdot \nabla_{\sigma} \pi d \sigma ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10155}%
$ \nabla^{N_D}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10157}%
$\displaystyle {\cal D}(\zeta) = - K_{HD}                        \left[ (-1)^{N_D/2} \nabla^{N_D}                               - \left( \frac{2}{a^2} \right)^{N_D/2}                        \right]                     \zeta ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10159}%
$\displaystyle {\cal D}(D) = - K_{HD}                        \left[ (-1)^{N_D/2} \nabla^{N_D}                               - \left( \frac{2}{a^2} \right)^{N_D/2}                        \right]                     D ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10161}%
$\displaystyle {\cal D}(T) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} T ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10163}%
$\displaystyle {\cal D}(q) = - (-1)^{N_D/2} K_{HD} \nabla^{N_D} q .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10165}%
$ N_D$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10167}%
$ \sim$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10170}%
$\displaystyle \DP{\pi}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10171}%
$\displaystyle = - \sum_{k=1}^{K} ( D_k + \Dvect{v}_k \cdot \nabla \pi )         \Delta  \sigma_k,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10172}%
$\displaystyle \dot{\sigma}_{k-1/2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10173}%
$\displaystyle = - \sigma_{k-1/2} \DP{\pi}{t}    - \sum_{l=k}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi )                  \Delta  \sigma_l,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10174}%
$\displaystyle \dot{\sigma}_{1/2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10175}%
$\displaystyle = \dot{\sigma}_{K+1/2} = 0.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10176}%
$\displaystyle \Dvect{v} \cdot \nabla \pi =   \frac{u}{a \cos \varphi} \DP{\pi}{\lambda}   + \frac{v}{a} \DP{\pi}{\varphi}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar10178}%
\begin{align*}\begin{split}   \Phi_{1} &amp; = \Phi_{s} + C_{p} ( \sigma_{1}^{-\kappa} - 1  ) T_{v,1} \\&amp; = \Phi_{s} + C_{p} \alpha_{1} T_{v,1}.  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10179}%
\begin{align*}\begin{split}   \Phi_k - \Phi_{k-1}      &amp; = C_{p}       \left[ \left( \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{\kappa}              - 1       \right] T_{v,k}        + C_{p}       \left[ 1-           \left( \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} } \right)^{\kappa}       \right] T_{v,k-1} \\&amp; = C_{p} \alpha_k T_{v,k} + C_{p} \beta_{k-1} T_{v,k-1}.  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10180}%
$\displaystyle \alpha_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10181}%
$\displaystyle = \left( \frac{ \sigma_{k-1/2} }                                { \sigma_k } \right)^{\kappa} -1 ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10182}%
$\displaystyle \beta_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10183}%
$\displaystyle = 1- \left( \frac{ \sigma_{k+1/2} }                                { \sigma_k } \right)^{\kappa} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10185}%
$\displaystyle \DP{\zeta_k}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10186}%
$\displaystyle =   \Dinv{a \cos \varphi}           \left\{               \DP{{v_A}_{,k}}{\lambda}             - \DP{({u_A}_{,k} \cos \varphi)}{\varphi}           \right\}         + {\cal D}(\zeta_k),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10188}%
$\displaystyle =   \Dinv{a \cos \varphi}           \left\{               \DP{{u_A}_{,k}}{\lambda}             + \DP{({v_A}_{,k} \cos \varphi)}{\varphi}           \right\}         - \nabla^{2}_{\sigma}          ( \Phi_k + C_{p} \hat{\kappa}_k \overline{T}_k \pi             + ($%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10189}%
$\displaystyle )_k )         + {\cal D}(D_k).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10190}%
\begin{align*}\begin{split}   {u_A}_{,k}     &amp; =   ( \zeta_k + f ) v_k          - \frac{1}{2 \Delta \sigma_k}               [   \dot{\sigma}_{k-1/2} ( u_{k-1} - u_k   )                + \dot{\sigma}_{k+1/2} ( u_k     - u_{k+1} ) ] \\&amp; \quad         - \frac{C_{p} \hat{\kappa}_k T_{v,k}^{\prime}}{a \cos \varphi}              \DP{\pi}{\lambda}          + {\cal F}_{\lambda, k},   \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10191}%
\begin{align*}\begin{split}   {v_A}_{,k}       &amp; = - ( \zeta_k + f ) u_k          - \frac{1}{2 \Delta \sigma_k}               [   \dot{\sigma}_{k-1/2} ( v_{k-1} - v_k   )                + \dot{\sigma}_{k+1/2} ( v_k   - v_{k+1} ) ] \\&amp; \quad         - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a}               \DP{\pi}{\mu}          + {\cal F}_{\varphi, k},   \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10192}%
\begin{align*}\begin{split}   \hat{\kappa}_k       &amp; = \frac{  \sigma_{k-1/2}(   \sigma^{\kappa}_{k-1/2}                                   - \sigma^{\kappa}_k      )                 + \sigma_{k+1/2}(   \sigma^{\kappa}_k                                   - \sigma^{\kappa}_{k+1/2}  ) }               { \sigma^{\kappa}_k ( \sigma_{k-1/2} - \sigma_{k+1/2} ) } \\&amp; = \frac{ \sigma_{k-1/2} \alpha_k + \sigma_{k+1/2} \beta_k }               { \Delta \sigma_k                                  },    \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10193}%
$\displaystyle %
T_{v,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10194}%
$\displaystyle = T_k - \overline{T}_k,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10195}%
$\displaystyle %
($%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10196}%
$\displaystyle )_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10197}%
$\displaystyle = \frac{v^{2}_k + v^{2}_k}{2}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar10199}%
\begin{align*}\begin{split}   \DP{T_k}{t}     &amp; = - \Dinv{a \cos \varphi}             \left\{   \DP{(u_k T^{\prime}_k)}{\lambda}                     + \DP{(v_k T^{\prime}_k \cos \varphi)}{\varphi}             \right\}         + \hat{H}_k  \\&amp; \quad         + \frac{Q_k}{C_{p}}         + {\cal D}(T_k)          + {\cal D}'(\Dvect{v}).  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10200}%
\begin{align*}\begin{split}    \hat{H}_k      &amp; \equiv            T_k' D_k          - \frac{1}{\Delta \sigma_k}               [   \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k   )                + \dot{\sigma}_{k+1/2} ( T_k   - \hat{T}_{k+1/2} ) ]                \\&amp; \quad          + \left\{ \alpha_k                     \left[   \sigma_{k-1/2} \Dvect{v}_k \cdot \nabla \pi                            - \sum_{l=k}^{K}                                 ( D_l + \Dvect{v}_l \cdot \nabla \pi )                                  \Delta  \sigma_l                     \right]            \right.                \\&amp; \quad          + \quad            \left. \beta_k                    \left[   \sigma_{k+1/2} \Dvect{v}_k \cdot \nabla \pi                           - \sum_{l=k+1}^{K}                                ( D_l + \Dvect{v}_l \cdot \nabla \pi )                                 \Delta  \sigma_l                    \right]            \right\}               \frac{1}{\Delta \sigma_k} T_{v,k} \\&amp; =    T_k' D_k           - \frac{1}{\Delta \sigma_k}               [   \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k   )                + \dot{\sigma}_{k+1/2} ( T_k   - \hat{T}_{k+1/2} ) ]                \\&amp; \quad          + \hat{\kappa}_k \Dvect{v}_k \cdot \nabla \pi T_{v,k}                 \\&amp; \quad          - \alpha_k \sum_{l=k}^{K}                        ( D_l + \Dvect{v}_l \cdot \nabla \pi )                         \Delta  \sigma_l                            \frac{T_{v,k}}{\Delta \sigma_k}                 \\&amp; \quad          - \beta_k \sum_{l=k+1}^{K}                       ( D_l + \Dvect{v}_l \cdot \nabla \pi )                        \Delta  \sigma_l                           \frac{T_{v,k}}{\Delta \sigma_k},   \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10201}%
\begin{align*}\begin{split}   \hat{T}_{k-1/2}     &amp; = \frac{           \left[             \left( \displaystyle                 \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{\kappa}               - 1             \right] \sigma_{k-1}^{\kappa} T_k          + \left[               1             - \left( \displaystyle                   \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} }               \right)^{\kappa}           \right] \sigma_k^{\kappa} T_{k-1}                   }         { \sigma_{k-1}^{\kappa} - \sigma_k^{\kappa}           }         \\&amp; = a_k T_k + b_{k-1} T_{k-1},   \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10202}%
$\displaystyle a_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10203}%
$\displaystyle = \alpha_k              \left[ 1- \left( \frac{ \sigma_k }{ \sigma_{k-1} }                         \right)^{\kappa} \right]^{-1},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10204}%
$\displaystyle b_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10205}%
$\displaystyle = \beta_k              \left[ \left( \frac{ \sigma_k }{ \sigma_{k+1} }                       \right)^{\kappa} - 1 \right]^{-1} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10207}%
$\displaystyle \DP{q_k}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10208}%
$\displaystyle =   - \Dinv{a \cos \varphi}               \left\{   \DP{(u_k q_k)}{\lambda}                       + \DP{(v_k q_k \cos \varphi)}{\varphi}               \right\}           + R_k            + S_{q,k}           + {\cal D}(q_k).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10209}%
$\displaystyle R_k  =    q_k D_k           - \frac{1}{2 \Delta \sigma_k}               \left[   \dot{\sigma}_{k-1/2} ( q_{k-1} - q_k   )                     + \dot{\sigma}_{k+1/2} ( q_k   - q_{k+1} ) \right].$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10212}%
$\displaystyle \DP{\tilde{\pi}_n^m}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10213}%
$\displaystyle = - \sum_{k=1}^{K} (\tilde{D}_n^m)_k \Delta  \sigma_k         + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}                  Z_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) w_j.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10214}%
$\displaystyle Z \equiv - \sum_{k=1}^{K} \Dvect{v}_k \cdot \nabla \pi.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar10216}%
\begin{align*}\begin{split}    \DP{\tilde{\zeta}_n^m}{t}       &amp; = \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}              im (v_A)_{ij}            Y_n^{m *} ( \lambda_i, \varphi_j )           \frac{w_j}{a \cos \varphi_j}            \\&amp; \quad +             \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}              (u_A)_{ij} \cos \varphi_j            \DP{}{\varphi}  Y_n^{m *} ( \lambda_i, \varphi_j )            \frac{w_j}{a \cos \varphi_j}            \\&amp; \quad +           \tilde{\cal D}_{M,n}^m \tilde{\zeta}_n^m  ,  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

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\lthtmldisplayA{alignstar10217}%
\begin{align*}\begin{split}    \DP{\tilde{D}_n^m}{t}       &amp; = \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}              im (u_A)_{ij}            Y_n^{m *} ( \lambda_i, \varphi_j )           \frac{w_j}{a \cos \varphi_j}            \\&amp; \quad -            \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}              (v_A)_{ij}            \cos \varphi_j             \DP{}{\varphi}  Y_n^{m *} ( \lambda_i, \varphi_j )            \frac{w_j}{a \cos \varphi_j}            \\&amp; \quad -           \frac{n(n+1)}{a^{2}}            \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}              (\mbox{\sl KE})_{ij}            Y_n^{m *} ( \lambda_i, \varphi_j )            w_j           \\&amp; \quad +            \frac{n(n+1)}{a^{2}}             ( \Phi_n^m + C_{p} \hat{\kappa}_k \bar{T}_k \pi_n^m )             +  \tilde{\cal D}_{M,n}^m \tilde{D}_n^m .  \end{split}\end{align*}%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10218}%
$\displaystyle \tilde{\cal D}_{M,n}^m = - K_{HD} \left[                              \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2}                             - \left( \frac{2}{a} \right)^{N_D}                             \right]  .$%
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\lthtmldisplayA{alignstar10220}%
\begin{align*}\begin{split}    \DP{\tilde{T}_n^m}{t}   &amp; = - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}            im u_{ij} T'_{ij}          Y_n^{m *} ( \lambda_i, \varphi_j )         \frac{w_j}{a \cos \varphi_j}          \\&amp; \quad       + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}            v_{ij} T'_{ij}          \cos \varphi_j           \DP{}{\varphi}  Y_n^{m *} ( \lambda_i, \varphi_j )          \frac{w_j}{a \cos \varphi_j}          \\&amp; \quad       + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}            \left( \hat{H}_{ij} + \frac{Q_{ij}}{C_{p}} \right)          Y_n^{m *} ( \lambda_i, \varphi_j )          w_j         \\&amp; \quad       + \tilde{\cal D}_{H,n}^m \tilde{T}_n^m         \\&amp; \quad       + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}            {\cal D}'_{ij}(\Dvect{v})          Y_n^{m *} ( \lambda_i, \varphi_j )          w_j .  \end{split}\end{align*}%
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$\displaystyle \tilde{\cal D}_{H,n}^m     = - K_{HD} \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2} .$%
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\lthtmldisplayA{alignstar10223}%
\begin{align*}\begin{split}    \DP{\tilde{q}_n^m}{t}   &amp; = - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}           im u_{ij} q_{ij}         Y_n^{m *} ( \lambda_i, \varphi_j )        \frac{w_j}{a \cos \varphi_j}         \\&amp; \quad       + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}           v_{ij} q_{ij}         \cos \varphi_j          \DP{}{\varphi}  Y_n^{m *} ( \lambda_i, \varphi_j )         \frac{w_j}{a \cos \varphi_j}         \\&amp; \quad       + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}           \left( \hat{R}_{ij} + S_{q,ij} \right)         Y_n^{m *} ( \lambda_i, \varphi_j )         w_j        \\&amp; \quad       + \tilde{\cal D}_{H,n}^m \tilde{q}_n^m .  \end{split}\end{align*}%
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$ \Delta t$%
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10229}%
$ 2 \Delta t$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10231}%
$ {\cal F}_\lambda, {\cal F}_\varphi, Q, S_q$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10235}%
$ {\cal A}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10236}%
$\displaystyle \hat{\cal A}^{t+\Delta t}      =  \bar{\cal A}^{t-\Delta t}     + 2 \Delta t        \dot{\cal A}_{adv}\left( {\cal A}^{t} \right)     + 2 \Delta t        \dot{\cal A}_{dif}\left( \hat{\cal A}^{t+\Delta t} \right)     + 2 \Delta t        \dot{\cal A}_{phy}\left( \bar{\cal A}^{t-\Delta t} \right) .$%
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$ \dot{\cal A}_{adv} $%
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{\newpage\clearpage
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$ \dot{\cal A}_{dif} $%
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{\newpage\clearpage
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$ \dot{\cal A}_{phy} $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10244}%
$ \hat{\cal A}^{t+\Delta t} $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10246}%
$ \dot{\cal A}_{dis} $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10248}%
$ \dot{\cal A}_{cnd} $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10250}%
$ {\cal A}^{t+\Delta t} $%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10251}%
$\displaystyle {\cal A}^{t+\Delta t}      =  \hat{\cal A}^{t+\Delta t}     + 2 \Delta t        \dot{\cal A}_{dis}\left( \hat{\cal A}^{t+\Delta t} \right)     + 2 \Delta t        \dot{\cal A}_{cnd}\left( \hat{\cal A}^{t+\Delta t} \right) .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10252}%
$\displaystyle \bar{\cal A}^{t}     = ( 1-2 \epsilon_f ) {\cal A}^{t}     +  \epsilon_f          \left( \bar{\cal A}^{t-\Delta t} + {\cal A}^{t+\Delta t} \right)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10254}%
$ \bar{\cal A}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10256}%
$ \epsilon_f$%
\lthtmlinlinemathZ
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10259}%
$ T=\overline{T}_k$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10261}%
$ \Dvect{D}=\{ D_{k} \}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10263}%
$ \Dvect{T}=\{ T_{k} \}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10264}%
$\displaystyle \DP{\pi}{t} =           \left( \frac{\partial \pi}{\partial t} \right)_{\rm NG}      - \Dvect{C} \cdot \Dvect{D}  ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10265}%
$\displaystyle \frac{\partial \Dvect{D}}{\partial t} =            \left( \DP{\Dvect{D}}{t} \right)_{\rm NG}             - \nabla^{2}_{\sigma} ( \Dvect{\Phi}_{S}                                    + \underline{W} \Dvect{T}                                   + \Dvect{G} \pi )             + {\cal D}_M \Dvect{D} ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10266}%
$\displaystyle \DP{\Dvect{T}}{t}$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10267}%
$\displaystyle =   \left( \DP{\Dvect{T}}{t}       \right)_{\rm NG}            - \underline{h} \Dvect{D}          + {\cal D}_H \Dvect{T} .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10268}%
$\displaystyle \left( \DP{\pi}{t} \right)^{\rm NG}    =   - \sum_{k=1}^{K} \Dvect{v}_{k} \cdot \nabla \pi          \Delta  \sigma_{k}     =   Z_{k} ,$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10269}%
$\displaystyle \dot{\sigma}^{\rm NG}_{k-1/2}  = - \sigma_{k-1/2} \left( \DP{\pi}{t} \right)^{\rm NG}    - \sum_{l=k}^{K} \Dvect{v}_{l} \cdot \nabla \pi        \Delta  \sigma_{l} ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10270}%
$\displaystyle \left( \DP{D}{t} \right)^{\rm NG}    =             \Dinv{a \cos \varphi}             \left\{                 \DP{{u_A}_{,k}}{\lambda}               + \DP{({v_A}_{,k} \cos \varphi)}{\varphi}             \right\}           - \nabla^{2}_{\sigma} ($%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10271}%
$\displaystyle )_{k}            - \nabla^{2}_{\sigma} \sum_{k=1}^{K} W_{kl} ( T_{v,l}-T_{l} )           + {\cal D}(D_{k}) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10272}%
$\displaystyle \left( \DP{T_{k}}{t} \right)^{\rm NG}        =   - \frac{1}{a(1-\mu^{2})}                 \DP{U_k T'_k}{\lambda}           - \frac{1}{a}                \DP{V_k T'_k}{\mu}           + H_{k}            + {\cal D}(T_{k})            + {\cal D}^{\prime}(\Dvect{v}) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10273}%
\begin{align*}\begin{split}   H_k &amp; =     T_{k}^{\prime} D_{k}  \\&amp; \quad            - \frac{1}{\Delta \sigma_{k}}               [   \dot{\sigma}_{k-1/2} ( \hat{T^{\prime}}_{k-1/2}                                           - T^{\prime}_{k}   )                + \dot{\sigma}_{k+1/2} ( T^{\prime}_{k}                                            - \hat{T^{\prime}}_{k+1/2} ) ]                 \\&amp; \quad            - \frac{1}{\Delta \sigma_{k}}               [   \dot{\sigma}^{\rm NG}_{k-1/2} ( \hat{\overline{T}}_{k-1/2}                                           - \overline{T}_{k}   )                + \dot{\sigma}^{\rm NG}_{k+1/2} ( \overline{T}_{k}                                            - \hat{\overline{T}}_{k+1/2} ) ]                 \\&amp; \quad            + \hat{\kappa}_{k} T_{v,k} \Dvect{v}_{k} \cdot \nabla \pi                 \\&amp; \quad            - \frac{\alpha_{k}}{\Delta \sigma_{k} } T_{v,k}              \sum_{l=k}^{K} \Dvect{v}_{l} \cdot \nabla \pi                 \Delta \sigma_{l}            - \frac{\beta_{k}}{\Delta \sigma_{k} } T_{v,k}              \sum_{l=k+1}^{K} \Dvect{v}_{l} \cdot \nabla \pi                 \Delta \sigma_{l}                 \\&amp; \quad            - \frac{\alpha_{k}}{\Delta \sigma_{k} } T'_{v,k}              \sum_{l=k}^{K} D_l  \Delta \sigma_{l}            - \frac{\beta_{k}}{\Delta \sigma_{k} } T'_{v,k}              \sum_{l=k+1}^{K} D_l  \Delta \sigma_{l} .  \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10274}%
$\displaystyle C_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10275}%
$\displaystyle = \Delta \sigma_{k} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10276}%
$\displaystyle W_{kl}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10277}%
$\displaystyle = C_{p} \alpha_{l} \delta_{k \geq l}           + C_{p} \beta_{l} \delta_{k-1 \geq l} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10278}%
$\displaystyle G_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10279}%
$\displaystyle = \hat{\kappa}_{k} C_{p} \overline{T}_{k} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10280}%
$\displaystyle \underline{h}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10281}%
$\displaystyle = \underline{Q}\underline{S} - \underline{R} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10282}%
$\displaystyle Q_{kl}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10283}%
$\displaystyle =            \frac{1}{\Delta \sigma_{k}}               ( \hat{\overline{T}}_{k-1/2} - \overline{T}_{k} ) \delta_{k=l}           + \frac{1}{\Delta \sigma_{k}}               ( \overline{T}_{k} - \hat{\overline{T}}_{k+1/2}  )  \delta_{k+1=l}  ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10284}%
$\displaystyle S_{kl}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10285}%
$\displaystyle =  \sigma_{k-1/2} \Delta \sigma_{l}             - \Delta \sigma_{l} \delta_{k \leq l }  ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10286}%
$\displaystyle R_{kl}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10287}%
$\displaystyle = - \left(  \frac{ \alpha_{k} }{ \Delta \sigma_{k} }                        \Delta \sigma_{l} \delta_{k \leq l}                      + \frac{ \beta_{k} }{ \Delta \sigma_{k} }                        \Delta \sigma_{l} \delta_{k+1 \leq l}                 \right) \overline{T}_{k} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10289}%
$ \delta_{k \leq l}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10291}%
$ k \leq l$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10292}%
$\displaystyle \delta_{t} {\cal A}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10293}%
$\displaystyle \equiv         \frac{1}{2 \Delta t}            \left( {\cal A}^{t+\Delta t} - {\cal A}^{t-\Delta t} \right) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10294}%
$\displaystyle \overline{\cal A}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10295}%
$\displaystyle \equiv \frac{1}{2} \left(   {\cal A}^{t+\Delta t}                                   + {\cal A}^{t-\Delta t} \right)       =      {\cal A}^{t-\Delta t} + \delta_{t} {\cal A} \Delta t .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10296}%
$\displaystyle \delta_{t} \pi =           \left( \DP{\pi}{t} \right)_{\rm NG}        - \Dvect{C} \cdot \overline{ \Dvect{D} }^{t} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10297}%
$\displaystyle \delta_{t} \Dvect{D} =           \left( \DP{\Dvect{D}}{t} \right)_{\rm NG}             - \nabla^{2}_{\sigma} ( \Dvect{\Phi}_{S}                                    + \underline{W}                                       \overline{ \Dvect{T} }^{t}                                   + \Dvect{G}                                   \overline{\pi}^{t} )            + {\cal D}_M ( \Dvect{D}^{t-\Delta t}                           + 2 \Delta t \delta_{t} \Dvect{D} ) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10298}%
$\displaystyle \delta_{t} \Dvect{T} =         \left( \DP{\Dvect{T}}{t} \right)_{\rm NG}            - \underline{h} \overline{ \Dvect{D} }^{t}           + {\cal D}_H ( \Dvect{T}^{t-\Delta t}                         + 2 \Delta t \delta_{t} \Dvect{T} ) .$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar10299}%
\begin{align*}\begin{split}  &amp;   \left\{   ( 1-2\Delta t {\cal D}_H )( 1-2\Delta t {\cal D}_M )                  \underline{I}                - ( \Delta t )^{2}                  (   \underline{W} \  \underline{h}                     + \Dvect{G} \Dvect{C}^{T} ) \nabla^{2}_{\sigma}      \right\}       \overline{ \Dvect{D} }^{t}         \\&amp; \qquad      =   ( 1-2\Delta t {\cal D}_H )( 1-\Delta t {\cal D}_M )             \Dvect{D}^{t-\Delta t}        + ( 1-2\Delta t {\cal D}_H ) \Delta t             \left( \DP{\Dvect{D}}{t} \right)_{\rm NG}     \\&amp; \qquad \quad        -  \Delta t \nabla^{2}_{\sigma}                         \left\{ \quad                             ( 1-2\Delta t {\cal D}_H ) \Dvect{\Phi}_{S}                            + \underline{W}                              \left[ ( 1-2\Delta t {\cal D}_H )                                      \Dvect{T}^{t-\Delta t}                                   + \Delta t                                        \left( \DP{\Dvect{T}}{t}                                            \right)_{\rm NG} \right]                    \right.   \\&amp; \qquad                    \left.  \hspace*{20mm}                            + ( 1-2\Delta t {\cal D}_H ) \Dvect{G}                              \left[ \pi^{t-\Delta t}                                    + \Delta t                                      \left( \DP{\pi}{t} \right)_{\rm NG}                               \right]                    \right\} .   \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10300}%
$\displaystyle \nabla^{2}_{\sigma} = - \frac{n(n+1)}{a^{2}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10302}%
$ \overline{ \Dvect{D}_n^m }^{t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10303}%
$\displaystyle D^{t+\Delta t} = 2\overline{ \Dvect{D} }^{t} - D^{t-\Delta t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10305}%
$ t+\Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10323}%
$ S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10324}%
$\displaystyle \DD{S}{z} &lt; 0$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10325}%
$\displaystyle \DP{T}{z} - \frac{RT}{C_p p} \DP{p}{z} + \frac{L}{C_p} \DP{q^*}{z} &lt; 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10329}%
$ L$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10331}%
$ q^{*}$%
\lthtmlinlinemathZ
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10336}%
$ (\hat{q}, \hat{T})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10338}%
$ (q, T)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10339}%
$\displaystyle q_{k-1}$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10340}%
$\displaystyle = q^* (T_{k-1}, p_{k-1}),$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10341}%
$\displaystyle q_{k}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10342}%
$\displaystyle = q^{*} (T_{k}, p_{k}),$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10343}%
$\displaystyle T_{k-1} - T_{k}     + \frac{L}{C_p}         \left\{ q^{*} (T_{k-1}, p_{k-1})  - q^{*} (T_{k}, p_{k}) \right\}    - \frac{R}{C_p} \frac{\Delta p_{k-1/2}}{p_{k-1/2}}        \frac{T_{k-1} + T_{k}}{2}  = 0,$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10344}%
$\displaystyle (C_p T_{k} + L q_{k}) \Delta p_{k}   + (C_p T_{k-1} + L q_{k-1}) \Delta p_{k-1} =   (C_p \hat{T}_{k} + L \hat{q}_{k}) \Delta p_{k}   + (C_p \hat{T}_{k-1} + L \hat{q}_{k-1}) \Delta p_{k-1}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10346}%
$ q_{k}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10348}%
$ q_{k-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10350}%
$ q \ll 1$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10352}%
$\displaystyle = q^{*} (\hat{T}_{k},p_{k})    + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k}}     \Delta T_{k},$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10354}%
$\displaystyle = q^{*} (\hat{T}_{k-1},p_{k-1})    + \left. \DP{q^{*}}{T} \right|_{\hat{T}_{k-1}}     \Delta T_{k-1}.$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10355}%
$\displaystyle \Delta T_{k}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10356}%
$\displaystyle = T_{k}   - \hat{T}_{k}   ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10357}%
$\displaystyle \Delta T_{k-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10358}%
$\displaystyle = T_{k-1} - \hat{T}_{k-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10360}%
$ \Delta T_{k-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10363}%
$\displaystyle (C_p T_{k-1} + L q_{k-1}) \Delta p_{k-1}     - (C_p \hat{T}_{k-1} + L \hat{q}_{k-1}) \Delta p_{k-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10364}%
$\displaystyle = - (C_p T_{k} + L q_{k}) \Delta p_{k}     + (C_p \hat{T}_{k} + L \hat{q}_{k}) \Delta p_{k},$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10365}%
$\displaystyle \left\{ \Delta T_{k-1}        + \frac{L}{C_p} (q_{k-1} - \hat{q}_{k-1} ) \right\}     \Delta p_{k-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10366}%
$\displaystyle = - \left\{ \Delta T_{k} + \frac{L}{C_p} (q_{k} - \hat{q}_{k}) \right\}     \Delta p_{k},$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10367}%
$\displaystyle \left[   \Delta T_{k-1}             + \frac{L}{C_p}               \left\{   q^{*} (\hat{T}_{k-1} )                       + \left. \DP{q^{*}}{T} \right|_{k-1}                        \Delta T_{k-1}                      - \hat{q}_{k-1}                         \right\}     \right] \Delta p_{k-1} \hspace{-7em}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10368}%
$\displaystyle \hspace{-7em}     = - \left[   \Delta T_{k}             + \frac{L}{C_p}               \left\{   q^{*} (T_{k})                       + \left. \DP{q^{*}}{T} \right|_{k}                        \Delta T_{k}                      - \hat{q}_{k}              \right\}     \right] \Delta p_{k},$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10369}%
$\displaystyle \left\{  1+                      \frac{L}{C_p}   \left. \DP{q^{*}}{T} \right|_{k-1}        \right\}         \Delta T_{k-1} \Delta p_{k-1}      + \frac{L}{C_p}               \left\{   q^{*} (\hat{T}_{k-1} )                       - \hat{q}_{k-1}                         \right\}        \Delta p_{k-1}       \hspace{-7em}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10370}%
$\displaystyle \hspace{-7em}    = - \left[   1            + \frac{L}{C_p}               \left. \DP{q^{*}}{T} \right|_{k}        \right] \Delta T_{k}  \Delta p_{k}      - \frac{L}{C_p}        \left\{  q^{*} (\hat{T}_{k})                       - \hat{q}_{k}        \right\}        \Delta p_{k}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10371}%
$\displaystyle \gamma_{k} \equiv \frac{L}{C_p} \left. \DP{q^{*}}{T} \right|_{k}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10372}%
$\displaystyle \left\{  1 + \gamma_{k-1} \right\}      \Delta T_{k-1} \Delta p_{k-1}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10373}%
$\displaystyle \hspace{3em} =       - \left(  1            + \gamma_{k}        \right) \Delta T_{k}  \Delta p_{k}      + \frac{L}{C_p}         \left[            \left\{ \hat{q}_{k-1} - q^{*} (\hat{T}_{k-1} ) \right\}             \Delta p_{k-1}          + \left\{ \hat{q}_{k} - q^{*} (\hat{T}_{k}) \right\}            \Delta p_{k}        \right]$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10374}%
$\displaystyle \Delta \hat{Q} \equiv             \left\{ \hat{q}_{k-1} - q^{*} (\hat{T}_{k-1} ) \right\}             \Delta p_{k-1}          + \left\{ \hat{q}_{k} - q^{*} (\hat{T}_{k}) \right\}            \Delta p_{k}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10378}%
$\displaystyle =       - \left(  1            + \gamma_{k}        \right) \Delta T_{k}  \Delta p_{k}      + \frac{L}{C_p} \Delta \hat{Q},$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10380}%
$\displaystyle =       - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}        \frac{\Delta p_{k}}{\Delta p_{k-1}}        \Delta T_{k}      + \frac{1}{ 1 + \gamma_{k-1}}        \frac{L}{C_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10382}%
$ \Delta T_{k}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10386}%
$\displaystyle \hat{T}_{k-1} + \Delta T_{k-1}   - \left( \hat{T}_{k}  + \Delta T_{k} \right)  + \frac{L}{C_p}    \left\{   q^{*} (\hat{T}_{k-1})              + \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1}            - q^{*} (\hat{T}_{k})              - \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k}     \right\}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10387}%
$\displaystyle \qquad  - \frac{R}{C_p}    \frac{\Delta p_{k-1/2}}{p_{k-1/2}}    \frac{\hat{T}_{k-1} + \Delta T_{k-1} + \hat{T}_{k} + \Delta T_{k}}{2} = 0,$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10388}%
$\displaystyle \hat{T}_{k-1} - \hat{T}_{k}  + \frac{L}{C_p}    \left\{   q^{*} (\hat{T}_{k-1})              - q^{*} (\hat{T}_{k})       \right\}  - \frac{R}{C_p}    \frac{\Delta p_{k-1/2}}{p_{k-1/2}}    \frac{\hat{T}_{k-1} + \hat{T}_{k} }{2}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10389}%
$\displaystyle \qquad  + \Delta T_{k-1} - \Delta T_{k}  +  \frac{L}{C_p} \left. \DP{q^{*}}{T} \right|_{k-1} \Delta T_{k-1}  - \frac{L}{C_p} \left. \DP{q^{*}}{T} \right|_{k} \Delta T_{k}  - \frac{R}{C_p}    \frac{\Delta p_{k-1/2}}{p_{k-1/2}}    \frac{\Delta T_{k-1} + \Delta T_{k}}{2} = 0.$%
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$\displaystyle S_{k-1/2} \equiv     \hat{T}_{k-1} - \hat{T}_{k}  + \frac{L}{C_p}    \left\{   q^{*} (\hat{T}_{k-1})              - q^{*} (\hat{T}_{k})       \right\}  - \frac{R}{C_p}    \frac{\Delta p_{k-1/2}}{p_{k-1/2}}    \frac{\hat{T}_{k-1} + \hat{T}_{k} }{2}$%
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$\displaystyle S_{k-1/2} &gt; 0$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10392}%
$\displaystyle \Delta T_{k-1} - \Delta T_{k}  + \gamma_{k-1} \Delta T_{k-1}  - \gamma_{k} \Delta T_{k}  - \frac{R}{C_p}    \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}    \Delta T_{k-1}   - \frac{R}{C_p}    \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}    \Delta T_{k} = - S_{k-1/2},$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10393}%
$\displaystyle \left( - 1 - \gamma_{k}            - \frac{R}{C_p}            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}   \right)   \Delta T_{k}  + \left(        1 + \gamma_{k-1}      - \frac{R}{C_p}        \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}    \right)    \Delta T_{k-1}  = - S_{k-1/2}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10394}%
$\displaystyle \left( - 1 - \gamma_{k}            - \frac{R}{C_p}            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}   \right)   \Delta T_{k}$%
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$\displaystyle \hspace{2em}  + \left(        1 + \gamma_{k-1}      - \frac{R}{C_p}        \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}    \right) \left(       - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}        \frac{\Delta p_{k}}{\Delta p_{k-1}}        \Delta T_{k}      + \frac{1}{ 1 + \gamma_{k-1}}        \frac{L}{C_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}} \right)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10396}%
$\displaystyle \hspace{4em} = - S_{k-1/2}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10398}%
$ \kappa = R/C_p$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10402}%
$ \kappa$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10403}%
$\displaystyle \left( - 1 - \gamma_{k}            - \kappa            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}   \right)   \Delta T_{k}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10404}%
$\displaystyle \hspace{2em}  + \left(        1 + \gamma_{k-1}      - \kappa        \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}    \right) \left(       - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}        \frac{\Delta p_{k}}{\Delta p_{k-1}}        \Delta T_{k}      + \frac{1}{ 1 + \gamma_{k-1}}        \frac{L}{C_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}} \right)$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10408}%
$\displaystyle \left[       \left( - 1 - \gamma_{k}  \right)          - \kappa            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}      +         \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \left( - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}         \frac{\Delta p_{k}}{\Delta p_{k-1}}        \right)     \right]    \Delta T_{k}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10409}%
$\displaystyle \hspace{2em} = - S_{k-1/2}    - \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \frac{1}{ 1 + \gamma_{k-1}}        \frac{L}{C_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10413}%
$\displaystyle = \frac{ \displaystyle   - S_{k-1/2}    - \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \frac{1}{ 1 + \gamma_{k-1}}        \frac{L}{C_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}   }{ \displaystyle       \left( - 1 - \gamma_{k}  \right)          - \kappa            \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}      +         \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \left( - \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}         \frac{\Delta p_{k}}{\Delta p_{k-1}}        \right)    }$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10414}%
$\displaystyle = \frac{ \displaystyle    S_{k-1/2}    + \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \frac{1}{ 1 + \gamma_{k-1}}        \frac{L}{C_p} \Delta \hat{Q} \frac{1}{\Delta p_{k-1}}   }{ \displaystyle        \left( 1 + \gamma_{k}  \right)      + \kappa        \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}      +         \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \frac{ 1 + \gamma_{k} }{ 1 + \gamma_{k-1}}         \frac{\Delta p_{k}}{\Delta p_{k-1}}   }$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10415}%
$\displaystyle = \frac{ \displaystyle     (1 + \gamma_{k-1}) \Delta p_{k-1} S_{k-1/2}    + \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \frac{L}{C_p} \Delta \hat{Q}    }{ \displaystyle        \left( 1 + \gamma_{k}  \right) (1 + \gamma_{k-1}) \Delta p_{k-1}      + \kappa        \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        (1 + \gamma_{k-1}) \Delta p_{k-1}      +         \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        (1 + \gamma_{k})        \Delta p_{k}   }$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10416}%
$\displaystyle = \frac{ \displaystyle     (1 + \gamma_{k-1}) \Delta p_{k-1} S_{k-1/2}    + \left(             1 + \gamma_{k-1}           - \kappa             \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \right)        \frac{L}{C_p} \Delta \hat{Q}    }{ \displaystyle        \left( 1 + \gamma_{k}  \right) (1 + \gamma_{k-1})         (\Delta p_{k-1} + \Delta p_{k})      + \kappa        \frac{\Delta p_{k-1/2}}{2 p_{k-1/2}}        \left\{           (1 + \gamma_{k-1}) \Delta p_{k-1}         -  (1 + \gamma_{k}) \Delta p_{k}        \right\}   }$%
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\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10419}%
$ k_B$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10421}%
$ k_T$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10423}%
$\displaystyle T_k = \hat{T}_k      +  \frac{L}{C_p} (1-b) I                     \frac{( T^c_k - T_k )}                         {\displaystyle \sum_{k=k_B}^{k_T}                           ( T^c_k - T_k ) \frac{\Delta p_k}{g}  }$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10425}%
$\displaystyle q_k = \bar{q}_k^{t-\Delta t}      +  b I         \frac{( q^c_k - q_k )}                         {\displaystyle \sum_{k=k_B}^{k_T}                           ( q^c_k - q_k ) \frac{\Delta p_k}{g}  }$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10429}%
$ t-\Delta t$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10433}%
$\displaystyle I = \sum_{k=k_B}^{k_T}       ( \hat{q}_k - \bar{q}_k^{t-\Delta t} ) \frac{\Delta p_k}{g}  .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10435}%
$ T^c_k, q^c_k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10437}%
$\displaystyle b = \left\{           \begin{array}{ll}            \left( \frac{1-\langle \mbox{RH}\rangle}                        {1-\mbox{RH}_c} \right)^n              &amp; \langle \mbox{RH}\rangle \ge \mbox{RH}_c \\1              &amp; \langle \mbox{RH}\rangle &lt; \mbox{RH}_c \\\end{array}      \right.$%
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\lthtmlinlinemathA{tex2html_wrap_inline10439}%
$ \equiv \hat{q}/q^*(\hat{T},p)$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay10441}%
$\displaystyle \langle$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10442}%
$\displaystyle \rangle    = \frac{\displaystyle \sum_{k=k_B}^{k_T} \hat{q}                                             \frac{\Delta p_k}{g}  }           {\displaystyle \sum_{k=k_B}^{k_T} q^*(\hat{T},p)                                             \frac{\Delta p_k}{g}  }  .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10446}%
$ k$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10448}%
$ (T^a_k, q^a_k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10450}%
$ k+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10452}%
$ (T^a_{k+1}, q^a_{k+1})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10454}%
$\displaystyle T^a_{k+1} = T^a_k \left( \frac{p_{k+1}}{p_k} \right)^{\kappa}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10456}%
$\displaystyle q^a_{k+1} = q^a_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10458}%
$\displaystyle q^a_{k+1} &gt; q^*( T^a_{k+1}, p_{k+1} )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10460}%
$\displaystyle T^a_{k+1} \leftarrow         T^a_{k+1}         + \frac{L}{C_p} \frac{ q^a_{k+1} - q^*( T^a_{k+1},p_{k+1}) }                              {\displaystyle                                     1 + \frac{L}{C_p} \DP{q^*}{T} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10462}%
$\displaystyle q^a_{k+1} \leftarrow         q^a_{k+1}        - \frac{L}{C_p} \DP{q^*}{T}                         \frac{ q^a_{k+1} - q^*( T^a_{k+1},p_{k+1}) }                              {\displaystyle                                     1 + \frac{L}{C_p} \DP{q^*}{T} } ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10464}%
$ T^a_1=\hat{T}_1, q^a_1=\hat{q}_1 $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10466}%
$ T^a_k = \hat{T}_k, q^a_k = q^*( \hat{T}_k, p_k ) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10468}%
$ T^a_{k+1}, q^a_{k+1} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10470}%
$\displaystyle T^a_{k+1} &gt; \hat{T}_{k+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10476}%
$ T^a_{k_B} = \hat{T}_{k_B}, 
q^a_{k_B} = q^*( \hat{T}_{k_B}, p_{k_B} ) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10478}%
$ T^c_k = T^a_k, q^c_k = q^a_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10480}%
$ q^c_k = q^*( T^c_k, p_k ) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10484}%
$\displaystyle T^c_k &lt; \hat{T}_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10486}%
$ P$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10488}%
$\displaystyle P = (1-b) I .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10498}%
$\displaystyle \hat{q} &gt; q^*(\hat{T},p)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10500}%
$ \hat{T}, \hat{q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10502}%
$\displaystyle q = q^*( T,p )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10504}%
$\displaystyle C_p T + L q = C_p \hat{T} + L \hat{q}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10506}%
$\displaystyle T = \hat{T} + \frac{L}{C_p} \frac{ \hat{q} - q^*(\hat{T},p)        }                                  {\displaystyle                                     1 + \frac{L}{C_p} \DP{q^*}{T} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10508}%
$\displaystyle q = \hat{q} - \frac{L}{C_p} \DP{q^*}{T}                             \frac{ \hat{q} - q^*(\hat{T},p)        }                                  {\displaystyle                                     1 + \frac{L}{C_p} \DP{q^*}{T} } .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10512}%
$\displaystyle P = \frac{\Delta p}{g} ( \hat{q} - q ) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10514}%
$ T,q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10522}%
$ F_S^I$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10524}%
$ S_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10526}%
$ r_S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10528}%
$ \zeta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10530}%
$\displaystyle F_S^I (\lambda, \varphi)= - S_0 r_S^{-2} \cos \zeta .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10534}%
$\displaystyle \cos \zeta = \cos \varphi \cos \delta_S \cos H            + \sin \varphi \sin \delta_S$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10536}%
$ \delta_S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10538}%
$ H$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10540}%
$ \pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10543}%
$\displaystyle \overline{F_S^I} (\varphi)   \simeq - S_0 ( A_{ins} + B_{ins} \cos^2 \varphi ) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10545}%
$\displaystyle \overline{\cos \zeta} \simeq A_{\zeta} + B_{\zeta} \cos^2 \varphi .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10547}%
$ \mathcal{A}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10549}%
$\displaystyle \overline{F_S^I} (\varphi)   \simeq - S_0 (1 - \mathcal{A})( A_{ins} + B_{ins} \cos^2 \varphi )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10551}%
$ \overline{\cos \zeta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10553}%
$ \overline{\sec \zeta}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10560}%
$\displaystyle \overline{\sec \zeta} \simeq \frac{1}{A_{\zeta} + B_{\zeta} \cos^2 \varphi} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10608}%
$ \delta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10610}%
$ \lambda_{subsolar}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10613}%
$\displaystyle H  = \lambda - \lambda_{subsolar} * \frac{180}{\pi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10616}%
$\displaystyle \cos \zeta = \sin \varphi  \sin \delta
+ \cos \varphi \cdot \sqrt{1 - \sin^2 \delta} \cdot
\cos H$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10619}%
$\displaystyle F_S^I (\lambda, \varphi) = - S_0 (1 - \mathcal{A}) \cos \zeta$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10623}%
$ N_S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10625}%
$ F_S$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10628}%
$\displaystyle F_S(z)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10630}%
$\displaystyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10632}%
$\displaystyle \sum_i^{N_S} a_i
\left[
(1-\alpha_A) F_S^I
\exp \left( - \tau_{S,i}(z) \sec \zeta \right)
\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10634}%
$\displaystyle \left. - \alpha_g (1-\alpha_A) F_S^I
\exp \left( - \tau_{S,i}(0) \sec \zeta \right)
\exp \left( - ( \tau_{S,i}(0)-\tau_i(z) ) \sec \zeta_0 \right)
\right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10640}%
$ \zeta_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10642}%
$ \sec \zeta_0 = 1.66$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10644}%
$ \alpha_A$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10646}%
$ \alpha_g$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10648}%
$ \tau_{S,i}(z)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10650}%
$\displaystyle \tau_{S,i}(z) = \int_z^\infty k_{S,i} \rho q dz            + \int_z^\infty \bar{k}_{S,i} \rho dz$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10652}%
$ k_{S,i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10654}%
$ i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10656}%
$ \bar{k}_{S,i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10660}%
$ z$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10662}%
$ a_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10666}%
$\displaystyle F_S(0) = \sum_i^{N_S} a_i           (1-\alpha_g) (1-\alpha_A) F_S^I            \exp ( - \tau_{S,i}(0)\sec \zeta ) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10669}%
$ N_R$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10671}%
$ F_R$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10673}%
$\displaystyle F_R(z) =   \left( \pi B(T_g) - \pi B(T_s) \right) {\cal T}^f(z,0)           +        \pi B(T(z_T))                  {\cal T}^f(z,z_T)           - \int_0^{z_T} \DD{\pi B}{\xi} {\cal T}^f(z,\xi) d \xi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10675}%
$ {\cal T}^f(z_1,z_2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10677}%
$ z=z_1,z_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10679}%
$ \pi B \equiv \sigma_{SB} T^4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10683}%
$\displaystyle {\cal T}^f(z_1,z_2)     = \sum_{i=1}^{N_R} b_i         \exp \left( - \delta_R | \tau_{R,i}(z_1) - \tau_{R,i}(z_2) |              \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10685}%
$ \tau_i(z)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10687}%
$\displaystyle \tau_{R,i}(z) = \int_z^\infty k_{R,i} \rho q dz                + \int_z^\infty \bar{k}_{R,i} \rho dz$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10689}%
$ k_{R,i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10693}%
$ \bar{k}_{R,i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10699}%
$ b_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10703}%
$ \delta_R = 1.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10716}%
$ q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10721}%
$ \Delta p_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10723}%
$ \Delta q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10726}%
$\displaystyle \Delta p_s = g \int \hat{\rho} \Delta q d \hat{z}
= -g \int \hat{\rho} \Delta q \frac{d \hat{p}}{- \hat{\rho} g}
= \int \Delta q d \hat{p}
= \hat{p_s} \int \Delta q d \sigma$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10728}%
$ \hat{p_s}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10730}%
$ \hat{\rho}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10734}%
$ F_{evap} \sim L C (q^{*} - q_s) \rho_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10738}%
$ \Delta q_{evap}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10741}%
$\displaystyle \frac{\Delta q_{evap}}{\Delta t}
= \frac{ F_{evap} L^{-1} \rho_s^{-1}}{\Delta z}
= \frac{ F_{evap} L^{-1} \rho_s^{-1}}{\Delta p / (\rho_s g)}
= F_{evap} \frac{g}{L \Delta p}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10744}%
$\displaystyle \Delta q_{evap}
= F_{evap} \frac{g}{L \Delta p} \Delta t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10747}%
$\displaystyle \Delta q_{cond}
= S^{cond}_{q} \Delta t$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10749}%
$ S^{cond}_{q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10751}%
$\displaystyle \Delta p_s = F_{evap} \frac{g}{L} \Delta t +    \hat{p_s} \int S^{cond}_{q} \Delta t d \sigma$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10758}%
$\displaystyle \hat{\rho} \hat{q} = \rho q,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10762}%
$ \hat{q}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10764}%
$ \rho$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10768}%
$\displaystyle \frac{\hat{p_{s}} \sigma}{RT} \hat{q} = \frac{p_{s} \sigma}{RT} q$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10770}%
$ p_s = \hat{p_s} + \Delta p_s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10773}%
$\displaystyle \frac{\hat{p_{s}} \sigma}{RT} \hat{q}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10777}%
$\displaystyle \frac{(\hat{p_s} + \Delta p_s) \sigma}{RT} q,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10779}%
$\displaystyle \hat{p_{s}} \hat{q}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10783}%
$\displaystyle (\hat{p_s} + \Delta p_s) q,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10785}%
$\displaystyle q = \frac{\hat{p_s}}{\hat{p_s} + \Delta p_s} \hat{q}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10792}%
$\displaystyle T_{Bk} = \frac{T_{k+1/2} + T_{k-1/2}}{2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10794}%
$ 1 \leq k \leq KMAX$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10800}%
$\displaystyle (T_{k} - T_{k-1}) \cdot (T_{k+1} - T_{k}) &lt; 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10803}%
$ T_{k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10805}%
$ \hat{T}_{k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10808}%
$\displaystyle \hat{T}_{k} = T_{k} + S_{grst} (T_{Bk} - T_{k})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10811}%
$\displaystyle \Delta T_{k} \equiv \hat{T}_{k} - T_{k} = S_{grst} (T_{Bk} - T_{k})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10814}%
$\displaystyle \sum^{KMAX}_{k=1} c_v \Delta T_{k} \Delta p_{k}
= \sum^{KMAX}_{k=1} c_v S_{grst} (T_{Bk} - T_{k}) (p_s \Delta \sigma_{k})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10817}%
$\displaystyle \sum^{KMAX}_{k=1} \Delta T_{k} \Delta p_{k} = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10820}%
$\displaystyle \hat{T}_{k}
=   T_{k} + S_{grst} (T_{Bk} - T_{k})
- \frac{
\sum^{KMAX}_{k=1} S_{grst} (T_{Bk} - T_{k}) \Delta p_{k}
}{
\sum^{KMAX}_{k=1} \Delta p_{k}
}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10823}%
$\displaystyle \sum^{KMAX}_{k=1} \hat{T}_{k} \Delta p_{k}
=   \sum^{KMAX}_{k=1}
\left\{
S_{grst} (T_{Bk} - T_{k}) \Delta p_{k}
- \frac{
\sum^{KMAX}_{k=1} S_{grst} (T_{Bk} - T_{k}) \Delta p_{k}
}{
\sum^{KMAX}_{k=1} \Delta p_{k}
}
\Delta p_{k}
\right\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10826}%
$\displaystyle \hat{T}_{k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10830}%
$\displaystyle T_{k-1} + S_{grst} (T_{Bk-1} - T_{k-1})
- \frac{
\sum^{k+1}_{k=k-1} S_{grst} (T_{Bk} - T_{k}) \Delta p_{k}
}{
\sum^{k+1}_{k=k-1} \Delta p_{k}
},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10832}%
$\displaystyle \hat{T}_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10836}%
$\displaystyle T_{k} + S_{grst} (T_{Bk} - T_{k})
- \frac{
\sum^{k+1}_{k=k-1} S_{grst} (T_{Bk} - T_{k}) \Delta p_{k}
}{
\sum^{k+1}_{k=k-1} \Delta p_{k}
},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10838}%
$\displaystyle \hat{T}_{k+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10842}%
$\displaystyle T_{k+1} + S_{grst} (T_{Bk+1} - T_{k+1})
- \frac{
\sum^{k+1}_{k=k-1} S_{grst} (T_{Bk} - T_{k}) \Delta p_{k}
}{
\sum^{k+1}_{k=k-1} \Delta p_{k}
}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10844}%
$ \hat{T}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10847}%
$\displaystyle \hat{\hat{T}}_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10851}%
$\displaystyle \hat{T}_{k} + S_{grst} (T_{Bk} - \hat{T}_{k})
- \frac{
\sum^{k+2}_{k=k} S_{grst} (T_{Bk} - \hat{T}_{k}) \Delta p_{k}
}{
\sum^{k+2}_{k=k} \Delta p_{k}
},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10853}%
$\displaystyle \hat{\hat{T}}_{k+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10857}%
$\displaystyle \hat{T}_{k+1} + S_{grst} (T_{Bk+1} - \hat{T}_{k+1})
- \frac{
\sum^{k+2}_{k=k} S_{grst} (T_{Bk} - \hat{T}_{k}) \Delta p_{k}
}{
\sum^{k+2}_{k=k} \Delta p_{k}
},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10859}%
$\displaystyle \hat{T}_{k+2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10863}%
$\displaystyle T_{k+1} + S_{grst} (T_{Bk+2} - T_{k+2})
- \frac{
\sum^{k+2}_{k=k} S_{grst} (T_{Bk} - \hat{T}_{k}) \Delta p_{k}
}{
\sum^{k+2}_{k=k} \Delta p_{k}
}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10867}%
$ k+2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10869}%
$ \hat{T}_{k+2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10871}%
$ T_{k+2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10879}%
$ (k-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10882}%
$\displaystyle (T_{k} - T_{k-1}) \cdot (T_{k+1} - T_{k}) \geq 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10890}%
$\displaystyle e^{*}(T) = p^{*}_{0} \exp \left( - \frac{L}{RT} \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10892}%
$ p^{*}_{0} = 1.4 \times 10^{11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay10895}%
$\displaystyle q^{*}(T, p) = p^{*}_{0} \exp \left( - \frac{L}{RT} \right) \frac{1}{p}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10900}%
$ a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10902}%
$ \times 10^6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10904}%
$ g$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10906}%
$ ^{-2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10910}%
$ ^{-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10926}%
$ C_v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10932}%
$ R_v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10938}%
$ d_{\mathrm{H}_2\mathrm{O}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10944}%
$ \epsilon_v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10946}%
$ \delta_v = \epsilon_v^{-1} - 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10950}%
$ \sigma_{SB}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10954}%
$ ^{-4}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline10956}%
$ \times 10^{-8}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11013}%
$ _2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11017}%
$ \times 10^{10}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11063}%
$ -$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\appendix
\stepcounter{chapter}
\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11134}%
$ \tilde{P}_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11138}%
$ 4 \pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11145}%
$ \tilde{P}_n^m(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11147}%
$ -1 \le \mu \le 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11148}%
$\displaystyle \tilde{P}_n^m               \equiv \frac{ (1-\mu^2)^{\frac{|m|}{2}} }{2^n n!}                         \DD[n+|m|]{}{\mu} (\mu^2-1)^n .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11154}%
$ \tilde{P}_n^0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11156}%
$ \tilde{P}_n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11158}%
$ \tilde{P}^m_n(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11159}%
$\displaystyle \DD{}{\mu}              \left\{ (1-\mu^2) \DD{}{\mu} \tilde{P}_n^m \right\}             + \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\}                                             \tilde{P}_n^m  = 0 .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11166}%
$\displaystyle (n-|m|+1) \tilde{P}_{n+1}^m            - (2n+1) \mu \tilde{P}_n^m             +(n+|m|) \tilde{P}_{n-1}^m        = 0 .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11170}%
$ 0\le |m| \le n-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11172}%
$ m=n=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11173}%
$\displaystyle (1-\mu^2)  \DD{}{\mu}\tilde{P}_{n}^{m}               =  (n+|m|) \tilde{P}_{n-1}^m                     -  n \mu \tilde{P}_n^m .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11179}%
$ \tilde{P}_n^m(\mu) \  (n=|m|,|m+1, \cdots)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11180}%
$\displaystyle \int_{-1}^1 \tilde{P}_n^m(\mu) \tilde{P}_{n'}^m (\mu) d \mu             =  \frac{2}{2n+1} \frac{(n+|m|)!}{(n-|m|)!}  \delta_{nn'} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11182}%
$ m,n,n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11184}%
$ 0\le |m| \le n,n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11188}%
$ A(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11190}%
$ \{ \tilde{P}_n^m | n=|m|,|m+1|,\cdots \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11191}%
$\displaystyle A(\mu) = \sum_{n=|m|}^{\infty}                         \tilde{A}_n^m \tilde{P}_n^m(\mu),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11192}%
$\displaystyle \tilde{A}_n^m =  \frac{2n+1}{2} \frac{(n-|m|)!}{(n+|m|)!}                      \int_{-1}^1  A(\mu) \tilde{P}_n^m(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11197}%
$ P_n^m(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11200}%
$\displaystyle P_n^m              \equiv \sqrt{ \frac{(2n+1)(n-|m|)!}{(n+|m|)!} }                       \tilde{P}_n^m                 = \sqrt{ \frac{(2n+1)(n-|m|)!}{(n+|m|)!} }                          \frac{ (1-\mu^2)^{\frac{|m|}{2}} }{2^n n!}                          \DD[n+|m|]{}{\mu} (\mu^2-1)^n .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11210}%
$ P^m_n(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11211}%
$\displaystyle \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} P_n^m \right\}              + \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\} P_n^m  = 0 .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11218}%
$\displaystyle (n-|m|+1)               \sqrt{ \frac{1}{2n+3} \frac{(n+1+|m|)!}{(n+1-|m|)!} }              P_{n+1}^m              - (2n+1)               \sqrt{ \frac{1}{2n+1} \frac{(n+|m|)!}{(n-|m|)!} }             \mu P_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11219}%
$\displaystyle \hspace*{2cm}             +(n+|m|)               \sqrt{ \frac{1}{2n-1} \frac{(n-1+|m|)!}{(n-1-|m|)!} }             P_{n-1}^m        = 0 ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11220}%
$\displaystyle ∴ P_{n+1}^m              = \sqrt{ \frac{(2n+1)(2n+3)}{(n-|m|+1)(n+|m|+1)} }                 \mu P_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11221}%
$\displaystyle \hspace*{2.5cm}              - \sqrt{ \frac{(2n+1)(2n+3)}{(n-|m|+1)(n+|m|+1)} }                \sqrt{ \frac{(n-|m|)(n+|m|)}{(2n+1)(2n-1)} } P_{n-1}^m .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11228}%
$\displaystyle (1-\mu^2)  \DD{}{\mu} P_n^m               = (n+|m|)                \sqrt{ \frac{(n-|m|)(2n+1)}{(n+|m|)(2n-1)} } P_{n-1}^m                - n \mu P_n^m .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11234}%
$ P_n^m(\mu) \  (n=|m|,|m+1, \cdots)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11235}%
$\displaystyle \int_{-1}^1 P_n^m(\mu) P_{n'}^m (\mu) d \mu             =  2 \delta_{nn'} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11245}%
$ \{ P_n^m | n=|m|,|m+1|,\cdots \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11246}%
$\displaystyle A(\mu) = \sum_{n=|m|}^{\infty}                       \tilde{A}_n^m P_n^m(\mu),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11247}%
$\displaystyle \tilde{A}_n^m =  \frac{1}{2}                     \int_{-1}^1  A(\mu) P_n^m(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11258}%
$ \exp(im\lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11260}%
$ \int_0^{2 \pi} \exp(im\lambda) \exp(-im'\lambda) 
d \lambda = 2 \pi \delta_{mm'}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11262}%
$ m,m'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11265}%
$\displaystyle Y_n^m(\lambda, \varphi)                \equiv P_n^m(\sin \varphi) \exp(i m \lambda) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11271}%
$ Y^m_n(\lambda, \varphi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11272}%
$\displaystyle \left[ \frac{1}{\cos \varphi}                        \DP{}{\varphi}                        \left( \cos \varphi \DP{}{\varphi} \right)                       + \frac{1}{\cos^2 \varphi} \DP[2]{}{\lambda}                  + n(n+1) \right] Y_n^m =0 .$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11273}%
$\displaystyle \left[                     \DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right)                    + \frac{1}{1-\mu^2} \DP[2]{}{\lambda}                     + n(n+1)                 \right] Y_n^m =0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11280}%
$\displaystyle \int_{-1}^1 Y_n^m(\lambda, \varphi)                       Y_{n'}^{m'*} (\lambda, \varphi)           d (\sin \varphi) d \lambda             =  4 \pi \delta_{mm'} \delta_{nn'} .$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11282}%
$ m,m',n,n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11286}%
$ 0 \le |m'| \le n'$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11290}%
$ \{ Y_n^m | m=0,1,2,\cdots, \   n=|m|,|m+1|,\cdots \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11291}%
$\displaystyle A(\lambda, \varphi)             = \sum_{m=0}^{\infty} \sum_{n=|m|}^{\infty}                       \tilde{A}_n^m Y_n^m(\lambda,\varphi)         ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11292}%
$\displaystyle \tilde{A}_n^m =  \frac{1}{4 \pi}                     \int_{-1}^1  d (\sin \varphi)                      \int_0^{2 \pi} d \lambda A(\lambda, \varphi)                     Y_n^{m*}(\lambda, \varphi)$%
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11295}%
$ Y_n^m(\varphi,\lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11303}%
$\displaystyle \frac{1}{r \cos \varphi} \DP{Y_n^m}{\lambda}       =  \frac{1}{r \cos \varphi}               \DP{}{\lambda}              \left( P_n^m (\sin \varphi) \exp(im \lambda) \right)      =  \frac{im}{r \cos \varphi}             P_n^m (\sin \varphi) \exp(im \lambda) .$%
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\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11307}%
$\displaystyle \frac{1}{r} \DP{Y_n^m}{\varphi}       =     \frac{1}{r}              \DP{}{\varphi}              \left( P_n^m (\sin \varphi) \exp(im \lambda) \right)       =  \frac{\sqrt{1-\mu^2} }{r}             \DD{}{\mu} P_n^{m} (\mu) \exp(im \lambda) .$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar11309}%
\begin{align*}\begin{split}       \nabla_H^2      Y_n^m      &amp;\equiv  \frac{1}{r^2}            \left[ \DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right)                   + \frac{1}{1-\mu^2} \DP[2]{}{\lambda}             \right] Y_n^m \\&amp;=  \frac{1}{r^2}             \left[ \frac{1}{\cos \varphi}                    \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right)                   + \frac{1}{\cos^2 \varphi} \DP[2]{}{\lambda}             \right] Y_n^m \\&amp;=  - \frac{n(n+1)}{r^2} Y_n^m      \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11314}%
$ n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11318}%
$ (\lambda', \varphi')$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11322}%
$ \{ Y_n^m(\lambda', \varphi') | m=-n,-n+1, \cdots, n \} $%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11323}%
$\displaystyle Y^m_n(\lambda, \varphi)         = \sum_{m'=-n}^{n} A_n^{m'} Y^{m'*}_n(\lambda',\varphi') .$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11333}%
$ \nabla_H^2 \equiv 
\frac{1}{r^2}
\left[ \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right)
+ \frac{1}{\cos^2 \varphi} \DP[2]{}{\lambda} 
\right]    
$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11335}%
$ - \frac{n(n+1)}{r^2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11337}%
$ \nabla_H^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11339}%
$ \nabla_H^2 Y^m_n(\lambda, \varphi)
=- \frac{n(n+1)}{r^2} Y^m_n(\lambda,\varphi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11341}%
$ Y_n^m \exp(im\lambda)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11347}%
$ \{Y_n^m |n=0,1,2,\cdots,\   m= -n, -n+1,\cdots,n \}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11349}%
$ \{Y_n^m  | m= -n, -n+1,\cdots,n \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11351}%
$ \nabla^2_H f = -\frac{n(n+1)}{r^2} f$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11353}%
$ Y_n^m(\lambda',\varphi')$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11359}%
$ \nabla^{'2}_H Y_n^m = -\frac{n(n+1)}{r^2} Y_n^m $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11368}%
$ P_n,P_n^1,P_n^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap5149}%
\Depsf[10cm]{spectral/spl-spherical-harmonics-images/legendre1.ps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap5157}%
\Depsf[][7cm]{spectral/spl-spherical-harmonics-images/legendre2.ps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11376}%
$ \overline{P_n^1} = P_n^1/\sqrt{2},
\overline{P_n^2} = P_n^2/\sqrt{2}$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11379}%
$ D$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11383}%
$ \chi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11385}%
$ \psi$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11390}%
$ f(\lambda,\varphi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11394}%
$ {\displaystyle \frac{1}{r \cos \varphi} \DP{f}{\lambda} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11396}%
$ f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11400}%
$ {\displaystyle \frac{1}{r} \DP{f}{\varphi}
\left( = \frac{\cos \varphi}{r} \DP{f}{\mu} \right) }$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar11403}%
\begin{align*}\begin{split}             \nabla^2_H f           &amp;\equiv             \frac{1}{r^2}            \left[ \frac{1}{\cos \varphi}                    \DP{}{\varphi} \left( \cos \varphi \DP{}{\varphi} \right)                   + \frac{1}{\cos^2 \varphi} \DP[2]{}{\lambda}             \right] f  \\&amp;=           \frac{1}{r^2}           \left[             \DP{}{\mu}                 \left\{ (1-\mu^2) \DP{}{\mu} \right\}           + \frac{1}{1-\mu^2} \DP[2]{}{\lambda}           \right] f      \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11406}%
$ \Dvect{v}=(v_1,v_2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar11407}%
\begin{align*}\begin{split}       \mbox{div}_H \Dvect{v}          &amp;\equiv     \frac{1}{r \cos \varphi} \DP{v_1}{\lambda}           +  \frac{1}{r \cos \varphi} \DP{}{\varphi} (v_2 \cos \varphi)     \\&amp;=     \frac{1}{r \sqrt{1-\mu^2} } \DP{v_1}{\lambda}           +  \frac{1}{r} \DP{}{\mu}(  \sqrt{1-\mu^2} v_2 )         \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11409}%
$ \Dvect{v}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11411}%
$ r$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar11412}%
\begin{align*}\begin{split}       (\mbox{rot} \Dvect{v})_r       &amp;\equiv    \frac{1}{r \cos \varphi} \DP{v_2}{\lambda}           -  \frac{1}{r \cos \varphi} \DP{}{\varphi}(v_1 \cos \varphi)    \\&amp;=     \frac{1}{r \sqrt{1-\mu^2} } \DP{v_2}{\lambda}           -  \frac{1}{r} \DP{}{\mu}(\sqrt{1-\mu^2}v_1)           \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11417}%
$ u,\  v$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{alignstar11418}%
\begin{align*}\begin{split}      D   &amp;=     \frac{1}{r \cos \varphi} \DP{u}{\lambda}           +  \frac{1}{r \cos \varphi} \DP{}{\varphi} (v \cos \varphi) .     \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{alignstar11424}%
\begin{align*}\begin{split}       \zeta       &amp;=     \frac{1}{r \cos \varphi} \DP{v}{\lambda}           -  \frac{1}{r \cos \varphi} \DP{}{\varphi}(u \cos \varphi) .     \end{split}\end{align*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11432}%
$\displaystyle D$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11433}%
$\displaystyle \equiv  \nabla_H^2 \chi ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11434}%
$\displaystyle \zeta$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11435}%
$\displaystyle \equiv  \nabla_H^2 \psi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11440}%
$\displaystyle u$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11441}%
$\displaystyle =  - \frac{1}{r}             \DP{\psi}{\varphi}             + \frac{1}{r\cos\varphi}  \DP{\chi}{\lambda} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11442}%
$\displaystyle v$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11443}%
$\displaystyle =    \frac{1}{r\cos\varphi}  \DP{\psi}{\lambda}             + \frac{1}{r}             \DP{\chi}{\varphi}$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11450}%
$ n-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11452}%
$ -1 \leq \mu \leq  1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11454}%
$ P_n(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11463}%
$ \mu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11469}%
$ P_0 \sim P_{n-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11475}%
$ f(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11478}%
$\displaystyle \int_{-1}^1 f(\mu) P_n(\mu) d \mu = 0$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11487}%
$ f(x)=(x-1)^n (x+1)^n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11489}%
$ f=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11491}%
$ x=-1,1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11493}%
$ f'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11495}%
$ \alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11497}%
$ -1&lt;\alpha&lt;1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11499}%
$ f'(\alpha)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11501}%
$ f'=2nx(x^2-1)^{n-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11503}%
$ f'=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11505}%
$ x=-1,\alpha,1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11507}%
$ f''=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11509}%
$ x=-1,\beta_1,\beta_2,1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11511}%
$ -1 &lt; \beta_1 &lt; \beta_2 &lt; 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11513}%
$ f^{(n)}=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11515}%
$ -1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11517}%
$ 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11523}%
$ {\displaystyle 
P_n= \frac{1}{2^n n!}
\DD[n]{}{\mu} (\mu^2-1)^n  }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11531}%
$ {\displaystyle x_j=\cos \frac{j-1/2}{n}\pi }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11537}%
$ g(\lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11539}%
$ 0 \le \lambda &lt; 2 \pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11540}%
$\displaystyle g(\lambda) = \sum_{m=-M}^{m=M} g_m \exp( i m \lambda )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11542}%
$ M &lt; I$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11545}%
$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda         = \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11546}%
$\displaystyle \lambda_n = \frac{2\pi (n-1)}{I} \  (n=1,2,\cdots,I)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11547}%
$\displaystyle \sum_{n=1}^{I} \exp(i m \lambda_n)            = \left\{                 \begin{array}{ll}                     I &amp; \  \ (m=0) , \\0 &amp; \  \ (0 &lt; |m| &lt; I) , \\\end{array}             \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11550}%
$ I &gt; M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11552}%
$ |m|$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11554}%
$ m \neq 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11556}%
$ {\displaystyle \exp(i m \lambda_n) 
= \exp \left( \frac{2\pi i m (n-1)}{I} \right) }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11560}%
$ m(n-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11575}%
$\displaystyle \sum^{I}_{n=1} \exp \left\{ im \frac{2 \pi (n-1)}{I} \right\}          = \frac{1 - \left( e^{\frac{im 2 \pi}{I}}\right)^I}                 {1 - e^{ \frac{im 2 \pi}{I} } }          = \frac{1 -  e^{im 2 \pi} }                 {1 - e^{ \frac{im 2 \pi}{I} } }          = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11576}%
$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11577}%
$\displaystyle =  \sum_{m=-M}^{M} \frac{1}{2\pi} g_m                  \int_0^{2\pi} \exp(i m \lambda) d \lambda            =g_0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11579}%
$ \int_0^{2\pi} \exp(i m \lambda) d \lambda $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11581}%
$ m=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11582}%
$\displaystyle \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11583}%
$\displaystyle =  \frac{1}{I}             \sum_{n=1}^{I} \sum_{m=-M}^{M}             g_m \exp(i m \lambda_n)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11584}%
$\displaystyle =  g_0            + \sum_{m=-M, m\neq0}^{M} \frac{g_m}{I}               \sum_{n=1}^{I}                 \left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11585}%
$\displaystyle \sum_{n=1}^{I}           \left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1} =0          \  \ \  \ (m \neq 0)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11586}%
$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda          = \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11591}%
$ 2J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11595}%
$ {\displaystyle \int_{-1}^1 f d \mu }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11597}%
$ P_J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11599}%
$ \mu_j（j=1,2,\cdots,J）$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11603}%
$ f(\mu_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11604}%
$\displaystyle \int^{1}_{-1} f(\mu) d \mu = 2 \sum_{j=1}^{J} f(\mu_j) w_j ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11605}%
$\displaystyle w_j = \frac{1}{2} \int_{-1}^{1}                        \frac{P_J(\mu)}{(\mu-\mu_j) P^{'}_J (\mu_i)}d \mu                    = \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 }  .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11609}%
$ \tilde{P_n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11619}%
$ K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11621}%
$ 0 \le K \le 2J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11624}%
$\displaystyle \int_{-1}^1 \tilde{P}_n(\mu) \tilde{P}_{n'}(\mu) d \mu          = \frac{2}{2n+1} \delta_{nn'} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11626}%
$ L(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11629}%
$\displaystyle L(\mu) \equiv                \sum_{j=1}^{J} f(\mu_j)                \prod_{k=1,k \neq j}^{J}                 \frac{ \mu-\mu_k}{\mu_j-\mu_k} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11633}%
$ L(\mu_j) = f(\mu_j) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11637}%
$ 0 \le K \le J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11641}%
$ J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11643}%
$ L=f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11647}%
$ L-f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11653}%
$ \mu_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11655}%
$ f(\mu)-L(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11660}%
$ J \le K \le 2J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11662}%
$ \mu=\mu_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11670}%
$ \tilde{P}_J(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11672}%
$ f-L$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11676}%
$ K-J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11678}%
$ S(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11679}%
$\displaystyle f(\mu) - L(\mu) = \tilde{P}_J(\mu) S(\mu)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11689}%
$ \tilde{P}_J(\mu) S(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11690}%
$\displaystyle \int_{-1}^{1} f(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11691}%
$\displaystyle =  \int_{-1}^1 L(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11692}%
$\displaystyle =  \sum_{j=1}^{J} f(\mu_j)              \int^1_{-1}              \frac{ {\displaystyle                         \prod_{k=1}^{J}(\mu-\mu_k) }                   }                   { (\mu-\mu_j)                     {\displaystyle                         \prod_{k=1,k \neq j}^{J}(\mu_j-\mu_k) }                   }                  d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11693}%
$\displaystyle =  \sum_{j=1}^{J} f(\mu_j)               \int_{-1}^1              \frac{\tilde{P}_J(\mu)}                   {(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11694}%
$\displaystyle =  2 \sum_{j=1}^{J} f(\mu_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11698}%
$ \tilde{P}_J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11706}%
$ {\displaystyle w_j
= \frac{1}{2} \int_{-1}^1
\frac{\tilde{P}_J(\mu)}
{(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu
}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11709}%
$\displaystyle (n+1) \tilde{P}_{n+1}(\mu)         = (2n+1) \mu \tilde{P}_n(\mu) - n \tilde{P}_{n-1}(\mu)        \  \ \  \ (n=0,1,2,\cdots)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11710}%
$\displaystyle (n+1) \left|               \begin{array}{ll}                 \tilde{P}_{n+1}(x) &amp; \tilde{P}_n(x) \\\tilde{P}_{n+1}(y) &amp; \tilde{P}_n(y)               \end{array}            \right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11711}%
$\displaystyle =             \left|               \begin{array}{ll}                 (2n+1)x \tilde{P}_n(x)-n\tilde{P}_{n-1}(x)                     &amp;   \tilde{P}_n(x) \\(2n+1)y \tilde{P}_n(y)-n\tilde{P}_{n-1}(y)                     &amp;   \tilde{P}_n(y)               \end{array}            \right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11712}%
$\displaystyle =             (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11713}%
$\displaystyle \quad               +n (- \tilde{P}_{n-1}(x) \tilde{P}_n(y)                + \tilde{P}_{n-1}(y) \tilde{P}_n(x) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11714}%
$\displaystyle =             (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)            +            n \left|                     \begin{array}{ll}                       \tilde{P}_{n}(x) &amp; \tilde{P}_{n-1}(x) \\\tilde{P}_{n}(y) &amp; \tilde{P}_{n-1}(y)                     \end{array}                  \right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11716}%
$ n=0,1,\cdots,n-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11717}%
$\displaystyle n \left|               \begin{array}{ll}                 \tilde{P}_n(x) &amp; \tilde{P}_{n-1}(x) \\\tilde{P}_n(y) &amp; \tilde{P}_{n-1}(y)               \end{array}            \right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11718}%
$\displaystyle =   \sum_{k=0}^{n-1} (2k+1)(x-y)\tilde{P}_k(x)\tilde{P}_k(y)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11720}%
$ n=J,x=\mu,y=\mu_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11722}%
$ \tilde{P}_J(\mu_j)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11723}%
$\displaystyle J  \tilde{P}_J (\mu)  \tilde{P}_{J-1} (\mu_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11724}%
$\displaystyle =   \sum_{k=0}^{J-1} (2k+1) (\mu-\mu_j)                \tilde{P}_k(\mu) \tilde{P}_k(\mu_j).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11725}%
$\displaystyle \frac{\tilde{P}_J (\mu) }{\mu-\mu_j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11726}%
$\displaystyle =  \frac{                     {\displaystyle                       \sum_{k=0}^{J-1} (2k+1)                        \tilde{P}_k(\mu) \tilde{P}_k(\mu_j) }                  }                  {J \tilde{P}_{J-1}(\mu_j)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11727}%
$\displaystyle w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11728}%
$\displaystyle =  \frac{1}{2} \int_{-1}^1                \frac{\tilde{P}_J(\mu)}                     {(\mu-\mu_j) \tilde{P}^{'}_J(\mu_j)}                d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11729}%
$\displaystyle =  \frac{1}{2 J \tilde{P}_{J-1}(\mu_j)                          \tilde{P}^{'}_{J} (\mu_j)}              \sum_{k=0}^{J-1}                    (2k+1) \tilde{P}_k(\mu_j)                    \int_{-1}^1 \tilde{P}_k(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11730}%
$\displaystyle =  \frac{1}{J \tilde{P}_{J-1}(\mu_j)                          \tilde{P}^{'}_{J} (\mu_j)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11732}%
$ k=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11734}%
$ \tilde{P}_0 = 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11735}%
$\displaystyle (1-\mu^2) \DP{\tilde{P}_n}{\mu}          = n \tilde{P}_{n-1}(\mu) - n \mu \tilde{P}_n(\mu)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11737}%
$ n=J, \mu=\mu_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11741}%
$\displaystyle =  \frac{1-\mu_j^2}{(J \tilde{P}_{J-1}(\mu_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11746}%
$\displaystyle \int_{-1}^1 P_n(\mu) P_n'(\mu) d \mu = 2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11748}%
$ {\displaystyle 
\tilde{P}_{J-1}
=\sqrt{ \frac{1}{2(\mbox{J}-1)+1} } P_{J-1} 
}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11749}%
$\displaystyle w_j = \frac{1-\mu_j^2}                   {(J  \sqrt{ \frac{1}{2J-1} } P_{J-1}(\mu_j))^2 }           = \frac{(2J-1)(1-\mu_j^2)}                   {(J  P_{J-1}(\mu_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11751}%
$\displaystyle w_j = \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11753}%
$\displaystyle \sum_{j=1}^{J}        \sum_{i=1}^{I}          P_n^m (\mu_j) P_{n'}^{m'} (\mu_j)          \exp(i m \lambda_i) \exp(-i m' \lambda_i) w_j       = I \delta_{nn'} \delta_{mm'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11755}%
$ i,j,m,m',n,n',I,J,M,N(m)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11757}%
$ 1 \le i \le I , 1 \le j \le J, $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11759}%
$ 0 \le |m|,|m'| \le M, |m| \le n \le N, |m'| \le n' \le N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11761}%
$ {\displaystyle M \le \left[ \frac{I}{2} \right] , 
N(m) \le J-1   }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11765}%
$ {\displaystyle \lambda_i=\frac{2\pi(i-1)}{I} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11771}%
$ [ \  ]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11774}%
$\displaystyle \quad          \sum_{j=1}^{J}             \sum_{i=1}^{I}             P_n^m (\mu_j) P_{n'}^{m'} (\mu_j)            \exp(i m \lambda_i) \exp(-i m' \lambda_{i})             w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11775}%
$\displaystyle =           I \sum_{j=1}^{J}              P_n^m (\mu_j) P_{n'}^{m'} (\mu_j) w_j \delta_{mm'} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11777}%
$\displaystyle =           I \sum_{j=1}^{J}              P_n^m (\mu_j) P_{n'}^{m} (\mu_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11778}%
$\displaystyle =           \frac{I}{2} \int_{-1}^1              P_n^m (\mu) P_{n'}^{m} (\mu) d\mu .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11780}%
$\displaystyle =             I \delta_{nn'} \delta_{mm'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11784}%
$ \{ f_k(\mu) \}(k=0,1,2,\cdots) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11786}%
$ \   [ a ,b ] \  $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11788}%
$ w(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11790}%
$ \lambda_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11792}%
$ {\displaystyle  
\left( \int_a^b f_k (\mu) f_{k'} (\mu) w(\mu) d \mu
= \lambda_k \delta_{kk'} \right)   }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11794}%
$ \mu_j, \mu_{j'}(1 \le j,j' \le J)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11796}%
$ f_J(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11798}%
$ w_j=w(\mu_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11799}%
$\displaystyle \sum_{j=0}^{J-1}                f_k (\mu_j) f_{k'} (\mu_{j}) w_j                  = \lambda_k  \delta_{kk'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11800}%
$\displaystyle \sum_{k=0}^{J-1}                \frac{f_k (\mu_j) f_k (\mu_{j'}) }{\lambda_k}                  = \frac{1}{w_j}  \delta_{jj'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11802}%
$ \{ P_n \}(n=0,1,2,\cdots,J-1) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11805}%
$\displaystyle \sum_{n=0}^{J-1} P_n (\mu_j) P_n (\mu_{j'})                   = \frac{1}{w_j}  \delta_{jj'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11807}%
$ P_{J}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11809}%
$ \{ P_n^m \} (n=|m|,|m|+1,|m|+2,\cdots,N)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11813}%
$ f_k（k=0,1,2,\cdots,K-1）$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11829}%
$ n=0,1,2,\cdots,N &lt; J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11835}%
$ J=63, N=42$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11839}%
$ J=63, N=21$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11840}%
$\displaystyle \sum_{i=0}^{I-1}          \exp(i m \lambda_i) \exp(-i m' \lambda_i)           = I \delta_{mm'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11841}%
$\displaystyle \sum_{m=-\frac{I}{2}+1}^{\frac{I}{2}}          \exp(i m \lambda_i) \exp(-i m \lambda_{i'})          = I \delta_{ii'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11845}%
$ I=2M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11849}%
$ I=2M+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11853}%
$ {\displaystyle  -\frac{I-1}{2} \sim \frac{I-1}{2} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11859}%
$ {\displaystyle \frac{I}{3} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11863}%
$ I=128$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11865}%
$ M=42$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11867}%
$ I=64$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11869}%
$ M=21$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11883}%
$ A_{ij} \; (i=1,2,\cdots,I, \; j=1,2,\cdots,J)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11885}%
$ \tilde{A}_n^m \;
(m=-M,-M+1, \cdots,M, \; n=|m|,|m|+1,\cdots,N(m))$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11889}%
$\displaystyle =    \frac{1}{I}                 \sum_{i=1}^{I} \sum_{j=1}^{J}                   A_{ij} Y_n^{m*} (\lambda_i,\varphi_j) w_j ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11891}%
$\displaystyle =  \frac{(2J-1)(1-\sin^2 \varphi_j)}                       {(J P_{J-1}(\sin \varphi_j))^2 } .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11893}%
$ {\displaystyle \sum_{m=-M}^{M} \sum_{n=|m|}^{N} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11895}%
$ {\displaystyle \sum_{m,n} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11897}%
$ {\displaystyle \sum_{i=1}^{I} \sum_{j=1}^{J} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11899}%
$ {\displaystyle \sum_{i,j} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11901}%
$\displaystyle g$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11902}%
$\displaystyle \equiv  \DP{f}{\lambda}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11905}%
$\displaystyle g_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11906}%
$\displaystyle \equiv         \left[         \DP{}{\lambda} \left(         \sum_{m,n} \tilde{f}_n^m Y_n^m (\lambda, \varphi)         \right)                                    \right]_{ij} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11908}%
$\displaystyle =   \sum_{m,n} im \tilde{f}_n^m          Y_n^m (\lambda_i,\varphi_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11916}%
$ {\displaystyle (g_{ij}=) 
\sum_{m,n} im \tilde{f}_n^m Y_n^m
= \sum_{m,n} \tilde{g}_n^m Y_n^m }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11918}%
$ {\displaystyle \sum_{i,j} Y_n^{m*}(\lambda_i,\varphi_j)w_j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11920}%
$ im' \tilde{f}_{n'}^{m'} 
= \tilde{g}_{n'}^{m'} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11921}%
$\displaystyle \tilde{g}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11922}%
$\displaystyle =   im \tilde{f}_n^m .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11924}%
$\displaystyle =    \frac{1}{I}                 \sum_{i,j} im f_{ij} Y_n^{m*} (\lambda_i, \varphi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11925}%
$\displaystyle h  \equiv \frac{g}{r \cos^2 \varphi}     =      \frac{1}{r \cos^2 \varphi} \DP{f}{\lambda}            \  \     \left[ =                      \DP{}{x} \left( \frac{f}{\cos \varphi} \right)                    \right]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11929}%
$ h$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11930}%
$\displaystyle h_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11931}%
$\displaystyle =  \frac{1}{r \cos^2 \varphi_i} g_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11932}%
$\displaystyle ∴\  h_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11933}%
$\displaystyle =              \frac{1}{r \cos^2 \varphi_j}             \sum_{m,n} im \tilde{f}_n^m Y_n^m (\lambda_i, \varphi_j) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11934}%
$\displaystyle \tilde{h}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11935}%
$\displaystyle =  \widetilde{          \left[             \DP{}{\lambda} \left( \frac{f}{r \cos^2 \varphi} \right)          \right]_n^m           }       = im \widetilde{ \left( \frac{f}{r \cos^2 \varphi} \right)_n^m }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11936}%
$\displaystyle =  \frac{1}{I}               \sum_{i,j}                  im                 \left( \frac{f}{r \cos^2 \varphi} \right)_{ij}                 Y_n^{m*} (\lambda_i, \varphi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11937}%
$\displaystyle =  \frac{1}{I}               \sum_{i,j} im f_{ij}                 Y_n^{m*} (\lambda_i \varphi_j)                  \frac{w_j}{r \cos^2 \varphi_j} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11939}%
$\displaystyle p \equiv \DP{f}{\varphi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11942}%
$\displaystyle p_{ij} \equiv \left[ \DP{}{\varphi}                         \left( \sum_{m,n} \tilde{f}_n^m Y_n^m \right)                 \right]_{ij} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11943}%
$\displaystyle p_{ij} = \sum_{m,n} \tilde{f}_n^m                     \left. \DD{P_n^m}{\varphi} \right|_j                     \exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11944}%
$\displaystyle p_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11945}%
$\displaystyle =  \frac{1}{I} \sum_{i,j} p_{ij} Y_n^{m*} w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11946}%
$\displaystyle =  \frac{1}{I} \sum_{i,j}                 \left(                       \sum_{m',n'} \tilde{f}_{n'}^{m'}                       \left. \DD{P_{n'}^{m'}}{\varphi} \right|_j                      \exp(i m' \lambda_i)                \right)               P_n^m(\varphi_j) \exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11947}%
$\displaystyle =  - \frac{1}{I} \sum_{i,j}                \left(                       \sum_{m',n'} \tilde{f}_{n'}^{m'}                       P_{n'}^{m'}(\varphi_j)                      \exp(i m' \lambda_i)               \right)              \left. \DD{P_n^m}{\varphi}\right|_j               \exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11948}%
$\displaystyle =  - \frac{1}{I} \sum_{i,j}                f_{ij}               \left. \DD{P_n^m}{\varphi}\right|_j               \exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11949}%
$\displaystyle \qquad            \sum_{i=1}^I \sum_{j=1}^J            f_{n'}^{m'}            P_{n}^{m}(\varphi_j)  \exp(im \lambda_i)            \left. \DD{P_{n'}^{m'}}{\varphi}\right|_j             \exp(-im' \lambda_i)            w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11950}%
$\displaystyle =  - \sum_{i=1}^I \sum_{j=1}^J            f_{n'}^{m'}            \left. \DD{P_{n}^{m}}{\varphi}\right|_j             \exp(-im \lambda_i)            P_{n'}^{m'}(\varphi_j)  \exp(im' \lambda_i)            w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11953}%
$\displaystyle \quad 		    \sum_{i} \sum_{j}                     f_{n'}^{m'}                     P_{n}^{m}(\varphi_j)  \exp(im \lambda_i)                     \left. \DD{P_{n'}^{m'}}{\varphi}\right|_j                      \exp(-im' \lambda_i)                     w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11954}%
$\displaystyle =  I \sum_{j}                    f_{n'}^{m'}                    P_{n}^{m}(\varphi_j)                    \left.\DD{P_{n'}^{m'}}{\varphi}\right|_j                   w_j \delta_{m m'}                  = I \sum_{j}                    f_{n'}^{m}                    P_{n}^{m}(\varphi_j)                    \left.\DD{P_{n'}^{m}}{\varphi}\right|_j                    w_j \delta_{m m'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11955}%
$\displaystyle =  \frac{I}{2}                     \int_{-1}^{1}                     f_{n'}^{m}                    P_{n}^{m}(\varphi)                     \DD{P_{n'}^m}{\varphi}  d \varphi \delta_{m m'} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11957}%
$\displaystyle = - \frac{I}{2}                     \int_{-1}^{1}                     f_{n'}^{m}                    P_{n'}^{m}(\varphi)                     \DD{P_n^m}{\varphi}  d \varphi \delta_{m m'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11958}%
$\displaystyle =  - I \sum_{j}                    f_{n'}^{m}                    P_{n'}^{m}(\varphi_j)                    \left.\DD{P_{n}^{m}}{\varphi}\right|_j                    w_j  \delta_{mm'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11959}%
$\displaystyle =  - \sum_{i} \sum_{j}                     f_{n'}^{m'}                     P_{n'}^{m'}(\varphi_j)  \exp(im' \lambda_i)                     \left. \DD{P_{n}^{m}}{\varphi}\right|_j                      \exp(-im \lambda_i)                     w_j .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11960}%
$\displaystyle q \equiv \cos^2\varphi \DP{f}{\varphi} = \cos^2\varphi\   p$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11961}%
$\displaystyle q_{ij} = \cos^2\varphi_j \sum_{m,n} \tilde{f}_n^m                           \left. \DD{P_n^m}{\varphi} \right|_j                           \exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11962}%
$\displaystyle \tilde{q}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11963}%
$\displaystyle =  \frac{1}{I} \sum_{i,j} q_{ij} Y_n^{m*} w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11964}%
$\displaystyle =  \frac{1}{I} \sum_{i,j}                \left(                       \cos^2\varphi_j                       \sum_{m',n'} \tilde{f}_{n'}^{m'}                       \left. \DD{P_{n'}^{m'}}{\varphi} \right|_j                      \exp(i m' \lambda_i)               \right)               P_n^m(\varphi_j) \exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11965}%
$\displaystyle =  - \frac{1}{I} \sum_{i,j}                \left(                       \sum_{m',n'} \tilde{f}_{n'}^{m'}                       P_{n'}^{m'}(\varphi_j)                      \exp(i m' \lambda_i)               \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11966}%
$\displaystyle \quad 	     \hspace{1.5cm} \times              \left. \DD{}{\varphi}                          \left( \cos^2\varphi P_n^m \right) \right|_j               \exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11967}%
$\displaystyle =  - \frac{1}{I} \sum_{i,j}                f_{ij}               \left. \DD{}{\varphi} \left( \cos^2\varphi P_n^m \right) \right|_j              \exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11968}%
$\displaystyle \quad       \sum_{i=1}^I \sum_{j=1}^J       f_{n'}^{m'} \cos^2\varphi_j       P_{n}^{m}(\varphi_j)  \exp(im \lambda_i)       \left. \DD{P_{n'}^{m'}}{\varphi}\right|_j        \exp(-im' \lambda_i)       w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11969}%
$\displaystyle =  - \sum_{i=1}^I \sum_{j=1}^J       f_{n'}^{m'}       \left. \DD{}{\varphi}        \left(\cos^2\varphi P_n^m \right) \right|_j        \exp(-im \lambda_i)       P_{n'}^{m'}(\varphi_j)  \exp(im' \lambda_i)       w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11976}%
$ \chi_n^m,\psi_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11978}%
$ u_{ij},v_{ij}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11979}%
$\displaystyle u  = - \frac{1}{r}            \DP{\psi}{\varphi}           + \frac{1}{r\cos\varphi} \DP{\chi}{\lambda}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11980}%
$\displaystyle u_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11981}%
$\displaystyle =  \sum_{m,n}             \left(                - \frac{1}{r} \tilde{\psi}_n^m                           \left. \DD{P_n^m}{\varphi} \right|_j                + \frac{1}{r\cos\varphi_j} im \tilde{\chi}_n^m P_n^m(\sin \varphi_j)             \right)             \exp(im \lambda_i) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11982}%
$\displaystyle v  =  \frac{1}{r\cos\varphi} \DP{\psi}{\lambda}          + \frac{1}{r}            \DP{\chi}{\varphi}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11983}%
$\displaystyle v_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay11984}%
$\displaystyle = \sum_{m,n}             \left(                 \frac{1}{r\cos\varphi_j} im \tilde{\psi}_n^m P_n^m (\sin \varphi_j)               + \frac{1}{r} \tilde{\chi}_n^m                    \left. \DD{P_n^m}{\varphi} \right|_{j}             \right)             \exp(im \lambda_i) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline11987}%
$ g = \DP{f}{\lambda}$%
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$\displaystyle \tilde{g}_n^m = im \tilde{f}_n^m .$%
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$ h = \nabla_H^2 f$%
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$\displaystyle h_{ij}    = \left. \left(\nabla_H^2 \sum_{m,n}                    \tilde{f}_n^m Y_n^m \right)\right|_{ij}    = - \sum_{m,n}         \frac{n(n+1)}{r^2}          \left. \tilde{f}_n^m Y_n^m \right|_{ij}$%
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$\displaystyle h_{ij} =  \sum_{m,n}         \left. \tilde{h}_n^m Y_n^m \right|_{ij}$%
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$ {\displaystyle \sum_{i,j} Y_{n'}^{m'*}|_{ij} }$%
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{\newpage\clearpage
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$\displaystyle \tilde{f}_{n'}^{m'}                    = -\frac{n(n+1)}{r^2} \tilde{h}_{n'}^{m'}$%
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\stepcounter{section}
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$ P_n^m(\sin \varphi) \exp(im \lambda)$%
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$ (n,m)$%
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{\newpage\clearpage
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$ J=K=M =N_{tr} $%
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$ K=2N_{tr},\  J=M=N_{tr} $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12031}%
$ K=J+M$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12033}%
$ K=42, J=M=21$%
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\stepcounter{subsection}
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\lthtmlinlinemathA{tex2html_wrap_inline12040}%
$ \Leftrightarrow$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12046}%
$ 1+ (2 \times1 +1) + \cdots 
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12048}%
$ 128 \times 64 = 8192 $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12050}%
$ (2 \times 21 +1) \times (21+1)= 946$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12052}%
$ 64 \times 64 = 4096 $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12054}%
$ 3/4$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12056}%
$ M,N(m)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12058}%
$ M=42, N=42$%
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\lthtmlinlinemathA{tex2html_wrap_inline12060}%
$ I,J$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12062}%
$ M,N$%
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{\newpage\clearpage
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$ -M \le m \le M,\   |m| \le n \le N(m)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12074}%
$ 1 \le i \le I,1 \le j \le J$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay12076}%
$\displaystyle \equiv \sum_{m=-M}^M \sum_{n=|m|}^N                         \tilde{A}_n^m P_n^{m}(\sin \varphi_j)                              \exp(im \lambda_i)$%
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$ \tilde{A}_n^m  ( -M \le m \le M, \  |m| \le n \le N) $%
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\lthtmlinlinemathA{tex2html_wrap_indisplay12080}%
$\displaystyle =  \frac{1}{I}                        \sum_{i=1}^I \sum_{j=1}^J                            A_{ij} P_n^{m}(\sin \varphi_j)                            \exp(-im \lambda_i)     w_j$%
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$ \varphi_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12088}%
$\displaystyle =  \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J                            \left(                              \sum_{m'=-M}^{M} \sum_{n'=|m'|}^N                                  \tilde{A}_{n'}^{m'}                                  P_{n'}^{m'}(\sin \varphi_j)                                 \exp(im' \lambda)                           \right)                           P_n^{m}(\sin \varphi_j)                           \exp(-im \lambda_i)     w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12089}%
$\displaystyle =  \frac{1}{I} \sum_{m'=-M}^{M} \sum_{n'=|m'|}^N                            \tilde{A}_{n'}^{m'}                         \sum_{i=1}^I \exp(i(m'-m) \lambda)                         \sum_{j=1}^J                            P_n^{m}(\sin \varphi_j)                           P_{n'}^{m'}(\sin \varphi_j)  w_j$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12095}%
$ -M \le m \le M,\   |m| \le n \le N$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12098}%
$\displaystyle \sum_{i=1}^I \exp(i(m'-m) \lambda) = I \delta_{mm'} ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12099}%
$\displaystyle \sum_{j=1}^J P_n^{m}(\sin \varphi_j)                         P_{n'}^{m}(\sin \varphi_j)  w_j                          = \delta_{nn'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12101}%
$ |m'-m|$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12103}%
$ 2M$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12107}%
$ I \ge 2M+1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12109}%
$ n+n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12113}%
$ I \delta_{mm'}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12117}%
$ P_n^m P_{n'}^{m'} $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12121}%
$ 2J-1 \ge {\rm max}[n+n'] = 2 {\rm max}[N]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12123}%
$ {\rm max}[n+n']$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12127}%
$ {\rm max}[N]$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12135}%
$ \frac{I-1}{2}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12139}%
$ \frac{I}{2}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12147}%
$ n=|m|, |m|+1, \cdots, |m|+J-1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12149}%
$ P_n^m P_{n'}^{m} $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12153}%
$ 2J+2|m|-2$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12165}%
$ B,C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12167}%
$ B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12169}%
$ C$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12174}%
$\displaystyle A= BC ,$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12175}%
$\displaystyle B= \sum_{m=-M}^{M} \sum_{n=|m|}^{N}              \left( \tilde{B}_n^m \exp(im \lambda) \right)               P_n^m(\sin \varphi) ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12176}%
$\displaystyle C= \sum_{m=-M}^{M} \sum_{n=|m|}^{N}              \left( \tilde{C}_n^m \exp(im \lambda) \right)               P_n^m(\sin \varphi)$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12178}%
$ A,B,C$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12184}%
$ \tilde{B}_n^m = \tilde{B}_n^{m*},  {\it etc.}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12192}%
$ \tilde{B}_n^m,\tilde{C}_n^m$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12198}%
$ 0 \le m \le M,\   |m| \le n \le N$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12200}%
$\displaystyle \equiv  \widetilde{(BC)_n^m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12201}%
$\displaystyle =  \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J                  B_{ij} C_{ij}                  P_n^{m}(\sin \varphi_j) \exp(-im \lambda_i)     w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12202}%
$\displaystyle =  \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J                \left(  \sum_{m'=-M}^{M} \sum_{n'=|m'|}^N                        \tilde{B}_{n'}^{m'}                                    \exp(im' \lambda_i)                               P_{n'}^{m'}(\sin \varphi_j)                  \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12203}%
$\displaystyle \quad                \times                \left(  \sum_{m''=-M}^{M} \sum_{n''=|m''|}^N                         \tilde{C}_{n''}^{m''}                                    \exp(im'' \lambda_i)                              P_{n''}^{m''}(\sin \varphi_j)                  \right)                 P_n^{m}(\sin \varphi_j) \exp(-im \lambda_i)   w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12204}%
$\displaystyle =  \frac{1}{I}                \sum_{m'=-M}^{M} \sum_{n'=|m'|}^N               \sum_{m''=-M}^{M} \sum_{n''=|m''|}^N                  \tilde{B}_{n'}^{m'} \tilde{C}_{n''}^{m''}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12205}%
$\displaystyle \quad               \times                \sum_{i=1}^I                      \exp(i(m'+m''-m) \lambda_i)                 \sum_{j=1}^J                              P_{n'}^{m'}(\sin \varphi_j)                              P_{n''}^{m''}(\sin \varphi_j)                             P_n^{m}(\sin \varphi_j)       w_j .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12215}%
$ I \ge 3M+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12219}%
$ 2J-1 \ge {\rm max}[n+n'+n''] = 3 {\rm max}[N]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12221}%
$ {\rm max}[n+n'+n'']$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12223}%
$ n+n'+n''$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12239}%
$ {\displaystyle M= \left[ \frac{I-1}{3} \right] }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12243}%
$ N_{{\rm max}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12245}%
$ N_{{\rm max}}=M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12247}%
$ N_{{\rm max}}=2M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12251}%
$ J \ge \frac{3N_{{\rm max}}+1}{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12259}%
$ M=21, N=42$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12269}%
$ {\displaystyle M= \left[ \frac{I}{2} \right] }$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12287}%
$ J \ge \frac{2N_{{\rm max}}+1}{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12291}%
$ M=64$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12293}%
$ N=64$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12295}%
$ J=65$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12297}%
$ I=128, J=65$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12301}%
$ M=32$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12305}%
$ J \ge 65$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline12307}%
$ I=64, J=65$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

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