\batchmode


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


\usepackage{Dennou6}


\Dtitle[DCPAM3 ��2�� Υ����]{DCPAM3 \\[5mm]
          ��2�� \ Υ���� \\[70mm]}
  \Dauthor[�ϵ�ή����Ǿ�����]{�ϵ�ή����Ǿ�����}
  %
\Dver{dcpam3-20060118} \Dmodify{2006}{01}{18}
  \Ddate[\Dmoddate ]{\DmoddateJ (DCPAM3-\Dversion )}

%
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\footnotetext[0]{#1}
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} 


\makeindex




\usepackage[dvips]{color}


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A_{ij} Y_n^{m*} (\lambda_i,  \phi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline584}%
${\displaystyle 
      \left(\tilde{A}^m_n \exp(im\lambda) \right)^* 
      = \tilde{A}^{-m}_n \exp(-im\lambda) }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline586}%
$m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline590}%
$n-m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline592}%
$\phi=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline598}%
$A_{ij}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline600}%
$A_{ij}^{even}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline602}%
$A_{ij}^{odd}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline604}%
$A{i,J-j}=A_{ij}^{even}-A_{ij}^{odd}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline606}%
$A_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline608}%
$A_{i,j}+A_{i,J-j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline610}%
$A_{i,j}-A_{i,J-j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline612}%
$j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline614}%
$J/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7245}%
$\displaystyle A_{ij} =  \sum_{m=0}^{M} \sum_{n=m}^{N}
\Re \tilde{A}_n^m Y_n^m(\lambda_i, \phi_j)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7246}%
$\displaystyle \tilde{A}_n^m =
\left\{
\begin{array}{ll}
{\displaystyle \frac{1}{I}
\sum_{i=1}^{I} \sum_{j=1}^{J}  }
A_{ij} Y_n^{m*}(\lambda_i, \phi_j) w_j
&   \ \ \ m=0, m \le n \le N \\
{\displaystyle \frac{2}{I}
\sum_{i=1}^{I} \sum_{j=1}^{J}  }
A_{ij} Y_n^{m*}(\lambda_i,\phi_j) w_j
&   \ \ \ 1 \le m \le M, m \le n \le N
\end{array}
\right.$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline620}%
$A(\lambda,\phi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7254}%
$\displaystyle A(\lambda,\phi)
\equiv \sum_{m=-M}^{M} \sum_{n=|m|}^{N}
\tilde{A}_n^m Y_n^m (\lambda, \phi)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline628}%
$\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7258}%
$\displaystyle \left( \DP{f}{\lambda} \right)_{ij}$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7260}%
$\displaystyle \sum_{m=-M}^M
\sum_{n=|m|}^N im \tilde{f}_n^m
Y_n^m (\lambda_i,\phi_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7261}%
$\displaystyle \widetilde{ \left( \DP{f}{\lambda} \right)_n^m }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7262}%
$\textstyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7263}%
$\displaystyle \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J
im f_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline630}%
$\mu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7266}%
$\displaystyle \left(\DP{f}{\mu}\right)_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7268}%
$\displaystyle \sum_{m=-M}^M \sum_{n=|m|}^N \tilde{f}_n^m
\left. \DD{P_n^m}{\mu} \right|_j
\exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7269}%
$\displaystyle \widetilde{\left(\DP{f}{\mu}\right)_n^m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7271}%
$\displaystyle - \frac{1}{I} \sum_{i=1}^I \sum_{j=1}^J f_{ij}
\left. \DD{P_n^m}{\mu}\right|_j
\exp(-im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{Dchapter}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7274}%

% latex2html id marker 7274
\setcounter{footnote}{8}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7276}%

% latex2html id marker 7276
\setcounter{footnote}{8}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath734}%
\begin{displaymath}
  \frac{\partial \pi}{\partial t}
 = - \sum_{k=1}^{K} ( D_k + \Dvect{v}_k \cdot \nabla \pi ) 
       \Delta  \sigma_k
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath741}%
\begin{displaymath}
  \dot{\sigma}_{k-1/2}
 = - \sigma_{k-1/2} \frac{\partial \pi}{\partial t}
   - \sum_{l=k}^{K} ( D_l + \Dvect{v}_l \cdot \nabla \pi )          
       \Delta  \sigma_l
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath751}%
\begin{displaymath}
  \dot{\sigma}_{1/2} = \dot{\sigma}_{K+1/2} = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7285}%
$\displaystyle \Phi_{1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7287}%
$\displaystyle \Phi_{s} + C_{p} ( \sigma_{1}^{-\kappa} - 1  ) T_{v,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7289}%
$\displaystyle \Phi_{s} + C_{p} \alpha_{1} T_{v,1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7291}%
$\displaystyle \Phi_k - \Phi_{k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7293}%
$\displaystyle C_{p}
\left[ \left( \frac{ \sigma_{k-1/2} }{ \sigma_k } \right)^{\kappa}
- 1 \right] T_{v,k}
+ C_{p}
\left[ 1-
\left( \frac{ \sigma_{k-1/2} }{ \sigma_{k-1} } \right)^{\kappa}
\right] T_{v,k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7295}%
$\displaystyle C_{p} \alpha_k T_{v,k} + C_{p} \beta_{k-1} T_{v,k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7297}%
$\displaystyle \alpha_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7299}%
$\displaystyle \left( \frac{ \sigma_{k-1/2} }
{ \sigma_k } \right)^{\kappa} -1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7300}%
$\displaystyle \beta_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7302}%
$\displaystyle 1- \left( \frac{ \sigma_{k+1/2} }
{ \sigma_k } \right)^{\kappa} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath798}%
\begin{displaymath}
  \frac{\partial \zeta_k}{\partial t} 
     =   \frac{1}{a(1-\mu^{2})} \frac{\partial (\mbox{\sl VA})_k}
                                      {\partial \lambda}
          - \frac{1}{a} \frac{\partial (\mbox{\sl UA})_k}{\partial \mu}
          + {\cal D}(\zeta_k) 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath812}%
\begin{displaymath}
  \frac{\partial D}{\partial t} 
   =   \frac{1}{a(1-\mu^{2})}
           \frac{\partial (\mbox{\sl UA})_k}{\partial \lambda}
          + \frac{1}{a} \frac{\partial (\mbox{\sl VA})_k}{\partial \mu}
          - \nabla^{2}_{\sigma}
           ( \Phi_k + C_{p} \hat{\kappa}_k \bar{T}_k \pi 
             + (\mbox{\sl KE})_k )
          + {\cal D}(D_k) 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7307}%
$\displaystyle (\mbox{\sl UA})_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7309}%
$\displaystyle ( \zeta_k + f ) V_k
- \frac{1}{2 \Delta \sigma_k}
[   \dot{\sigma}_{k-1/2} ( U_{k-1} - U_k   )
+ \dot{\sigma}_{k+1/2} ( U_k   - U_{k+1} ) ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7310}%
$\displaystyle - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a}
\frac{\partial \pi}{\partial \lambda}
+ F_{\lambda k} \cos \varphi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7312}%
$\displaystyle (\mbox{\sl VA})_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7314}%
$\displaystyle - ( \zeta_k + f ) U_k
- \frac{1}{2 \Delta \sigma_k}
[   \dot{\sigma}_{k-1/2} ( V_{k-1} - V_k   )
+ \dot{\sigma}_{k+1/2} ( V_k   - V_{k+1} ) ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7315}%
$\displaystyle - \frac{C_{p} \hat{\kappa}_k T_{v,k}'}{a}
( 1 - \mu^{2} ) \frac{\partial \pi}{\partial \mu}
+ F_{\varphi k} \cos \varphi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7317}%
$\displaystyle \hat{\kappa}_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7319}%
$\displaystyle \frac{  \sigma_{k-1/2}(   \sigma^{\kappa}_{k-1/2}
- \sigma^{\kappa}_k      )
+ \sigma_{k+1/2}(   \sigma^{\kappa}_k
- \sigma^{\kappa}_{k+1/2}  ) }
{ \sigma^{\kappa}_k
( \sigma_{k-1/2} - \sigma_{k+1/2} )         }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7321}%
$\displaystyle \frac{ \sigma_{k-1/2} \alpha_k + \sigma_{k+1/2} \beta_k }
{ \Delta \sigma_k                                    }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath886}%
\begin{displaymath}
T_{v,k} = T_k - \bar{T}_k
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath890}%
\begin{displaymath}
     (\mbox{\sl KE})_k =  \frac{U^{2}_k+V^{2}_k}{2(1-\mu^{2}) } 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7326}%
$\displaystyle \frac{\partial T_k}{\partial t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7328}%
$\displaystyle - \frac{1}{a(1-\mu^{2})}
\frac{\partial U_k T_k'}{\partial \lambda}
- \frac{1}{a}
\frac{\partial V_k T_k'}{\partial \mu}
+ \hat{H}_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7329}%
$\displaystyle + \frac{Q_k}{C_{p}}
+ {\cal D}(T_k)
+ {\cal D}'(\Dvect{v})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7331}%
$\displaystyle \hat{H}_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7333}%
$\displaystyle T_k' D_k
- \frac{1}{\Delta \sigma_k}
[   \dot{\sigma}_{k-1/2} ( \hat{T}_{k-1/2} - T_k   )
+ \dot{\sigma}_{k+1/2} ( T_k   - \hat{T}_{k+1/2} ) ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7334}%
$\displaystyle + \left\{ \alpha_k
\left[ \sigma_{k-1/2} \Dvect{v}_k \cdot \nabla \pi
- \sum_{l=k}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta  \sigma_l
\right]
\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7335}%
$\displaystyle + \left. \beta_k
\left[ \sigma_{k+1/2} \Dvect{v}_k \cdot \nabla \pi
- \sum_{l=k+1}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta  \sigma_l
\right]
\right\}
\frac{1}{\Delta \sigma_k} T_{v,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7338}%
$\displaystyle + \hat{\kappa}_k \Dvect{v}_k \cdot \nabla \pi T_{v,k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7339}%
$\displaystyle - \alpha_k \sum_{l=k}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta  \sigma_l
\frac{T_{v,k}}{\Delta \sigma_k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7340}%
$\displaystyle - \beta_k \sum_{l=k+1}^{K}
( D_l + \Dvect{v}_l \cdot \nabla \pi )
\Delta  \sigma_l
\frac{T_{v,k}}{\Delta \sigma_k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7342}%
$\displaystyle \hat{T}_{k-1/2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7344}%
$\displaystyle \frac{ \left[ \left( \frac{ \sigma_{k-1/2} }
{ \sigma_k } \right)^{\kappa}
- 1 \right] \sigma_{k-1}^{\kappa} T_k
+ \left[ 1-
\left( \frac{ \sigma_{k-1/2} }
{ \sigma_{k-1} } \right)^{\kappa}
\right] \sigma_k^{\kappa} T_{k-1}         }
{ \sigma_{k-1}^{\kappa} - \sigma_k^{\kappa}           }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7346}%
$\displaystyle a_k T_k + b_{k-1} T_{k-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7348}%
$\displaystyle a_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7350}%
$\displaystyle \alpha_k
\left[ 1- \left( \frac{ \sigma_k }{ \sigma_{k-1} }
\right)^{\kappa} \right]^{-1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7351}%
$\displaystyle b_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7353}%
$\displaystyle \beta_k
\left[ \left( \frac{ \sigma_k }{ \sigma_{k+1} }
\right)^{\kappa} - 1 \right]^{-1} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath995}%
\begin{displaymath}
  \frac{\partial q_k}{\partial t}
      =   - \frac{1}{a(1-\mu^{2})} 
               \frac{\partial U_k q_k}{\partial \lambda}
          - \frac{1}{a}
               \frac{\partial V_k q_k}{\partial \mu}
          + R_k 
          + S_{q,k}
          + {\cal D}(q_k) 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1010}%
\begin{displaymath}
R_k  =  q_k D_k 
       - \frac{1}{2 \Delta \sigma_k} 
             [   \dot{\sigma}_{k-1/2} ( q_{k-1} - q_k   )
               + \dot{\sigma}_{k+1/2} ( q_k   - q_{k+1} ) ]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7357}%

% latex2html id marker 7357
\setcounter{footnote}{9}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7361}%
$\displaystyle \DP{\tilde{\pi}_n^m}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7363}%
$\displaystyle - \sum_{k=1}^{K} (\tilde{D}_n^m)_k \Delta  \sigma_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7364}%
$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
Z_{ij} Y_n^{m *} ( \lambda_i, \mu_j ) w_j  ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1128}%
\begin{displaymath}
Z \equiv - \sum_{k=1}^{K} \Dvect{v}_k \cdot \nabla \pi .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7368}%
$\displaystyle \DP{\tilde{\zeta}_n^m}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7370}%
$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im (\mbox{\sl VA})_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7371}%
$\textstyle +$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7372}%
$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
(\mbox{\sl UA})_{ij}
(1-\mu_j^2)
\frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7374}%
$\displaystyle \tilde{\cal D}_{M,n}^m \tilde{\zeta}_n^m  ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7376}%
$\displaystyle \DP{\tilde{D}_n^m}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7378}%
$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im (\mbox{\sl UA})_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7379}%
$\textstyle -$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7380}%
$\displaystyle \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
(\mbox{\sl VA})_{ij}
(1-\mu_j^2)
\frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7382}%
$\displaystyle \frac{n(n+1)}{a^{2}}
\frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
(\mbox{\sl KE})_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7384}%
$\displaystyle \frac{n(n+1)}{a^{2}}
( \Phi_n^m + C_{p} \hat{\kappa}_k \bar{T}_k \pi_n^m )
+  \tilde{\cal D}_{M,n}^m \tilde{D}_n^m  ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1212}%
\begin{displaymath}
\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]  .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7388}%
$\displaystyle \DP{\tilde{T}_n^m}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7390}%
$\displaystyle - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im U_{ij} T'_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7391}%
$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
V_{ij} T'_{ij}
(1-\mu_j^2)
\frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7392}%
$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
\left( \hat{H}_{ij} + \frac{Q_{ij}}{C_{p}} \right)
Y_n^{m *} ( \lambda_i, \mu_j )
w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7393}%
$\displaystyle + \tilde{\cal D}_{H,n}^m \tilde{T}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7394}%
$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
{\cal D}'_{ij}(\Dvect{v})
Y_n^{m *} ( \lambda_i, \mu_j )
w_j ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1276}%
\begin{displaymath}
\tilde{\cal D}_{H,n}^m 
   = - K_{HD} \left( \frac{-n(n+1)}{a^{2}} \right)^{N_D/2} .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7398}%
$\displaystyle \DP{\tilde{q}_n^m}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7400}%
$\displaystyle - \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
im U_{ij} q_{ij}
Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7401}%
$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
V_{ij} q_{ij}
(1-\mu_j^2)
\frac{\partial }{\partial \mu}  Y_n^{m *} ( \lambda_i, \mu_j )
\frac{w_j}{a(1-\mu_j^{2})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7402}%
$\displaystyle + \frac{1}{I} \sum_{i=1}^{I} \sum_{j=1}^{J}
\left( \hat{R}_{ij} + S_{q,ij} \right)
Y_n^{m *} ( \lambda_i, \mu_j )
w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7403}%
$\displaystyle + \tilde{\cal D}_{H,n}^m \tilde{q}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7404}%

% latex2html id marker 7404
\setcounter{footnote}{9}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1889}%
$\Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1891}%
$2 \Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1893}%
${\cal F}_\lambda, {\cal F}_\varphi, Q, S_q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1897}%
${\cal A}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1397}%
\begin{displaymath}
  \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)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1899}%
$ \dot{\cal A}_{adv} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1901}%
$ \dot{\cal A}_{dif} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1903}%
$ \dot{\cal A}_{phy} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1905}%
$ \hat{\cal A}^{t+\Delta t} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1907}%
$ \dot{\cal A}_{dis} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1909}%
$ \dot{\cal A}_{cnd} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1911}%
$ {\cal A}^{t+\Delta t} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1429}%
\begin{displaymath}
  {\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)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1443}%
\begin{displaymath}
  \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)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1913}%
$\bar{\cal A}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1915}%
$\epsilon_f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1917}%
$T=\bar{T}_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1919}%
$NG$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1921}%
$\Dvect{D}=\{ D_{k} \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1923}%
$\Dvect{T}=\{ T_{k} \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1460}%
\begin{displaymath}
   \DP{\pi}{t} =��
          \left( \frac{\partial \pi}{\partial t} \right)_{NG}  
     - \Dvect{C} \cdot \Dvect{D}  ,
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1469}%
\begin{displaymath}
  \frac{\partial \Dvect{D}}{\partial t} = 
          \left( \DP{\Dvect{D}}{t} \right)_{NG}  
          - \nabla^{2}_{\sigma} ( \Dvect{\Phi}_{S} 
                                  + \underline{W} \Dvect{T}
                                  + \Dvect{G} \pi )  
          + {\cal D}_M \Dvect{D} ,
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7432}%
$\displaystyle \DP{\Dvect{T}}{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7434}%
$\displaystyle \left( \DP{\Dvect{T}}{t}       \right)_{NG}
- \underline{h} \Dvect{D}
+ {\cal D}_H \Dvect{T} ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1496}%
\begin{displaymath}
  \left( \DP{\pi}{t} \right)^{NG}
   =   - \sum_{k=1}^{K} \Dvect{v}_{k} \cdot \nabla \pi  
       \Delta  \sigma_{k} \nonumber \\
   =   Z_{k}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1508}%
\begin{displaymath}
  \dot{\sigma}^{NG}_{k-1/2}
 = - \sigma_{k-1/2} \left( \DP{\pi}{t} \right)^{NG}
   - \sum_{l=k}^{K} \Dvect{v}_{l} \cdot \nabla \pi
       \Delta  \sigma_{l}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1522}%
\begin{displaymath}
  \left( \DP{D}{t} \right)^{NG}
   =   \frac{1}{a(1-\mu^{2})}
            \DP{(\mbox{\sl UA})_{k}}{\lambda}
          + \frac{1}{a} \DP{(\mbox{\sl VA})_{k}}{\mu}
          - \nabla^{2}_{\sigma} (\mbox{\sl KE})_{k} 
          - \nabla^{2}_{\sigma} \sum_{k=1}^{K} W_{kl} ( T_{v,l}-T_{l} )
          + {\cal D}(D_{k}) 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1550}%
\begin{displaymath}
  \left( \DP{T_{k}}{t} \right)^{NG} 
      \equiv   - \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}) 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7440}%
$\displaystyle H_k$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7442}%
$\displaystyle T_{k}^{\prime} D_{k}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7443}%
$\displaystyle - \frac{1}{\Delta \sigma_{k}}
[   \dot{\sigma}_{k-1/2} ( \hat{T^{\prime}}_{k-1/2}
- T^{\prime}_{k}   )
+ \dot{\sigma}_{k+1/2} ( T^{\prime}_{k}
- \hat{T^{\prime}}_{k+1/2} ) ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7444}%
$\displaystyle - \frac{1}{\Delta \sigma_{k}}
[   \dot{\sigma}^{NG}_{k-1/2} ( \hat{\bar{T}}_{k-1/2}
- \bar{T}_{k}   )
+ \dot{\sigma}^{NG}_{k+1/2} ( \bar{T}_{k}
- \hat{\bar{T}}_{k+1/2} ) ]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7445}%
$\displaystyle + \hat{\kappa}_{k} T_{v,k} \Dvect{v}_{k} \cdot \nabla \pi$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7446}%
$\displaystyle - \frac{\alpha_{k}}{\Delta \sigma_{k} } T_{v,k}
\sum_{l=k}^{K} \Dvect{v}_{l} \cdot \nabla \pi
\Delta \sigma_{l}
- \frac{\beta_{k}}{\Delta \sigma_{k} } T_{v,k}
\sum_{l=k+1}^{K} \Dvect{v}_{l} \cdot \nabla \pi
\Delta \sigma_{l}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7447}%
$\displaystyle - \frac{\alpha_{k}}{\Delta \sigma_{k} } T'_{v,k}
\sum_{l=k}^{K} D_l  \Delta \sigma_{l}
- \frac{\beta_{k}}{\Delta \sigma_{k} } T'_{v,k}
\sum_{l=k+1}^{K} D_l  \Delta \sigma_{l}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1637}%
\begin{displaymath}
  C_{k} = \Delta \sigma_{k}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1642}%
\begin{displaymath}
  W_{kl} = C_{p} \alpha_{l} \delta_{k \geq l}
         + C_{p} \beta_{l} \delta_{k-1 \geq l}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1651}%
\begin{displaymath}
  G_{k} = \hat{\kappa}_{k} C_{p} \bar{T}_{k}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1659}%
\begin{displaymath}
\underline{h} = \underline{Q}\underline{S} - \underline{R}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1665}%
\begin{displaymath}
  Q_{kl} = \frac{1}{\Delta \sigma_{k}} 
             ( \hat{\bar{T}}_{k-1/2} - \bar{T}_{k} ) \delta_{k=l} 
         + \frac{1}{\Delta \sigma_{k}} 
             ( \bar{T}_{k} - \hat{\bar{T}}_{k+1/2}  ) \delta_{k+1=l} 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1682}%
\begin{displaymath}
  S_{kl} = \sigma_{k-1/2} \Delta \sigma_{l} 
           - \Delta \sigma_{l} \delta_{k \leq l } 
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1689}%
\begin{displaymath}
  R_{kl} = - \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) \bar{T}_{k} .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1925}%
$\delta_{k \leq l}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1927}%
$ k \leq l$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1704}%
\begin{displaymath}
  \delta_{t} {\cal A} \equiv \frac{1}{2 \Delta t} 
        \left( {\cal A}^{t+\Delta t} - {\cal A}^{t-\Delta t} \right)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7459}%
$\displaystyle \overline{\cal A}^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7461}%
$\displaystyle \frac{1}{2} \left( {\cal A}^{t+\Delta t}
+ {\cal A}^{t-\Delta t} \right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7463}%
$\displaystyle {\cal A}^{t-\Delta t} + \delta_{t} {\cal A} \Delta t   ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1729}%
\begin{displaymath}
  \delta_{t} \pi =
          \left( \DP{\pi}{t} \right)_{NG}  
     - \Dvect{C} \cdot \overline{ \Dvect{D} }^{t}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1739}%
\begin{displaymath}
  \delta_{t} \Dvect{D} =
          \left( \DP{\Dvect{D}}{t} \right)_{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} )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1762}%
\begin{displaymath}
  \delta_{t} \Dvect{T} =
        \left( \DP{\Dvect{T}}{t} \right)_{NG}  
         - \underline{h} \overline{ \Dvect{D} }^{t} 
         + {\cal D}_H ( \Dvect{T}^{t-\Delta t}
                        + 2 \Delta t \delta_{t} \Dvect{T} )
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7468}%
$\displaystyle \left\{ ( 1-2\Delta t {\cal D}_H )( 1-2\Delta t {\cal D}_M )
\underline{I}
- ( \Delta t )^{2}  ( \underline{W} \ \underline{h}
+ \Dvect{G} \Dvect{C}^{T} ) \nabla^{2}_{\sigma}
\right\}
\overline{ \Dvect{D} }^{t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7469}%
$\displaystyle = ( 1-2\Delta t {\cal D}_H )( 1-\Delta t {\cal D}_M )
\Dvect{D}^{t-\Delta t}
+ ( 1-2\Delta t {\cal D}_H ) \Delta t
\left( \DP{\Dvect{D}}{t} \right)_{NG}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7470}%
$\displaystyle -  \Delta t \nabla^{2}_{\sigma}
\left\{  ( 1-2\Delta t {\cal D}_H ) \Dvect{\Phi}_{S}
+ \underline{W}
\left[ ( 1-2\Delta t {\cal D}_H )
\Dvect{T}^{t-\Delta t}
+ \Delta t
\left( \DP{\Dvect{T}}{t}
\right)_{NG} \right]
\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7471}%
$\displaystyle \left.  \hspace*{20mm}
+ ( 1-2\Delta t {\cal D}_H ) \Dvect{G}
\left[ \pi^{t-\Delta t}
+ \Delta t
\left( \DP{\pi}{t} \right)_{NG}
\right]
\right\} .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1887}%
\begin{displaymath}
    \nabla^{2}_{\sigma} = - \frac{n(n+1)}{a^{2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1929}%
$\overline{ \Dvect{D}_n^m }^{t}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1827}%
\begin{displaymath}
   D^{t+\Delta t} = 2\overline{ \Dvect{D} }^{t} - D^{t-\Delta t}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1931}%
$t+\Delta t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{Dchapter}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7479}%

% latex2html id marker 7479
\setcounter{footnote}{9}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\stepcounter{section}
\appendix
\stepcounter{Dchapter}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7487}%

% latex2html id marker 7487
\setcounter{footnote}{9}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7488}%

% latex2html id marker 7488
\setcounter{footnote}{9}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2403}%
$\tilde{P}_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2407}%
$4 \pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2413}%
$\tilde{P}_n^m(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2415}%
$-1 \le \mu \le 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7501}%
$\displaystyle \tilde{P}_n^m
\equiv \frac{ (1-\mu^2)^{\frac{|m|}{2}} }{2^n n!}
\DD[n+|m|]{}{\mu} (\mu^2-1)^n$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2421}%
$\tilde{P}_n^0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2423}%
$\tilde{P}_n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2425}%
$\tilde{P}^m_n(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7508}%
$\displaystyle \DD{}{\mu}
\left\{ (1-\mu^2) \DD{}{\mu} \tilde{P}_n^m \right\}
+ \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\}
\tilde{P}_n^m  = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7513}%
$\displaystyle (n-|m|+1) \tilde{P}_{n+1}^m
- (2n+1) \mu \tilde{P}_n^m
+(n+|m|) \tilde{P}_{n-1}^m        = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2435}%
$0\le |m| \le n-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2437}%
$m=n=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7518}%
$\displaystyle (1-\mu^2)  \DD{}{\mu}\tilde{P}_{n}^{m}
=  (n+|m|) \tilde{P}_{n-1}^m
-  n \mu \tilde{P}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2443}%
$\tilde{P}_n^m(\mu) \ (n=|m|,|m+1, \cdots)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7523}%
$\displaystyle \int_{-1}^1 \tilde{P}_n^m(\mu) \tilde{P}_{n'}^m (\mu) d \mu
=  \frac{2}{2n+1} \frac{(n+|m|)!}{(n-|m|)!}  \delta_{nn'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2445}%
$m,n,n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2447}%
$0\le |m| \le n,n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2451}%
$A(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2453}%
$ \{ \tilde{P}_n^m | n=|m|,|m+1|,\cdots \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7530}%
$\displaystyle A(\mu) = \sum_{n=|m|}^{\infty}
\tilde{A}_n^m \tilde{P}_n^m(\mu),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7531}%
$\displaystyle \tilde{A}_n^m =  \frac{2n+1}{2} \frac{(n-|m|)!}{(n+|m|)!}
\int_{-1}^1  A(\mu) \tilde{P}_n^m(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2457}%
$P_n^m(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7537}%
$\displaystyle P_n^m
\equiv \sqrt{ \frac{(2n+1)(n-|m|)!}{(n+|m|)!} }
\tilde{P}_n^m
= \sqrt{ \frac{(2n+1)(n-|m|)!}{(n+|m|)!} }
\frac{ (1-\mu^2)^{\frac{|m|}{2}} }{2^n n!}
\DD[n+|m|]{}{\mu} (\mu^2-1)^n   ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2469}%
$P^m_n(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7544}%
$\displaystyle \DD{}{\mu} \left\{ (1-\mu^2) \DD{}{\mu} P_n^m \right\}
+ \left\{ n(n+1) - \frac{m^2}{1-\mu^2} \right\} P_n^m  = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7549}%
$\displaystyle (n-|m|+1)
\sqrt{ \frac{1}{2n+3} \frac{(n+1+|m|)!}{(n+1-|m|)!} }
P_{n+1}^m
- (2n+1)
\sqrt{ \frac{1}{2n+1} \frac{(n+|m|)!}{(n-|m|)!} }
\mu P_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7550}%
$\displaystyle \hspace*{2cm}
+(n+|m|)
\sqrt{ \frac{1}{2n-1} \frac{(n-1+|m|)!}{(n-1-|m|)!} }
P_{n-1}^m        = 0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7551}%
$\displaystyle �� P_{n+1}^m
= \sqrt{ \frac{(2n+1)(2n+3)}{(n-|m|+1)(n+|m|+1)} }
\mu P_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7552}%
$\displaystyle \hspace*{2.5cm}
- \sqrt{ \frac{(2n+1)(2n+3)}{(n-|m|+1)(n+|m|+1)} }
\sqrt{ \frac{(n-|m|)(n+|m|)}{(2n+1)(2n-1)} } P_{n-1}^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7557}%
$\displaystyle (1-\mu^2)  \DD{}{\mu} P_n^m
= (n+|m|)
\sqrt{ \frac{(n-|m|)(2n+1)}{(n+|m|)(2n-1)} } P_{n-1}^m
- n \mu P_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2487}%
$P_n^m(\mu) \ (n=|m|,|m+1, \cdots)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2497}%
$ \{ P_n^m | n=|m|,|m+1|,\cdots \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7569}%
$\displaystyle A(\mu) = \sum_{n=|m|}^{\infty}
\tilde{A}_n^m P_n^m(\mu),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7570}%
$\displaystyle \tilde{A}_n^m =  \frac{1}{2}
\int_{-1}^1  A(\mu) P_n^m(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2505}%
$\exp(i m \lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2509}%
$\int_0^{2 \pi} \exp(im\lambda) \exp(-im'\lambda) 
                      d \lambda = 2 \pi \delta_{mm'}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2511}%
$m,m'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2517}%
$Y^m_n(\lambda, \phi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7586}%
$\displaystyle \left[ \frac{1}{\cos \phi}
\DP{}{\phi}
\left( \cos \phi \DP{}{\phi} \right)
+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
+ n(n+1) \right] Y_n^m =0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7588}%
$\displaystyle \left[
\DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right)
+ \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
+ n(n+1)
\right] Y_n^m =0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7593}%
$\displaystyle \int_{-1}^1 Y_n^m(\lambda, \phi)
Y_{n'}^{m'*} (\lambda, \phi)
d (\sin \phi) d \lambda
=  4 \pi \delta_{mm'} \delta_{nn'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2525}%
$m,m',n,n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2529}%
$0 \le |m'| \le n'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2533}%
$ \{ Y_n^m | m=0,1,2,\cdots, \  n=|m|,|m+1|,\cdots \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7600}%
$\displaystyle A(\lambda, \phi)
= \sum_{m=0}^{\infty} \sum_{n=|m|}^{\infty}
\tilde{A}_n^m Y_n^m(\lambda,\phi)         ,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7601}%
$\displaystyle \tilde{A}_n^m =  \frac{1}{4 \pi}
\int_{-1}^1  d (\sin \phi)
\int_0^{2 \pi} d \lambda A(\lambda, \phi)
Y_n^{m*}(\lambda, \phi)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2535}%
$Y_n^m(\phi,\lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7606}%
$\displaystyle \frac{1}{r \cos \phi} \DP{Y_n^m}{\lambda}
=  \frac{1}{r \cos \phi}
\DP{}{\lambda}
\left( P_n^m (\sin \phi) \exp(im \lambda) \right)
=  \frac{im}{r \cos \phi}
P_n^m (\sin \phi) \exp(im \lambda)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7609}%
$\displaystyle \frac{1}{r} \DP{Y_n^m}{\phi}
=     \frac{1}{r}
\DP{}{\phi}
\left( P_n^m (\sin \phi) \exp(im \lambda) \right)
=  \frac{\sqrt{1-\mu^2} }{r}
\DD{}{\mu} P_n^{m} (\mu) \exp(im \lambda)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7612}%
$\displaystyle \nabla_H^2
Y_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7614}%
$\displaystyle \frac{1}{r^2}
\left[ \DP{}{\mu} \left( (1-\mu^2) \DP{}{\mu} \right)
+ \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
\right] Y_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7616}%
$\displaystyle \frac{1}{r^2}
\left[ \frac{1}{\cos \phi}
\DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right)
+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
\right] Y_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7618}%
$\displaystyle - \frac{n(n+1)}{r^2} Y_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2539}%
$n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2543}%
$(\lambda', \phi')$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2547}%
$\{ Y_n^m(\lambda', \phi') | m=-n,-n+1, \cdots, n \} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7627}%
$\displaystyle Y^m_n(\lambda, \phi)
= \sum_{m'=-n}^{n} A_n^{m'} Y^{m'*}_n(\lambda',\phi')$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2553}%
$\nabla_H^2 \equiv 
           \frac{1}{r^2}
           \left[ \DP{}{\phi} \left( \cos \phi \DP{}{\phi} \right)
                  + \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda} 
           \right]    
        $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2555}%
$- \frac{n(n+1)}{r^2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2557}%
$\nabla_H^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2559}%
$\nabla_H^2 Y^m_n(\lambda, \phi)
        =- \frac{n(n+1)}{r^2} Y^m_n(\lambda,\phi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2561}%
$Y_n^m \exp(im\lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2567}%
$\{Y_n^m |n=0,1,2,\cdots,\  m= -n, -n+1,\cdots,n \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2569}%
$\{Y_n^m  | m= -n, -n+1,\cdots,n \}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2571}%
$\nabla^2_H f = -\frac{n(n+1)}{r^2} f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2573}%
$Y_n^m(\lambda',\phi')$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2579}%
$\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_inline2587}%
$P_n,P_n^1,P_n^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap2749}%
\Depsf[10cm]{zahyou/lg1.ps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap2757}%
\Depsf[][7cm]{zahyou/lg2.ps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2591}%
$ \overline{P_n^1} = P_n^1/\sqrt{2},
              \overline{P_n^2} = P_n^2/\sqrt{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7657}%

% latex2html id marker 7657
\setcounter{footnote}{14}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2950}%
$D$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2952}%
$\zeta$%
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$\chi$%
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$\psi$%
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$f(\lambda,\phi)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$x$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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${\displaystyle \frac{1}{r \cos \phi} \DP{f}{\lambda} }$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2966}%
$f$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2968}%
$y$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2970}%
${\displaystyle \frac{1}{r} \DP{f}{\phi}
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\lthtmlinlinemathZ
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{\newpage\clearpage
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$\displaystyle \nabla^2_H f$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7677}%
$\displaystyle \frac{1}{r^2}
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+ \frac{1}{\cos^2 \phi} \DP[2]{}{\lambda}
\right] f$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7679}%
$\displaystyle \frac{1}{r^2}
\left[
\DP{}{\mu}
\left\{ (1-\mu^2) \DP{}{\mu} \right\}
+ \frac{1}{1-\mu^2} \DP[2]{}{\lambda}
\right] f$%
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\stepcounter{subsection}
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\lthtmlinlinemathA{tex2html_wrap_inline2974}%
$\Dvect{v}=(v_1,v_2)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7683}%
$\displaystyle \mbox{div}_H \Dvect{v}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7685}%
$\displaystyle \frac{1}{r \cos \phi} \DP{v_1}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7687}%
$\displaystyle \frac{1}{r \sqrt{1-\mu^2} } \DP{v_1}{\lambda}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2976}%
$\Dvect{v}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2978}%
$r$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7691}%
$\displaystyle (\mbox{rot} \Dvect{v})_r$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7693}%
$\displaystyle \frac{1}{r \cos \phi} \DP{v_2}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7695}%
$\displaystyle \frac{1}{r \sqrt{1-\mu^2} } \DP{v_2}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7698}%
$\displaystyle (u,v) \equiv
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2982}%
$U,V$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7702}%
$\displaystyle D$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7704}%
$\displaystyle \frac{1}{r \cos \phi} \DP{u}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7706}%
$\displaystyle \frac{1}{r \cos^2 \phi} \DP{U}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7708}%
$\displaystyle \frac{1}{r (1-\mu^2)} \DP{U}{\lambda}
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7711}%
$\displaystyle (u,v) = \left( \frac{U}{\cos \phi}, \frac{V}{\cos \phi} \right)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7715}%
$\displaystyle \zeta$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7717}%
$\displaystyle \frac{1}{r \cos \phi} \DP{v}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7719}%
$\displaystyle \frac{1}{r \cos^2 \phi} \DP{V}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7721}%
$\displaystyle \frac{1}{r (1-\mu^2)} \DP{V}{\lambda}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7729}%
$\displaystyle \nabla_H^2 \chi$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7732}%
$\displaystyle \nabla_H^2 \psi$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7736}%
$\displaystyle U$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7738}%
$\displaystyle - \frac{1-\mu^2}{r}   \DP{\psi}{\mu}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7739}%
$\displaystyle V$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7741}%
$\displaystyle \frac{1}{r}             \DP{\psi}{\lambda}
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7742}%

% latex2html id marker 7742
\setcounter{footnote}{15}\fnsymbol{footnote}%
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\lthtmlpictureA{tex2html_wrap7743}%

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\setcounter{footnote}{15}\fnsymbol{footnote}%
\lthtmlpictureZ
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\stepcounter{section}
{\newpage\clearpage
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$ n-1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3098}%
$-1 \leq \mu \leq  1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3100}%
$ P_n(\mu)$%
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\stepcounter{subsection}
{\newpage\clearpage
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$P_0 \sim P_{n-1}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3120}%
$f(\mu)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7762}%
$\displaystyle \int_{-1}^1 f(\mu) P_n(\mu) d \mu = 0$%
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3128}%
$f(x)=(x-1)^n (x+1)^n$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3130}%
$f=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3132}%
$x=-1,1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3134}%
$f'$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3136}%
$\alpha$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3138}%
$-1<\alpha<1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3140}%
$f'(\alpha)=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3142}%
$f'=2nx(x^2-1)^{n-1}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3144}%
$f'=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3146}%
$x=-1,\alpha,1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3148}%
$f''=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3150}%
$x=-1,\beta_1,\beta_2,1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3152}%
$-1 < \beta_1 < \beta_2 < 1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3154}%
$f^{(n)}=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3156}%
$-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3158}%
$1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3164}%
${\displaystyle 
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3172}%
${\displaystyle x_j=\cos \frac{j-1/2}{n}\pi }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7790}%

% latex2html id marker 7790
\setcounter{footnote}{15}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7791}%

% latex2html id marker 7791
\setcounter{footnote}{15}\fnsymbol{footnote}%
\lthtmlpictureZ
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\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3591}%
$M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3593}%
$g(\lambda)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3595}%
$0 \le \lambda < 2 \pi$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7798}%
$\displaystyle g(\lambda) = \sum_{m=-M}^{m=M} g_m \exp( i m \lambda )$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3597}%
$M < I$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7802}%
$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7803}%
$\displaystyle \lambda_n = \frac{2\pi (n-1)}{I} \ (n=1,2,\cdots,I)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7805}%
$\displaystyle \sum_{n=1}^{I} \exp(i m \lambda_n)
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I & \ \ (m=0)  \\
0 & \ \ (0 < |m| < I) \\
\end{array}
\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3601}%
$I > M$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3603}%
$|m|$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3605}%
$m \neq 0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3607}%
${\displaystyle \exp(i m \lambda_n) 
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3611}%
$m(n-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7822}%
$\displaystyle \sum^{I}_{n=1} \exp \left\{ im \frac{2 \pi (n-1)}{I} \right\}
= \frac{1 - \left( e^{\frac{im 2 \pi}{I}}\right)^I}
{1 - e^{ \frac{im 2 \pi}{I} } }
= \frac{1 -  e^{im 2 \pi} }
{1 - e^{ \frac{im 2 \pi}{I} } }
= 0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7824}%
$\displaystyle \frac{1}{2\pi} \int_0^{2\pi} g(\lambda) d \lambda$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7826}%
$\displaystyle \sum_{m=-M}^{M} \frac{1}{2\pi} g_m
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3625}%
$\int_0^{2\pi} \exp(i m \lambda) d \lambda $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3627}%
$m=0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7830}%
$\displaystyle \frac{1}{I} \sum_{n=1}^{I} g(\lambda_n)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7832}%
$\displaystyle \frac{1}{I}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7834}%
$\displaystyle g_0
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\left( \exp(\frac{2 \pi i m}{I}) \right)^{n-1}$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7836}%
$\displaystyle \sum_{n=1}^{I}
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3631}%
$2J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3635}%
${\displaystyle \int_{-1}^1 f d \mu }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3637}%
$P_J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3639}%
$\mu_j��j=1,2,\cdots,J��$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3643}%
$f(\mu_j)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7849}%
$\displaystyle \int^{1}_{-1} f(\mu) d \mu = 2 \sum_{j=1}^{J} f(\mu_j) w_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7850}%
$\displaystyle w_j = \frac{1}{2} \int_{-1}^{1}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3647}%
$\tilde{P_n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3655}%
$K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3657}%
$0 \le K \le 2J-1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7861}%
$\displaystyle \int_{-1}^1 \tilde{P}_n(\mu) \tilde{P}_{n'}(\mu) d \mu
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\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3661}%
$L(\mu)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7865}%
$\displaystyle L(\mu) \equiv
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3667}%
$ L(\mu_j) = f(\mu_j) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3669}%
$L$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3671}%
$0 \le K \le J-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3675}%
$J-1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3677}%
$L=f$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3679}%
$L-f$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3685}%
$\mu_j$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3687}%
$f(\mu)-L(\mu)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3691}%
$0$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3693}%
$J \le K \le 2J-1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3695}%
$\mu=\mu_j$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3703}%
$\tilde{P}_J(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3705}%
$f-L$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3709}%
$K-J$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3711}%
$S(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7892}%
$\displaystyle f(\mu) - L(\mu) = \tilde{P}_J(\mu) S(\mu)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3721}%
$\tilde{P}_J(\mu) S(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7899}%
$\displaystyle \int_{-1}^{1} f(\mu) d \mu$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7901}%
$\displaystyle \int_{-1}^1 L(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7903}%
$\displaystyle \sum_{j=1}^{J} f(\mu_j)
\int^1_{-1}
\frac{ {\displaystyle
\prod_{k=1}^{J}(\mu-\mu_k) }
}
{ (\mu-\mu_j)
{\displaystyle
\prod_{k=1,k \neq j}^{J}(\mu_j-\mu_k) }
}
d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7905}%
$\displaystyle \sum_{j=1}^{J} f(\mu_j)
\int_{-1}^1
\frac{\tilde{P}_J(\mu)}
{(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7907}%
$\displaystyle 2 \sum_{j=1}^{J} f(\mu_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3725}%
$\tilde{P}_J$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3733}%
${\displaystyle w_j
         = \frac{1}{2} \int_{-1}^1
             \frac{\tilde{P}_J(\mu)}
                  {(\mu-\mu_j)\tilde{P}^{'}_J(\mu_j)} d \mu
   }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7916}%
$\displaystyle (n+1) \tilde{P}_{n+1}(\mu)
= (2n+1) \mu \tilde{P}_n(\mu) - n \tilde{P}_{n-1}(\mu)
\ \ \ \ (n=0,1,2,\cdots)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7918}%
$\displaystyle (n+1) \left|
\begin{array}{ll}
\tilde{P}_{n+1}(x) & \tilde{P}_n(x) \\
\tilde{P}_{n+1}(y) & \tilde{P}_n(y)
\end{array}
\right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7920}%
$\displaystyle \left|
\begin{array}{ll}
(2n+1)x \tilde{P}_n(x)-n\tilde{P}_{n-1}(x)
&   \tilde{P}_n(x) \\
(2n+1)y \tilde{P}_n(y)-n\tilde{P}_{n-1}(y)
&   \tilde{P}_n(y)
\end{array}
\right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7922}%
$\displaystyle (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7923}%
$\displaystyle +n (- \tilde{P}_{n-1}(x) \tilde{P}_n(y)
+ \tilde{P}_{n-1}(y) \tilde{P}_n(x) )$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7925}%
$\displaystyle (2n+1)(x-y)\tilde{P}_n(x)\tilde{P}_n(y)
+
n \left|
\begin{array}{ll}
\tilde{P}_{n}(x) & \tilde{P}_{n-1}(x) \\
\tilde{P}_{n}(y) & \tilde{P}_{n-1}(y)
\end{array}
\right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3737}%
$n=0,1,\cdots,n-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7928}%
$\displaystyle n \left|
\begin{array}{ll}
\tilde{P}_n(x) & \tilde{P}_{n-1}(x) \\
\tilde{P}_n(y) & \tilde{P}_{n-1}(y)
\end{array}
\right|$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7930}%
$\displaystyle \sum_{k=0}^{n-1} (2k+1)(x-y)\tilde{P}_k(x)\tilde{P}_k(y)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3739}%
$n=J,x=\mu,y=\mu_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3741}%
$\tilde{P}_J(\mu_j)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7934}%
$\displaystyle J  \tilde{P}_J (\mu)  \tilde{P}_{J-1} (\mu_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7936}%
$\displaystyle \sum_{k=0}^{J-1} (2k+1) (\mu-\mu_j)
\tilde{P}_k(\mu) \tilde{P}_k(\mu_j).$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7938}%
$\displaystyle \frac{\tilde{P}_J (\mu) }{\mu-\mu_j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7940}%
$\displaystyle \frac{
{\displaystyle
\sum_{k=0}^{J-1} (2k+1)
\tilde{P}_k(\mu) \tilde{P}_k(\mu_j) }
}
{J \tilde{P}_{J-1}(\mu_j)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7942}%
$\displaystyle w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7944}%
$\displaystyle \frac{1}{2} \int_{-1}^1
\frac{\tilde{P}_J(\mu)}
{(\mu-\mu_j) \tilde{P}^{'}_J(\mu_j)}
d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7946}%
$\displaystyle \frac{1}{2 J \tilde{P}_{J-1}(\mu_j)
\tilde{P}^{'}_{J} (\mu_j)}
\sum_{k=0}^{J-1}
(2k+1) \tilde{P}_k(\mu_j)
\int_{-1}^1 \tilde{P}_k(\mu) d \mu$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7948}%
$\displaystyle \frac{1}{J \tilde{P}_{J-1}(\mu_j)
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3743}%
$\tilde{P}_0 = 1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7951}%
$\displaystyle (1-\mu^2) \DP{\tilde{P}_n}{\mu}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3745}%
$n=J, \mu=\mu_j$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7957}%
$\displaystyle \frac{1-\mu_j^2}{(J \tilde{P}_{J-1}(\mu_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7961}%
$\displaystyle \int_{-1}^1 P_n(\mu) P_n'(\mu) d \mu = 2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3753}%
${\displaystyle 
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7964}%
$\displaystyle w_j = \frac{1-\mu_j^2}
{(J  \sqrt{ \frac{1}{2J-1} } P_{J-1}(\mu_j))^2 }
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{(J  P_{J-1}(\mu_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7967}%
$\displaystyle w_j = \frac{(2J-1)(1-\mu_j^2)}{(J P_{J-1}(\mu_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7968}%

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\setcounter{footnote}{18}\fnsymbol{footnote}%
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{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7969}%

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{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7970}%

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\stepcounter{section}
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\lthtmlinlinemathA{tex2html_wrap_indisplay7973}%
$\displaystyle \sum_{j=1}^{J}
\sum_{i=1}^{I}
P_n^m (\mu_j) P_{n'}^{m'} (\mu_j)
\exp(i m \lambda_i) \exp(-i m' \lambda_i) w_j
= I \delta_{nn'} \delta_{mm'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3971}%
$ i,j,m,m',n,n',I,J,M,N(m)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3973}%
$1 \le i \le I , 1 \le j \le J, $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3975}%
$0 \le |m|,|m'| \le M, |m| \le n \le N, |m'| \le n' \le N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3977}%
${\displaystyle M \le \left[ \frac{I}{2} \right] , 
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3981}%
${\displaystyle \lambda_i=\frac{2\pi(i-1)}{I} }$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3987}%
$[ \ ]$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7985}%
$\displaystyle \sum_{j=1}^{J}
\sum_{i=1}^{I}
P_n^m (\mu_j) P_{n'}^{m'} (\mu_j)
\exp(i m \lambda_i) \exp(-i m' \lambda_{i})
w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7987}%
$\displaystyle I \sum_{j=1}^{J}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7991}%
$\displaystyle I \sum_{j=1}^{J}
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7993}%
$\displaystyle \frac{I}{2} \int_{-1}^1
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay7997}%
$\displaystyle I \delta_{nn'} \delta_{mm'}$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3989}%
$ \{ f_k(\mu) \}(k=0,1,2,\cdots) $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3991}%
$\  [ a ,b ] \ $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3993}%
$w(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3995}%
$\lambda_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3997}%
${\displaystyle  
          \left( \int_a^b f_k (\mu) f_{k'} (\mu) w(\mu) d \mu
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3999}%
$\mu_j, \mu_{j'}(1 \le j,j' \le J)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4001}%
$f_J(\mu)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4003}%
$w_j=w(\mu_j)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8008}%
$\displaystyle \sum_{j=0}^{J-1}
f_k (\mu_j) f_{k'} (\mu_{j}) w_j
= \lambda_k  \delta_{kk'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8010}%
$\displaystyle \sum_{k=0}^{J-1}
\frac{f_k (\mu_j) f_k (\mu_{j'}) }{\lambda_k}
= \frac{1}{w_j}  \delta_{jj'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4005}%
$\{ P_n \}(n=0,1,2,\cdots,J-1) $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8014}%
$\displaystyle \sum_{n=0}^{J-1} P_n (\mu_j) P_n (\mu_{j'})
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\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4009}%
$P_{J}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4011}%
$\{ P_n^m \} (n=|m|,|m|+1,|m|+2,\cdots,N)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4013}%
$f_k��k=0,1,2,\cdots,K-1��$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4015}%
$k$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4029}%
$n=0,1,2,\cdots,N < J-1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4033}%
$J=63, N=42$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4037}%
$J=63, N=21$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8033}%
$\displaystyle \sum_{i=0}^{I-1}
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= I \delta_{mm'}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8035}%
$\displaystyle \sum_{m=-\frac{I}{2}+1}^{\frac{I}{2}}
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= I \delta_{ii'}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4041}%
$I=2M$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4045}%
$I=2M+1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4049}%
${\displaystyle  -\frac{I-1}{2} \sim \frac{I-1}{2} }$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4055}%
${\displaystyle \frac{I}{3} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4057}%
$I=128$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4059}%
$M=42$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4061}%
$I=64$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4063}%
$M=21$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap8050}%

% latex2html id marker 8050
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\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4675}%
$A_{ij} \; (i=1,2,\cdots,I, \; j=1,2,\cdots,J)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4677}%
$\tilde{A}_n^m \;
(m=-M,-M+1, \cdots,M, \; n=|m|,|m|+1,\cdots,N(m))$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8061}%
$\displaystyle A_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8063}%
$\displaystyle \sum_{m=-M}^{M} \sum_{n=|m|}^{N}
\tilde{A}_n^m
Y_n^m (\lambda_i,\phi_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8064}%
$\displaystyle \tilde{A}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8066}%
$\displaystyle \frac{1}{I}
\sum_{i=1}^{I} \sum_{j=1}^{J}
A_{ij} Y_n^{m*} (\lambda_i,\phi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8069}%
$\displaystyle \frac{(2J-1)(1-\sin^2 \phi_j)}
{(J P_{J-1}(\sin \phi_j))^2 }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4679}%
${\displaystyle \sum_{m=-M}^{M} \sum_{n=|m|}^{N} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4681}%
${\displaystyle \sum_{m,n} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4683}%
${\displaystyle \sum_{i=1}^{I} \sum_{j=1}^{J} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4685}%
${\displaystyle \sum_{i,j} }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4160}%
\begin{eqnarray*}
    g&\equiv& \DP{f}{\lambda}
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8078}%
$\displaystyle g_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8080}%
$\displaystyle \left[
\DP{}{\lambda} \left(
\sum_{m,n} \tilde{f}_n^m Y_n^m (\lambda, \phi)
\right)
\right]_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8084}%
$\displaystyle \sum_{m,n} im \tilde{f}_n^m
Y_n^m (\lambda_i,\phi_j)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4691}%
$g$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4693}%
${\displaystyle (g_{ij}=) 
                   \sum_{m,n} im \tilde{f}_n^m Y_n^m
                   = \sum_{m,n} \tilde{g}_n^m Y_n^m }$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4695}%
${\displaystyle \sum_{i,j} Y_n^{m*}(\lambda_i,\phi_j)w_j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4697}%
$im' \tilde{f}_{n'}^{m'} 
                              = \tilde{g}_{n'}^{m'} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8092}%
$\displaystyle \tilde{g}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8094}%
$\displaystyle im \tilde{f}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8098}%
$\displaystyle \frac{1}{I}
\sum_{i,j} im f_{ij} Y_n^{m*} (\lambda_i, \phi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4210}%
\begin{eqnarray*}
 h  \equiv \frac{g}{r \cos^2 \phi}
    =      \frac{1}{r \cos^2 \phi} \DP{f}{\lambda}
           \ \     \left[ = 
                    \DP{}{x} \left( \frac{f}{\cos \phi} \right) 
                  \right]                                          
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4701}%
$h$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4223}%
\begin{eqnarray*}
      h_{ij} &=& \frac{1}{r \cos^2 \phi_i} g_{ij}  \\
  ��\ h_{ij} &=&
            \frac{1}{r \cos^2 \phi_j}
            \sum_{m,n} im \tilde{f}_n^m Y_n^m (\lambda_i, \phi_j)
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8104}%
$\displaystyle \tilde{h}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8106}%
$\displaystyle \widetilde{
\left[
\DP{}{\lambda} \left( \frac{f}{r \cos^2 \phi} \right)
\right]_n^m
}
= im \widetilde{ \left( \frac{f}{r \cos^2 \phi} \right)_n^m }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8108}%
$\displaystyle \frac{1}{I}
\sum_{i,j}
im
\left( \frac{f}{r \cos^2 \phi} \right)_{ij}
Y_n^{m*} (\lambda_i, \phi_j) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8110}%
$\displaystyle \frac{1}{I}
\sum_{i,j} im f_{ij}
Y_n^{m*} (\lambda_i \phi_j)
\frac{w_j}{r \cos^2 \phi_j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4260}%
\begin{eqnarray*}
       p \equiv \DP{f}{\mu}
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8115}%
$\displaystyle p_{ij} \equiv \left[ \DP{}{\mu}
\left( \sum_{m,n} \tilde{f}_n^m Y_n^m \right)
\right]_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8117}%
$\displaystyle p_{ij} = \sum_{m,n} \tilde{f}_n^m
\left. \DD{P_n^m}{\mu} \right|_j
\exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4282}%
\begin{eqnarray*}
  p_n^m &=& \frac{1}{I} \sum_{i,j} p_{ij} Y_n^{m*} w_j \\
        &=& \frac{1}{I} \sum_{i,j} 
               \left( 
                     \sum_{m',n'} \tilde{f}_{n'}^{m'} 
                     \left. \DD{P_{n'}^{m'}}{\mu} \right|_j
                     \exp(i m' \lambda_i)
               \right)
              P_n^m(\mu_j) \exp(-im \lambda_i) w_j       \\
        &=& - \frac{1}{I} \sum_{i,j} 
              \left( 
                     \sum_{m',n'} \tilde{f}_{n'}^{m'} 
                     P_{n'}^{m'}(\mu_j)
                     \exp(i m' \lambda_i)
              \right)
             \left. \DD{P_n^m}{\mu}\right|_j 
             \exp(-im \lambda_i) w_j   \\
        &=& - \frac{1}{I} \sum_{i,j} 
              f_{ij} 
             \left. \DD{P_n^m}{\mu}\right|_j 
             \exp(-im \lambda_i) w_j                    \\
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8120}%
$\displaystyle \sum_{i=1}^I \sum_{j=1}^J
f_{n'}^{m'}
P_{n}^{m}(\mu_j)  \exp(im \lambda_i)
\left. \DD{P_{n'}^{m'}}{\mu}\right|_j
\exp(-im' \lambda_i)
w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8122}%
$\displaystyle - \sum_{i=1}^I \sum_{j=1}^J
f_{n'}^{m'}
\left. \DD{P_{n}^{m}}{\mu}\right|_j
\exp(-im \lambda_i)
P_{n'}^{m'}(\mu_j)  \exp(im' \lambda_i)
w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4337}%
\begin{eqnarray*}
                & & \sum_{i} \sum_{j}
                    f_{n'}^{m'}
                    P_{n}^{m}(\mu_j)  \exp(im \lambda_i)
                    \left. \DD{P_{n'}^{m'}}{\mu}\right|_j 
                    \exp(-im' \lambda_i)
                    w_j  \\
               &=& I \sum_{j} 
                  f_{n'}^{m'} 
                  P_{n}^{m}(\mu_j) 
                  \left.\DD{P_{n'}^{m'}}{\mu}\right|_j
                  w_j \delta_{m m'} 
                = I \sum_{j} 
                  f_{n'}^{m} 
                  P_{n}^{m}(\mu_j) 
                  \left.\DD{P_{n'}^{m}}{\mu}\right|_j 
                  w_j \delta_{m m'} \\
               &=& \frac{I}{2} 
                   \int_{-1}^{1} 
                   f_{n'}^{m}
                   P_{n}^{m}(\mu) 
                   \DD{P_{n'}^m}{\mu}  d \mu \delta_{m m'}        
           \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4377}%
\begin{eqnarray*}
                & & \sum_{i} \sum_{j}
                    f_{n'}^{m'}
                    P_{n}^{m}(\mu_j)  \exp(im \lambda_i)
                    \left. \DD{P_{n'}^{m'}}{\mu}\right|_j 
                    \exp(-im' \lambda_i)
                    w_j  \\
                &= & - \frac{I}{2} 
                   \int_{-1}^{1} 
                   f_{n'}^{m}
                   P_{n'}^{m}(\mu) 
                   \DD{P_n^m}{\mu}  d \mu \delta_{m m'}
                     \\
               &=& - I \sum_{j} 
                  f_{n'}^{m} 
                  P_{n'}^{m}(\mu_j) 
                  \left.\DD{P_{n}^{m}}{\mu}\right|_j 
                  w_j  \delta_{mm'} \\
               &=& - \sum_{i} \sum_{j}
                    f_{n'}^{m'}
                    P_{n'}^{m'}(\mu_j)  \exp(im' \lambda_i)
                    \left. \DD{P_{n}^{m}}{\mu}\right|_j 
                    \exp(-im \lambda_i)
                    w_j  
           \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4418}%
\begin{eqnarray*}
       q \equiv (1-\mu^2) \DP{f}{\mu} = (1-\mu^2) p
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4424}%
\begin{eqnarray*}
  q_{ij} = (1-\mu_j^2) \sum_{m,n} \tilde{f}_n^m 
                         \left. \DD{P_n^m}{\mu} \right|_j 
                         \exp(im \lambda_i)
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4431}%
\begin{eqnarray*}
  \tilde{q}_n^m &=& \frac{1}{I} \sum_{i,j} q_{ij} Y_n^{m*} w_j \\
        &=& \frac{1}{I} \sum_{i,j} 
              \left( 
                     (1-\mu_j^2)
                      \sum_{m',n'} \tilde{f}_{n'}^{m'} 
                     \left. \DD{P_{n'}^{m'}}{\mu} \right|_j
                     \exp(i m' \lambda_i)
              \right)
              P_n^m(\mu_j) \exp(-im \lambda_i) w_j       \\
        &=& - \frac{1}{I} \sum_{i,j} 
              \left( 
                     \sum_{m',n'} \tilde{f}_{n'}^{m'} 
                     P_{n'}^{m'}(\mu_j)
                     \exp(i m' \lambda_i)
              \right)                             \\
        & &  \hspace{1.5cm} \times
             \left. \DD{}{\mu} 
                        \left( (1-\mu^2) P_n^m \right) \right|_j 
             \exp(-im \lambda_i) w_j   \\
        &=& - \frac{1}{I} \sum_{i,j} 
              f_{ij} 
             \left. \DD{}{\mu} \left( (1-\mu^2)P_n^m \right) \right|_j
             \exp(-im \lambda_i) w_j   
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4467}%
\begin{eqnarray*}
  & & \sum_{i=1}^I \sum_{j=1}^J
      f_{n'}^{m'} (1-\mu_j^2)
      P_{n}^{m}(\mu_j)  \exp(im \lambda_i)
      \left. \DD{P_{n'}^{m'}}{\mu}\right|_j 
      \exp(-im' \lambda_i)
      w_j  \\
 &=& - \sum_{i=1}^I \sum_{j=1}^J
      f_{n'}^{m'}
      \left. \DD{}{\mu} 
      \left((1-\mu^2)P_n^m \right) \right|_j 
      \exp(-im \lambda_i)
      P_{n'}^{m'}(\mu_j)  \exp(im' \lambda_i)
      w_j  
 \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4705}%
$(U_{ij},V_{ij})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4707}%
$\tilde{D}_n^m, \;  \tilde{\zeta}_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8139}%
$\displaystyle D  =  \frac{1}{r(1-\mu^2)} \DP{U}{\lambda}
+ \frac{1}{r}             \DP{V}{\mu}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8141}%
$\displaystyle \tilde{D}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8143}%
$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
\frac{im}{r \cos^2 \phi_j} U_{ij} P_n^m
- \frac{1}{r} V_{ij} \left. \DD{P_n^m}{\mu} \right|_j
\right)
\exp(- im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8145}%
$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
im U_{ij} P_n^m
-  V_{ij}
\cos^2 \phi_j  \left. \DD{P_n^m}{\mu} \right|_j
\right)
\exp(- im \lambda_i)
\frac{w_j}{r \cos^2 \phi_j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8147}%
$\displaystyle \zeta  =  \frac{1}{r(1-\mu^2)} \DP{V}{\lambda}
- \frac{1}{r}             \DP{U}{\mu}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8149}%
$\displaystyle \tilde{\zeta}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8151}%
$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
\frac{im}{\cos^2 \phi}  V_{ij} P_n^m
+ \frac{1}{r} U_{ij} \left. \DD{P_n^m}{\mu} \right|_j
\right)
\exp(- im \lambda_i) w_j$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8153}%
$\displaystyle \frac{1}{I}
\sum_{ij}
\left(
im  V_{ij} P_n^m
+ U_{ij}
\cos^2 \phi_j \left. \DD{P_n^m}{\mu} \right|_j
\right)
\exp(- im \lambda_i)
\frac{w_j}{r \cos^2 \phi_j}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4713}%
$\chi_n^m,\psi_n^m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4715}%
$U_{ij},V_{ij}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8159}%
$\displaystyle U  = - \frac{1-\mu^2}{r}   \DP{\psi}{\mu}
+ \frac{1}{r}             \DP{\chi}{\lambda}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8161}%
$\displaystyle U_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8163}%
$\displaystyle \sum_{m,n}
\left(
- \frac{\cos^2 \phi_j}{r} \tilde{\psi}_n^m
\left. \DD{P_n^m}{\mu} \right|_j
+ \frac{1}{r} im \tilde{\chi}_n^m P_n^m(\sin \phi_j)
\right)
\exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8165}%
$\displaystyle \sum_{m,n}
\left(
- \frac{1}{r} \tilde{\psi}_n^m
\cos^2 \phi_j \left. \DD{P_n^m}{\mu} \right|_j
+ \frac{1}{r} im \tilde{\chi}_n^m P_n^m(\sin \phi_j)
\right)
\exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8167}%
$\displaystyle V  =  \frac{1}{r}             \DP{\psi}{\lambda}
+ \frac{1-\mu^2}{r}   \DP{\chi}{\mu}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8169}%
$\displaystyle V_{ij}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8171}%
$\displaystyle \sum_{m,n}
\left(
\frac{1}{r}  im \tilde{\psi}_n^m P_n^m (\sin \phi_j)
+ \frac{\cos^2 \phi_j}{r} \tilde{\chi}_n^m
\left. \DD{P_n^m}{\mu} \right|_{j}
\right)
\exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8173}%
$\displaystyle \sum_{m,n}
\left(
\frac{1}{r}  im \tilde{\psi}_n^m P_n^m (\sin \phi_j)
+ \frac{1}{r} \tilde{\chi}_n^m
\cos^2 \phi_j \left. \DD{P_n^m}{\mu} \right|_{j}
\right)
\exp(im \lambda_i)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap8174}%

% latex2html id marker 8174
\setcounter{footnote}{27}\fnsymbol{footnote}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4856}%
$g = \DP{f}{\lambda}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8178}%
$\displaystyle \tilde{g}_n^m = im \tilde{f}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4858}%
$h = \nabla_H^2 f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay8181}%
$\displaystyle \tilde{h}_n^m = -\frac{n(n+1)}{r^2} \tilde{f}_n^m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4817}%
\begin{eqnarray*}
  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} \\
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

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\end{document}
