<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% è¡¨é¡Œ  å›žè»¢çƒé¢ä¸Šã§ã® 2 æ¬¡å…ƒé&nbsp;†åœ§æµä½“ã®å®šå¼åŒ–
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% å±¥æ­´  2007/11/25 ç«¹åºƒ çœŸä¸€  æ–°è¦ä½œæˆ
%       2008/01/12 ç«¹åºƒ çœŸä¸€, å°å¸ƒæ–½ç¥ˆç¹”  ãƒŸã‚¹ã‚¿ã‚¤ãƒ—ä¿®æ­£
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\Dtitle[å›žè»¢çƒé¢é&nbsp;†åœ§æµä½“]
       {å›žè»¢çƒé¢ä¸Šã§ã®\\ 2 æ¬¡å…ƒé&nbsp;†åœ§æµä½“ã®å®šå¼åŒ–}
\Dauthor[ç«¹åºƒ çœŸä¸€]{ç«¹åºƒ çœŸä¸€, SPMODEL é–‹ç™ºã‚°ãƒ«ãƒ¼ãƒ—}
\Ddate{å¹³æˆ 20 å¹´ 1 æœˆ 12 æ—¥} 
\Dparskip

\begin{document}

\maketitle

ã“ã®æ–‡æ›¸ã§ã¯å›žè»¢ 2 æ¬¡å…ƒçƒé¢ä¸Šã®éžåœ§ç¸®æµä½“ã®æ”¯é…æ–¹ç¨‹å¼ã¨
æ¸¦åº¦æ–¹ç¨‹å¼ã‚’å°Žå‡ºã™ã‚‹ã“ã¨ã‚’è¡Œã†. 

å°Žå‡ºã®è¨ˆç®—ã®è¦‹é€šã—ã¯ 3 æ¬¡å…ƒç³»ã«ãŠã„ã¦æ¸¦åº¦æ–¹ç¨‹å¼ã‚’ãƒ™ã‚¯ãƒˆãƒ«å½¢å¼ã§
è¡¨ç¾ã—, ãã®å¾Œã«çƒé¢ã¸ã®æ‹˜æŸæ¡ä»¶ã‚’é©ç”¨ã—å‹•å¾„æˆåˆ†ã‚’æ›¸ãä¸‹ã™ã‚„ã‚Šæ–¹ãŒ
ãŠè–¦ã‚ã§ã‚ã‚‹(ç†è«–ãƒžãƒ‹ãƒ¥ã‚¢ãƒ«ç·šå½¢æ³¢å‹•ã€Œãƒ­ã‚¹ãƒ“ãƒ¼æ³¢(2æ¬¡å…ƒéžç™ºæ•£çƒé¢)ã€)ãŒ, 
ä»¥ä¸‹ã§ã¯ç·¯åº¦çµŒåº¦å‹•å¾„åº§æ¨™ã®å„æˆåˆ†ã®å¼ã‚’ã¾ãšæ›¸ãä¸‹ã—, 
å¼å¤‰å½¢ã—ã¦ã„ãã¨ã„ã†æ‰‹é–“ã®ã‹ã‹ã‚‹è¨ˆç®—ã‚’æ•¢ãˆã¦è¡Œã†. 

\section{å›žè»¢ 2 æ¬¡å…ƒçƒé¢ä¸Šã®éžåœ§ç¸®æµä½“ã®æ”¯é…æ–¹ç¨‹å¼ã®å°Žå‡º}

  ã¾ãš 3 æ¬¡å…ƒå›žè»¢ç³»ã®éžåœ§ç¸®æµä½“ã®æ”¯é…æ–¹ç¨‹å¼ã‹ã‚‰
  å›žè»¢ 2 æ¬¡å…ƒçƒé¢ä¸Šã®éžåœ§ç¸®æµä½“ã®æ”¯é…æ–¹ç¨‹å¼ã‚’å°Žå‡ºã™ã‚‹. 
  3 æ¬¡å…ƒå›žè»¢ç³»ã§ã®éžåœ§ç¸®æµä½“ã®æ”¯é…æ–¹ç¨‹å¼ã‚’ç·¯åº¦çµŒåº¦å‹•å¾„åº§æ¨™
  ($\lambda,\varphi, r)$ ã§æ›¸ãè¡¨ã™ã¨, 
  %
  \begin{eqnarray}
    \Deqlab{3 æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼çµŒåº¦}
    &amp; &amp; \DP{u}{t}
          +\frac{u}{r\cos\varphi}\DP{u}{\lambda}
          +\frac{v}{r}\DP{u}{\varphi}
          + w\DP{u}{r}
          + \frac{uw - uv\tan\varphi}{r} \nonumber\\
    &amp; &amp; \qquad
         - 2\Omega\sin\varphi v
          + 2\Omega\cos\varphi w 
          = -\frac{1}{\rho r\cos\varphi}\DP{p}{\lambda} \nonumber\\
    &amp; &amp;  \qquad 
          + \nu\left[
                    \frac{1}{r^2\cos^2\varphi}\DP[2]{u}{\lambda}
                  + \frac{1}{r^2}\DP[2]{u}{\varphi}
                  - \frac{\tan\varphi}{r^2}\DP{u}{\varphi}
                  + \frac{1}{r}\DP[2]{}{r}(ru) \right.\nonumber\\
    &amp; &amp;  \qquad\qquad  \left. 
                  + \frac{2}{r^2\cos\varphi}\DP{w}{\lambda}
                  -\frac{2\sin\varphi}{r^2\cos^2\varphi}\DP{v}{\lambda}
                  - \frac{u}{r^2\cos^2\varphi}, \right],\\
    \Deqlab{3 æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼ç·¯åº¦}
    &amp; &amp; \DP{v}{t}
          +\frac{u}{r\cos\varphi}\DP{v}{\lambda}
          +\frac{v}{r}\DP{v}{\varphi}
          + w\DP{v}{r}
          + \frac{vw + u^2\tan\varphi}{r}
          + 2\Omega\sin\varphi u
          = -\frac{1}{\rho r}\DP{p}{\varphi}\nonumber\\
    &amp; &amp;  \qquad 
          + \nu\left[
                    \frac{1}{r^2\cos^2\varphi}\DP[2]{v}{\lambda}
                  + \frac{1}{r^2}\DP[2]{v}{\varphi}
                  - \frac{\tan\varphi}{r^2}\DP{v}{\varphi}
                  + \frac{1}{r}\DP[2]{}{r}(rv) \right.\nonumber\\
    &amp; &amp;  \qquad\qquad  \left. 
                  + \frac{2}{r^2}\DP{w}{\varphi}
                  +\frac{2\sin\varphi}{r^2\cos^2\varphi}\DP{u}{\lambda}
                  - \frac{v}{r^2\cos^2\varphi}, \right],\\
    \Deqlab{3 æ¬¡å…ƒé‹å‹•æ–¹ç¨‹å¼å‹•å¾„}
    &amp; &amp; \DP{w}{t}
          +\frac{u}{r\cos\varphi}\DP{w}{\lambda}
          +\frac{v}{r}\DP{w}{\varphi}
          + w\DP{w}{r}
          - \frac{u^2+v^2}{r}
          - 2\Omega\cos\varphi u
          = -\frac{1}{\rho}\DP{p}{r} \nonumber\\
    &amp; &amp;  \qquad 
          + \nu\left[
                    \frac{1}{r^2\cos^2\varphi}\DP[2]{w}{\lambda}
                  + \frac{1}{r^2}\DP[2]{w}{\varphi}
                  - \frac{\tan\varphi}{r^2}\DP{w}{\varphi}
                  + \frac{1}{r}\DP[2]{}{r}(rw) \right.\nonumber\\
    &amp; &amp;  \qquad\qquad  \left. 
                  - \frac{2}{r^2}\DP{v}{\varphi}
                  + \frac{2v\tan\varphi }{r^2}
                  - \frac{2}{r^2\cos\varphi}\DP{u}{\lambda}
                  - \frac{2w}{r^2}, \right],\\
    \Deqlab{3 æ¬¡å…ƒé€£ç¶šã®å¼}
    &amp; &amp; \frac{1}{r\cos\varphi}\DP{u}{\lambda} 
       +\frac{1}{r\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
       + \frac{1}{r^2}\DP{}{r}(r^2w)=0.
  \end{eqnarray}
  %
  ã“ã“ã§ $u,v,w$ ã¯ãã‚Œãžã‚Œé€Ÿåº¦ã®ç·¯åº¦, çµŒåº¦, å‹•å¾„æˆåˆ†ã§ã‚ã‚‹. 
  $\Omega$ ã¯ç³»ã®å›žè»¢è§’é€Ÿåº¦ã§ã‚ã‚Šãã®å›žè»¢è»¸ã¯åŒ—æ¥µå‘ãã§ã‚ã‚‹.
  çƒé¢ã¸ã®æ‹˜æŸæ¡ä»¶ã¨ã—ã¦, $u,v \propto r, \ w=0$ ã‚’é©ç”¨ã—, 
  $ r \rightarrow a $ ã®æ¥µé™ã‚’ã¨ã‚‹ã¨, 
  $w$ ã®ã¤ã„ãŸé&nbsp;…ã¯æ¶ˆãˆ, ç²˜æ€§é&nbsp;…ã®
  $\Ddsty \frac{1}{r}\DP[2]{}{r}(ru) = \frac{2u}{a^2}$,
  $\Ddsty \frac{1}{r}\DP[2]{}{r}(rv) = \frac{2v}{a^2}$
  ã¨ãªã‚‹ã®ã§, 
  %
  \begin{eqnarray}
    \Deqlab{é‹å‹•æ–¹ç¨‹å¼çµŒåº¦}
    &amp; &amp; \DP{u}{t}
          +\frac{u}{a\cos\varphi}\DP{u}{\lambda}
          +\frac{v}{a}\DP{u}{\varphi}
          - \frac{uv\tan\varphi}{a}
          - 2\Omega\sin\varphi v
          = -\frac{1}{\rho a\cos\varphi}\DP{p}{\lambda}
           + f_\lambda, \\
    \Deqlab{é‹å‹•æ–¹ç¨‹å¼ç·¯åº¦}
    &amp; &amp; \DP{v}{t}
          +\frac{u}{a\cos\varphi}\DP{v}{\lambda}
          +\frac{v}{a}\DP{v}{\varphi}
          + \frac{u^2\tan\varphi}{a}
          + 2\Omega\sin\varphi u
          = -\frac{1}{\rho a}\DP{p}{\varphi}
           + f_\varphi,\\
    \Deqlab{é€£ç¶šã®å¼}
    &amp; &amp; \frac{1}{a\cos\varphi}\DP{u}{\lambda} 
       +\frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi) =0.
  \end{eqnarray}
  ãŸã&nbsp;ã—
  $f_\lambda,f_\varphi$ ã¯ç²˜æ€§é&nbsp;…ã§ã‚ã‚Š, 
  %
  \begin{eqnarray}
   &amp; &amp; f_\lambda =\nu\left[
                    \frac{1}{a^2\cos^2\varphi}\DP[2]{u}{\lambda}
                  + \frac{1}{a^2}\DP[2]{u}{\varphi}
                  - \frac{\tan\varphi}{a^2}\DP{u}{\varphi}
                  + \frac{2u}{a^2}
                  -\frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{v}{\lambda}
                  - \frac{u}{a^2\cos^2\varphi}, \right]\\
   &amp; &amp; f_\varphi = \nu\left[
                    \frac{1}{a^2\cos^2\varphi}\DP[2]{v}{\lambda}
                  + \frac{1}{a^2}\DP[2]{v}{\varphi}
                  - \frac{\tan\varphi}{a^2}\DP{v}{\varphi}
                  + \frac{2v}{a^2}
                  +\frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{u}{\lambda}
                  - \frac{v}{a^2\cos^2\varphi}\right].
  \end{eqnarray}
  %
  çƒé¢ã¸ã®æ‹˜æŸæ¡ä»¶ã‚’é©ç”¨ã—ãŸ  $u,v \propto r, \ w=0$ ã¯
  $r$ ãŒæœ‰é™ã®å…ƒã§ã¯æ­£ã—ã„è§£ã«ã¯ãªã‚‰ãªã„ã“ã¨ã«æ³¨æ„ã•ã‚ŒãŸã„.
  é‹å‹•æ–¹ç¨‹å¼ã®å‹•å¾„æˆåˆ†ã¨ã¤ã˜ã¤ã¾ã‚’ã‚ã‚ã›ã‚‹ã“ã¨ãŒã§ããªããªã‚‹. 
  $r\rightarrow a$ ã®æ¥µé™ã§ã¯å‹•å¾„æ–¹å‘ã«ã¯å£ã‹ã‚‰ã®å¿œåŠ›ã«ã‚ˆã‚Š
  çƒé¢ã¸é‹å‹•ãŒæ‹˜æŸã•ã‚Œã¦ã„ã‚‹ã¨è€ƒãˆã¦é‹å‹•æ–¹ç¨‹å¼ã®å‹•å¾„æˆåˆ†ã¯æ‰±ã‚ãªã„. 


\section{å›žè»¢ 2 æ¬¡å…ƒçƒé¢ä¸Šã®éžåœ§ç¸®æµä½“ã®æ¸¦åº¦æ–¹ç¨‹å¼ã®å°Žå‡º}

  ã“ã“ã§ã¯å›žè»¢ã™ã‚‹ 2 æ¬¡å…ƒçƒé¢ä¸Šã®éžåœ§ç¸®æµä½“ã®æ”¯é…æ–¹ç¨‹å¼ã‹ã‚‰
  æ¸¦åº¦æ–¹ç¨‹å¼ã‚’å°Žå‡ºã™ã‚‹. 
  æ”¯é…æ–¹ç¨‹å¼ã¯å›žè»¢ç³»ã§ã® 2 æ¬¡å…ƒçƒé¢ä¸Šã®é‹å‹•æ–¹ç¨‹å¼ãŠã‚ˆã³é€£ç¶šã®å¼
  \Deqref{é‹å‹•æ–¹ç¨‹å¼çµŒåº¦},\Deqref{é‹å‹•æ–¹ç¨‹å¼ç·¯åº¦},\Deqref{é€£ç¶šã®å¼}ã§ã‚ã‚‹. 
  %
  æ¸¦åº¦æ–¹ç¨‹å¼ã¯\Deqref{é‹å‹•æ–¹ç¨‹å¼çµŒåº¦}ã« 
  $\Ddsty -\frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi$,
  \Deqref{é‹å‹•æ–¹ç¨‹å¼ç·¯åº¦}ã« 
  $\Ddsty \frac{1}{a\cos\varphi}\DP{}{\lambda}$ ã‚’ä½œç”¨ã•ã›ã‚‹ã“ã¨ã«ã‚ˆã‚Š
  æ±‚ã¾ã‚‹. ä¸¡å¼ã®æ™‚é–“å¤‰åŒ–é&nbsp;…ã‹ã‚‰ã¯
  %
  \begin{eqnarray*}
    &amp; &amp; -\frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi \DP{u}{t} 
        +\frac{1}{a\cos\varphi}\DP{}{\lambda}\DP{v}{t} 
    = \DP{}{t}\left[-\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
                    +\frac{1}{a\cos\varphi}\DP{v}{\lambda} \right]\\
    &amp; &amp; = \DP{\zeta}{t}
  \end{eqnarray*}
  %
  ã“ã“ã§ $\Ddsty \zeta = -\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
                        +\frac{1}{a\cos\varphi}\DP{v}{\lambda}$
  ã¯æ¸¦åº¦(ã®å‹•å¾„æˆåˆ†)ã§ã‚ã‚‹. 

  éžç·šå½¢é&nbsp;…ã‹ã‚‰ã¯
  %
  \begin{eqnarray*}
    &amp; &amp; -\frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi
        \left[  \frac{u}{a\cos\varphi}\DP{u}{\lambda}
               + \frac{v}{a}\DP{u}{\varphi}
               - \frac{uv\tan\varphi}{a} \right] \\
    &amp;&amp;  +\frac{1}{a\cos\varphi}\DP{}{\lambda}
        \left[  \frac{u}{a\cos\varphi}\DP{v}{\lambda}
              + \frac{v}{a}\DP{v}{\varphi}
              + \frac{u^2\tan\varphi}{a} \right]
  \end{eqnarray*}
  %
  æ¸¦åº¦ã®ç§»æµã®å½¢ã¨é€£ç¶šã®å¼ã®å½¢ã‚’æ„è­˜ã—ã¦å¤‰å½¢ã—ã¦ã„ã. 
  ç¬¬ 1 é&nbsp;…ç›®ã¯
  \begin{eqnarray*}
    &amp; &amp; -\frac{1}{a\cos\varphi}\DP{}{\varphi}\left(\cos\varphi
           \frac{u}{a\cos\varphi}\DP{u}{\lambda}\right)
       = -\frac{1}{a\cos\varphi}\DP{}{\varphi}\left(
         \frac{u}{a}\DP{u}{\lambda}\right) \\
    &amp; &amp; = -\frac{1}{a\cos\varphi}
              \frac{u}{a}\DP{}{\lambda}\left(\DP{u}{\varphi}\right)
          -\frac{1}{a^2\cos\varphi}
               \left(\DP{u}{\lambda}\DP{u}{\varphi}\right) \\
    &amp; &amp;  = -\frac{u}{a\cos\varphi}
              \frac{1}{a}\DP{}{\lambda}
                 \left[\DP{}{\varphi}\frac{u\cos\varphi}{\cos\varphi}\right]
          -\frac{1}{a^2\cos\varphi}\DP{u}{\lambda}\DP{u}{\varphi}\\
    &amp; &amp;  = -\frac{u}{a\cos\varphi}
              \frac{1}{a}\DP{}{\lambda}
                 \left[\frac{1}{\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
                       +\frac{u\cos\varphi\sin\varphi}{\cos^2\varphi}\right]
          -\frac{1}{a^2\cos\varphi}\DP{u}{\lambda}\DP{u}{\varphi}\\
    &amp; &amp;  = -\frac{u}{a\cos\varphi}
              \DP{}{\lambda}\left[\frac{1}{a\cos\varphi}
                                   \DP{}{\varphi}(u\cos\varphi)\right]
           -\frac{u}{a\cos\varphi}
                       \frac{\tan\varphi}{a}\DP{u}{\lambda}
          -\frac{1}{a^2\cos\varphi}\DP{u}{\lambda}\DP{u}{\varphi}
  \end{eqnarray*}
  %
  ç¬¬ 2 é&nbsp;…ç›®ã¯
  \begin{eqnarray*}
    &amp; &amp; - \frac{1}{a\cos\varphi}\DP{}{\varphi}
            \left(\cos\varphi\frac{v}{a}\DP{u}{\varphi}\right)
     =  - \frac{v}{a}\DP{}{\varphi}\left(\frac{1}{a}\DP{u}{\varphi}\right)
        - \frac{1}{a}\DP{u}{\varphi}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp; 
     =  - \frac{v}{a}\DP{}{\varphi}
           \left[\frac{1}{a}\DP{}{\varphi}
                  \left(\frac{u\cos\varphi}{\cos\varphi}\right)\right]
        - \frac{1}{a}\DP{u}{\varphi}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp;
     =  - \frac{v}{a}\DP{}{\varphi}
           \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
                 +\frac{u\cos\varphi}{a}\frac{-\sin\varphi}{-\cos\varphi^2}
           \right]
        - \frac{1}{a}\DP{u}{\varphi}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp;
     =  - \frac{v}{a}\DP{}{\varphi}
           \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)\right]
        - \frac{v}{a}\DP{}{\varphi} \left[\frac{u\tan\varphi}{a}\right]
        - \frac{1}{a}\DP{u}{\varphi}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)
  \end{eqnarray*}
  %
  ç¬¬ 3 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;   -\frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi 
             \left(- \frac{uv\tan\varphi}{a}\right)
       =   \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}
              \DP{}{\varphi}(v\cos\varphi) 
          +\frac{v\cos\varphi}{a\cos\varphi}
             \DP{}{\varphi}\left(\frac{u\tan\varphi}{a}\right)\\
    &amp; &amp; =  \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi) 
          +\frac{v}{a}\DP{}{\varphi}\left(\frac{u\tan\varphi}{a}\right)
  \end{eqnarray*}
  %
  ç¬¬ 4 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;     \frac{1}{a\cos\varphi}\DP{}{\lambda}
               \left(\frac{u}{a\cos\varphi}\DP{v}{\lambda}\right)
         = \frac{u}{a\cos\varphi}\DP{}{\lambda}
                  \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
          + \frac{1}{a^2\cos^2\varphi}\DP{u}{\lambda}\DP{v}{\lambda}
  \end{eqnarray*}
  %
  ç¬¬ 5 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;    \frac{1}{a\cos\varphi}\DP{}{\lambda}
                  \left(\frac{v}{a}\DP{v}{\varphi}\right)
        =   \frac{v}{a}\frac{1}{a\cos\varphi}\DP{}{\varphi}
               \left(\DP{v}{\lambda}\right)
          + \frac{1}{a^2\cos\varphi}\DP{v}{\lambda}\DP{v}{\varphi} \\
    &amp; &amp;
        =   \frac{v}{a}\frac{1}{a\cos\varphi}\DP{}{\varphi}
                \left(\frac{\cos\varphi}{\cos\varphi}\DP{v}{\lambda}\right)
           + \frac{1}{a^2\cos\varphi}\DP{v}{\lambda}\DP{v}{\varphi} \\
    &amp; &amp; =   \frac{v}{a}\frac{1}{a}\DP{}{\varphi}
                \left(\frac{1}{\cos\varphi}\DP{v}{\lambda}\right)
           -\frac{v}{a}\frac{1}{a\cos\varphi}
                \left(\frac{\sin\varphi}{\cos\varphi}\DP{v}{\lambda}\right)
           + \frac{1}{a^2\cos\varphi}\DP{v}{\lambda}\DP{v}{\varphi} \\
    &amp; &amp; =   \frac{v}{a}\DP{}{\varphi}
                \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
           -\frac{v\tan\varphi}{a^2\cos\varphi}\DP{v}{\lambda}
           + \frac{1}{a^2\cos\varphi}\DP{v}{\lambda}\DP{v}{\varphi}\\
    &amp; &amp; =   \frac{v}{a}\DP{}{\varphi}
                \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
           + \frac{1}{a^2\cos\varphi}\DP{v}{\lambda}
               \left(\DP{v}{\varphi} - v\tan\varphi\right)\\
    &amp; &amp; =   \frac{v}{a}\DP{}{\varphi}
                \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
           + \frac{1}{a\cos\varphi}\DP{v}{\lambda}
             \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
  \end{eqnarray*}
  %
  ç¬¬ 6 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;   \frac{1}{a\cos\varphi}\DP{}{\lambda}
             \left(\frac{u^2\tan\varphi}{a}\right)
        =   \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}\DP{u}{\lambda}
         +  \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}\DP{u}{\lambda}
  \end{eqnarray*}
  %
  å…¨ã¦ã®é&nbsp;…ã‚’è¶³ã—ã‚ã‚ã›ã‚‹ã¨, 
  \begin{eqnarray*}
    &amp;= &amp; -\frac{u}{a\cos\varphi}
          \DP{}{\lambda}\left[\frac{1}{a\cos\varphi}
                               \DP{}{\varphi}(u\cos\varphi)\right]
          -\frac{u}{a\cos\varphi}
                   \frac{\tan\varphi}{a}\DP{u}{\lambda}
          -\frac{1}{a^2\cos\varphi}\DP{u}{\lambda}\DP{u}{\varphi}\\
    &amp; &amp;  - \frac{v}{a}\DP{}{\varphi}
           \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)\right]
        - \frac{v}{a}\DP{}{\varphi} \left[\frac{u\tan\varphi}{a}\right]
        - \frac{1}{a}\DP{u}{\varphi}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp; +  \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi) 
          +\frac{v}{a}\DP{}{\varphi}\left(\frac{u\tan\varphi}{a}\right)\\
    &amp; &amp; + \frac{u}{a\cos\varphi}\DP{}{\lambda}
                  \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
          + \frac{1}{a^2\cos^2\varphi}\DP{u}{\lambda}\DP{v}{\lambda}\\
    &amp; &amp;   + \frac{v}{a}\DP{}{\varphi}
                \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
           + \frac{1}{a\cos\varphi}\DP{v}{\lambda}
             \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp;  +   \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}\DP{u}{\lambda}
         +  \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}\DP{u}{\lambda}\\
    &amp;=&amp; 
         -\frac{u}{a\cos\varphi}
          \DP{}{\lambda}\left[\frac{1}{a\cos\varphi}
                               \DP{}{\varphi}(u\cos\varphi)\right]
          -\frac{1}{a^2\cos\varphi}\DP{u}{\lambda}\DP{u}{\varphi}\\
    &amp; &amp;  - \frac{v}{a}\DP{}{\varphi}
           \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)\right]
        - \frac{1}{a}\DP{u}{\varphi}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp; +  \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}
            \DP{}{\varphi}(v\cos\varphi)  \\
    &amp; &amp; + \frac{u}{a\cos\varphi}\DP{}{\lambda}
                  \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
          + \frac{1}{a^2\cos^2\varphi}\DP{u}{\lambda}\DP{v}{\lambda}\\
    &amp; &amp;   + \frac{v}{a}\DP{}{\varphi}
                \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
           + \frac{1}{a\cos\varphi}\DP{v}{\lambda}
             \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)\\
    &amp; &amp;  +   \frac{u\tan\varphi}{a}\frac{1}{a\cos\varphi}\DP{u}{\lambda}\\
    &amp;=&amp; 
         \frac{u}{a\cos\varphi}\DP{}{\lambda}
              \left[   \frac{1}{a\cos\varphi}\DP{v}{\lambda}
                     - \frac{1}{a\cos\varphi}
                          \DP{}{\varphi}(u\cos\varphi)\right]
        + \frac{v}{a}\DP{}{\varphi}
           \left[\frac{1}{a\cos\varphi}\DP{v}{\lambda}
                 - \frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)\right]\\
   &amp; &amp;  +  \frac{u\tan\varphi}{a}
              \left[ \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                    + \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)\right]
          - \frac{1}{a}\DP{u}{\varphi}
            \left[    \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                    + \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
            \right]\\
   &amp; &amp;    + \frac{1}{a\cos\varphi}\DP{v}{\lambda}
               \left[   \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                      + \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
               \right]\\
    &amp;=&amp;     \left[
               \frac{u}{a\cos\varphi}\DP{}{\lambda} + \frac{v}{a}\DP{}{\varphi}
            \right]
            \left[   \frac{1}{a\cos\varphi}\DP{v}{\lambda}
                     - \frac{1}{a\cos\varphi}
                          \DP{}{\varphi}(u\cos\varphi)
            \right]\\
   &amp; &amp;  +  \left[\frac{1}{a\cos\varphi}\DP{v}{\lambda}
              - \frac{1}{a}\DP{u}{\varphi} + \frac{u\tan\varphi}{a} \right]
              \left[ \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                    + \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
              \right]\\
    &amp;=&amp;     \left[
               \frac{u}{a\cos\varphi}\DP{}{\lambda} + \frac{v}{a}\DP{}{\varphi}
            \right]
            \left[   \frac{1}{a\cos\varphi}\DP{v}{\lambda}
                     - \frac{1}{a\cos\varphi}
                          \DP{}{\varphi}(u\cos\varphi)
            \right]\\
   &amp; &amp;  +  \left[\frac{1}{a\cos\varphi}\DP{v}{\lambda}
                     - \frac{1}{a\cos\varphi}
                          \DP{}{\varphi}(u\cos\varphi)
           \right]
              \left[ \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                    + \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
           \right]\\
    &amp;=&amp;     \left[
               \frac{u}{a\cos\varphi}\DP{}{\lambda} + \frac{v}{a}\DP{}{\varphi}
            \right] \zeta
        +  \zeta
              \left[ \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                    + \frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
              \right]\\
    &amp;=&amp;     \left[
               \frac{u}{a\cos\varphi}\DP{}{\lambda} + \frac{v}{a}\DP{}{\varphi}
            \right] \zeta
  \end{eqnarray*}
  %
  æœ€å¾Œã«é€£ç¶šã®å¼\Deqref{é€£ç¶šã®å¼}ã‚’ç”¨ã„ãŸ. 

  ã‚³ãƒªã‚ªãƒªåŠ›ã®é&nbsp;…ã‹ã‚‰ã¯
  \begin{eqnarray*}
    &amp; &amp; 
       -\frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi(- 2\Omega\sin\varphi v)
       + \frac{1}{a\cos\varphi}\DP{}{\lambda}(2\Omega\sin\varphi u) \\
    &amp; &amp;
     =   2\Omega\frac{\sin\varphi}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)
       + 2\Omega\frac{v\cos\varphi}{a\cos\varphi}\DP{}{\varphi}(\sin\varphi)
       + 2\Omega\frac{\sin\varphi}{a\cos\varphi}\DP{u}{\lambda}\\
    &amp; &amp;
     =   2\Omega\sin\varphi
           \left[ \frac{1}{a\cos\varphi}\DP{u}{\lambda}
                 +\frac{1}{a\cos\varphi}\DP{}{\varphi}(v\cos\varphi)\right]
       + 2\Omega\frac{v}{a}\cos\varphi
     = 2\Omega\frac{v}{a}\cos\varphi.
  \end{eqnarray*}
  %
  åœ§åŠ›é&nbsp;…ã¯
  %
  \begin{eqnarray*}
   &amp; &amp;
       - \frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi
           \left(\frac{1}{\rho a\cos\varphi}\DP{p}{\lambda}\right)
       + \frac{1}{a\cos\varphi}\DP{}{\lambda}    
           \left(\frac{1}{\rho a}\DP{p}{\varphi}\right)\\
   &amp; &amp;
    =   - \frac{1}{a\cos\varphi}\DP{}{\varphi}
           \left(\frac{1}{\rho a}\DP{p}{\lambda}\right)
       + \frac{1}{\rho a^2\cos\varphi}
           \frac{\partial^2 p}{\partial\lambda\partial\varphi}\\
   &amp; &amp;
    =   - \frac{1}{\rho a^2\cos\varphi}
           \frac{\partial^2 p}{\partial\lambda\partial\varphi}
       + \frac{1}{\rho a^2\cos\varphi}
           \frac{\partial^2 p}{\partial\lambda\partial\varphi}
    = 0.
  \end{eqnarray*}

  ç²˜æ€§é&nbsp;…ã¯, ã¾ãš $f_\lambda$ ã® 2,3,6 é&nbsp;…ç›®ã‚’ã¾ã¨ã‚ã¦ 
  %
  \begin{eqnarray*}
   &amp; &amp;      \frac{1}{a^2}\DP[2]{u}{\varphi}
          - \frac{\tan\varphi}{a^2}\DP{u}{\varphi}
          - \frac{u}{a^2\cos^2\varphi}
       =    \frac{1}{a^2}\DP[2]{u}{\varphi}
          - \frac{1}{a^2}\DP{}{\varphi}(u\tan\varphi) \\
   &amp; &amp; =    \frac{1}{a^2}\DP[]{}{\varphi} 
              \left(\DP[]{u}{\varphi} - u\tan\varphi\right)
       =    \frac{1}{a^2}\DP[]{}{\varphi} 
              \left[\frac{1}{\cos\varphi}\DP[]{}{\varphi}(u\cos\varphi)\right]
  \end{eqnarray*}
  %
  $f_\varphi$ ã‚‚åŒæ§˜ã«, 
  %
  \begin{eqnarray*}
    &amp; &amp;      \frac{1}{a^2}\DP[2]{v}{\varphi}
           - \frac{\tan\varphi}{a^2}\DP{v}{\varphi}
           - \frac{v}{a^2\cos^2\varphi}
       =    \frac{1}{a^2}\DP[]{}{\varphi} 
              \left[\frac{1}{\cos\varphi}\DP[]{}{\varphi}(v\cos\varphi)\right]
  \end{eqnarray*}
  %
  ã—ãŸãŒã£ã¦, 
  \begin{eqnarray*}
    &amp; &amp;   - \frac{1}{a\cos\varphi}\DP{}{\varphi}(\cos\varphi f_\lambda)
          + \frac{1}{a\cos\varphi}\DP{f_\varphi}{\lambda} \\
    &amp;=&amp;    - \frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi
              \left\{
                   \frac{1}{a^2\cos^2\varphi}\DP[2]{u}{\lambda}
                 + \frac{1}{a^2}\DP[]{}{\varphi} 
                      \left[\frac{1}{\cos\varphi}
                            \DP[]{}{\varphi}(u\cos\varphi)\right]
                 + \frac{2u}{a^2}
                 - \frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{v}{\lambda}
              \right\} \\
    &amp; &amp;    + \frac{1}{a\cos\varphi}\DP{}{\lambda}
              \left\{
                    \frac{1}{a^2\cos^2\varphi}\DP[2]{v}{\lambda}
                  + \frac{1}{a^2}\DP[]{}{\varphi} 
                       \left[\frac{1}{\cos\varphi}
                               \DP[]{}{\varphi}(v\cos\varphi)\right]
                  + \frac{2v}{a^2}
                  +\frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{u}{\lambda}
              \right\}
  \end{eqnarray*}
  %
  ã“ã‚Œã‚ˆã‚Šæ¸¦åº¦ã®å½¢ã‚’æ„è­˜ã—ãªãŒã‚‰å¤‰å½¢ã—ã¦ã„ã. ç¬¬ 1 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;  - \frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi
              \left\{
                   \frac{1}{a^2\cos^2\varphi}\DP[2]{u}{\lambda}    
              \right\}
     =   - \frac{1}{a\cos\varphi}\DP{}{\varphi}
              \left[
                   \frac{1}{a^2\cos\varphi}\DP[2]{u}{\lambda}    
              \right]\\
    &amp;=&amp;   - \frac{1}{a\cos\varphi}\DP[2]{}{\lambda}\DP{}{\varphi}
              \left(\frac{u}{a^2\cos\varphi}\right)
     =   - \frac{1}{a\cos\varphi}\DP[2]{}{\lambda}\DP{}{\varphi}
              \left(\frac{u\cos\varphi}{a^2\cos^2\varphi}\right)\\
    &amp;=&amp;   - \frac{1}{a\cos\varphi}\DP[2]{}{\lambda}
              \left[  \frac{1}{a^2\cos^2\varphi}
                        \DP{}{\varphi}(u\cos\varphi)
                    + u\cos\varphi
                       \frac{2\sin\varphi\cos\varphi}{a^2\cos^4\varphi}
              \right]\\
    &amp;=&amp;   - \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
              \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
              \right]
         - \frac{1}{a\cos\varphi}\DP[2]{}{\lambda}
                    \left(\frac{2u\sin\varphi}{a^2\cos^2\varphi}\right) \\
    &amp;=&amp;   - \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
              \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
              \right]
         - \frac{1}{a\cos\varphi}\DP[]{}{\lambda}
                 \left(\frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{u}{\lambda}
                 \right).
  \end{eqnarray*}
  %
  ç¬¬ 2 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
   &amp; &amp;  - \frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi
                 \left\{ \frac{1}{a^2}\DP[]{}{\varphi} 
                      \left[\frac{1}{\cos\varphi}
                            \DP[]{}{\varphi}(u\cos\varphi)\right]\right\}\\
   &amp;=&amp; - \frac{1}{a^2\cos\varphi}\DP{}{\varphi}\left\{
                 \cos\varphi\DP[]{}{\varphi} 
                      \left[\frac{1}{a\cos\varphi}
                            \DP[]{}{\varphi}(u\cos\varphi)\right]\right\}.
  \end{eqnarray*}
  %
  ç¬¬ 3 é&nbsp;…ç›®ã¯
  \begin{eqnarray*}
      - \frac{1}{a\cos\varphi}\DP{}{\varphi}
         \left(\cos\varphi \frac{2u}{a^2}\right)
    = - \frac{2}{a^2}\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi).
  \end{eqnarray*}
  %
  ç¬¬ 4 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;   - \frac{1}{a\cos\varphi}\DP{}{\varphi}\cos\varphi
               \left(
                 - \frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{v}{\lambda}
               \right)
      =    \frac{1}{a\cos\varphi}\DP{}{\varphi}
               \left(
                  \frac{2\sin\varphi}{a^2\cos\varphi}\DP{v}{\lambda}
               \right).
  \end{eqnarray*}
  %
  ç¬¬ 5 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;  \frac{1}{a\cos\varphi}\DP{}{\lambda}
            \left[\frac{1}{a^2\cos^2\varphi}\DP[2]{v}{\lambda}\right]
     =   \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
            \left[\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right].
  \end{eqnarray*}
  %
  ç¬¬ 6 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;    \frac{1}{a\cos\varphi}\DP{}{\lambda}
              \left\{
                 \frac{1}{a^2}\DP[]{}{\varphi} 
                     \left[\frac{1}{\cos\varphi}
                           \DP[]{}{\varphi}(v\cos\varphi)\right]
              \right\}
     =     \frac{1}{a^3\cos\varphi}\DP[]{}{\varphi}
                  \left[\frac{1}{\cos\varphi}
                        \DP[]{}{\varphi}\left(\cos\varphi\DP{v}{\lambda}\right)
                  \right]\\
    &amp;=&amp;    \frac{1}{a^3\cos\varphi}\DP[]{}{\varphi}
                  \left[\frac{1}{\cos\varphi}
                     \DP[]{}{\varphi}
                       \left(
                          \frac{\cos\varphi}{\cos\varphi^2}\DP{v}{\lambda}
                       \right)
                  \right]\\
    &amp;=&amp;    \frac{1}{a^3\cos\varphi}\DP[]{}{\varphi}
                  \left[\frac{1}{\cos\varphi}
                  \left(
                      \cos^2\varphi\DP[]{}{\varphi}
                         \left(\frac{1}{\cos\varphi}\DP{v}{\lambda}\right)
                    + \frac{1}{\cos\varphi}\DP{v}{\lambda}
                        (-2\sin\varphi\cos\varphi)
                  \right)\right]\\
    &amp;=&amp;    \frac{1}{a^3\cos\varphi}\DP[]{}{\varphi}
                  \left[   \cos\varphi\DP[]{}{\varphi}
                              \left(\frac{1}{\cos\varphi}\DP{v}{\lambda}\right)
                         -\frac{2\sin\varphi}{\cos\varphi}\DP{v}{\lambda}
                  \right]\\
    &amp;=&amp;    \frac{1}{a^2\cos\varphi}\DP[]{}{\varphi}
                  \left[\cos\varphi\DP[]{}{\varphi}
                          \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
                  \right]
         - \frac{1}{a\cos\varphi}\DP[]{}{\varphi}\left(
                         \frac{2\sin\varphi}{a^2\cos\varphi}\DP{v}{\lambda}
                  \right).
  \end{eqnarray*}
  %
  ç¬¬ 7 é&nbsp;…ç›®ã¯
  %
  \begin{eqnarray*}
    &amp; &amp;   \frac{1}{a\cos\varphi}\DP{}{\lambda}\left(\frac{2v}{a^2}\right)
     =   \frac{2}{a^2} \frac{1}{a\cos\varphi}\DP{v}{\lambda}.
  \end{eqnarray*}
  %
  ç¬¬ 8 é&nbsp;…ç›®ã¯ãã®ã¾ã¾ã§
  %
  \begin{eqnarray*}
        \frac{1}{a\cos\varphi}\DP{}{\lambda}
            \left(
             \frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{u}{\lambda}\right).
  \end{eqnarray*}
  %
  å…¨ã¦ã¾ã¨ã‚ã‚‹ã¨
  %
  \begin{eqnarray*}
    &amp; &amp;   - \frac{1}{a\cos\varphi}\DP{}{\varphi}(\cos\varphi f_\lambda)
          + \frac{1}{a\cos\varphi}\DP{f_\varphi}{\lambda} \\
    &amp;=&amp;   - \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
              \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
              \right]
         - \frac{1}{a\cos\varphi}\DP[]{}{\lambda}
                 \left(\frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{u}{\lambda}
                 \right)\\
    &amp; &amp; - \frac{1}{a^2\cos\varphi}\DP{}{\varphi}\left\{
                  \cos\varphi\DP[]{}{\varphi} 
                       \left[\frac{1}{a\cos\varphi}
                             \DP[]{}{\varphi}(u\cos\varphi)\right]\right\}\\
    &amp; &amp; - \frac{2}{a^2}\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
        +  \frac{1}{a\cos\varphi}\DP{}{\varphi}
                \left(
                   \frac{2\sin\varphi}{a^2\cos\varphi}\DP{v}{\lambda}
                \right) \\
    &amp; &amp; +  \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
             \left[\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right]
        +   \frac{1}{a^2\cos\varphi}\DP[]{}{\varphi}
                   \left[\cos\varphi\DP[]{}{\varphi}
                           \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
                   \right] \\
    &amp; &amp; - \frac{1}{a\cos\varphi}\DP[]{}{\varphi}\left(
                          \frac{2\sin\varphi}{a^2\cos\varphi}\DP{v}{\lambda}
                   \right)
        + \frac{2}{a^2} \frac{1}{a\cos\varphi}\DP{v}{\lambda}
        + \frac{1}{a\cos\varphi}\DP{}{\lambda}
             \left(
              \frac{2\sin\varphi}{a^2\cos^2\varphi}\DP{u}{\lambda}\right)\\
    %
    &amp;=&amp;   - \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
              \left[\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
              \right]\\
    &amp; &amp; - \frac{1}{a^2\cos\varphi}\DP{}{\varphi}\left\{
                  \cos\varphi\DP[]{}{\varphi} 
                       \left[\frac{1}{a\cos\varphi}
                             \DP[]{}{\varphi}(u\cos\varphi)\right]\right\}\\
    &amp; &amp; - \frac{2}{a^2}\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
        +  \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
             \left[\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right]\\
    &amp; &amp; +   \frac{1}{a^2\cos\varphi}\DP[]{}{\varphi}
                   \left[\cos\varphi\DP[]{}{\varphi}
                           \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}\right)
                   \right]
        + \frac{2}{a^2} \frac{1}{a\cos\varphi}\DP{v}{\lambda}\\
    %
    &amp;=&amp;   \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
             \left[  \frac{1}{a\cos\varphi}\DP{v}{\lambda}
                   - \frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
             \right]\\
    &amp; &amp;  + \frac{1}{a^2\cos\varphi}\DP[]{}{\varphi}
                   \left[\cos\varphi\DP[]{}{\varphi}
                     \left(\frac{1}{a\cos\varphi}\DP{v}{\lambda}
                           -\frac{1}{a\cos\varphi}
                               \DP[]{}{\varphi}(u\cos\varphi)
                     \right)
                   \right]\\
    &amp; &amp;  + \frac{2}{a^2}
             \left[  \frac{1}{a\cos\varphi}\DP{v}{\lambda}
                    -\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
             \right] \\
    %
    &amp;=&amp;   \frac{1}{a^2\cos^2\varphi}\DP[2]{\zeta}{\lambda}
         + \frac{1}{a^2\cos\varphi}\DP[]{}{\varphi}
                   \left(\cos\varphi\DP{\zeta}{\varphi}\right)
         + \frac{2\zeta}{a^2}. 
  \end{eqnarray*}

  ã“ã‚Œã‚ˆã‚Š, æ¸¦åº¦æ–¹ç¨‹å¼ã¯
  %
  \begin{eqnarray*}
    \Deqlab{æ¸¦åº¦æ–¹ç¨‹å¼æ¸¦åº¦}
    &amp; &amp;\DP{\zeta}{t} 
      + \left[
            \frac{u}{a\cos\varphi}\DP{}{\lambda} + \frac{v}{a}\DP{}{\varphi}
        \right] \zeta
      + 2\Omega\frac{v}{a}\cos\varphi \\
    &amp; &amp; \hspace*{30mm}
     =  \nu\left[ \frac{1}{a^2\cos^2\varphi}\DP[2]{\zeta}{\lambda}
         + \frac{1}{a^2\cos\varphi}\DP[]{}{\varphi}
                   \left(\cos\varphi\DP{\zeta}{\varphi}\right)
         + \frac{2\zeta}{a^2}\right]. 
  \end{eqnarray*}
  %
  ã“ã“ã§æµç·šé–¢æ•° $\psi$ ã‚’
  %
  \begin{equation}
    \Deqlab{æµç·šé–¢æ•°}
    u = -\frac{1}{a}\DP{\psi}{\varphi}, \quad 
    v = \frac{1}{a\cos\varphi}\DP{\psi}{\lambda},
  \end{equation}
  %
  ã¨ã—ã¦å®šç¾©ã™ã‚‹ã¨, é€£ç¶šã®å¼\Deqref{é€£ç¶šã®å¼}ã‚’è‡ªå‹•çš„ã«æº€ãŸã™. 
  æ¸¦åº¦ $\zeta$ ã¯
  %
  \begin{eqnarray*}
    \zeta &amp;=&amp; -\frac{1}{a\cos\varphi}\DP{}{\varphi}(u\cos\varphi)
                 +\frac{1}{a\cos\varphi}\DP{v}{\lambda}\\
          &amp;=&amp; \frac{1}{a^2\cos\varphi}\DP{}{\varphi}
                \left(\cos\varphi\DP{\psi}{\varphi}\right)
            + \frac{1}{a^2\cos^2\varphi}\DP[2]{\psi}{\lambda}
          =  \Dlapla \psi.
  \end{eqnarray*}
  %
  ã“ã“ã§ $\Ddsty \Dlapla= \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
                         +\frac{1}{a^2\cos\varphi}\DP{}{\varphi}
                            \cos\varphi\DP{}{\varphi}$
  ã¯çƒé¢ä¸Šã®æ°´å¹³ãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã§ã‚ã‚‹. 
  ã“ã‚Œã‚‰ã‚’æ¸¦åº¦æ–¹ç¨‹å¼\Deqref{æ¸¦åº¦æ–¹ç¨‹å¼æ¸¦åº¦}ã«ä»£å…¥ã™ã‚‹ã¨, 
  %
  \begin{eqnarray*}
    &amp; &amp; \DP{\Dlapla\psi}{t} 
      + \left[
           - \frac{1}{a}\DP{\psi}{\varphi}\frac{1}{a\cos\varphi}\DP{}{\lambda}
           + \frac{1}{a\cos\varphi}\DP{\psi}{\lambda}\frac{1}{a}\DP{}{\varphi}
        \right] \Dlapla\psi
      + \frac{2\Omega}{a^2}\DP{\psi}{\lambda}
    = \nu\left(\Dlapla + \frac{2}{a^2}\right)\Dlapla\psi. 
  \end{eqnarray*}
  %
  ã•ã‚‰ã« $\sin$ ç·¯åº¦ $\mu=\sin\varphi$ ã‚’å°Žå…¥ã™ã‚‹ã¨, 
  $\Ddsty \DP{}{\mu}= \frac{1}{\cos\varphi}\DP{}{\varphi}$
  ã§ã‚ã‚‹ã‹ã‚‰, 
  \begin{eqnarray*}
    &amp; &amp; \DP{\Dlapla\psi}{t} 
      + \left[
           - \frac{1}{a^2}\DP{\psi}{\mu}\DP{}{\lambda}
           + \frac{1}{a^2}\DP{\psi}{\lambda}\DP{}{\mu}
        \right] \Dlapla\psi
      + \frac{2\Omega}{a^2}\DP{\psi}{\lambda}
    =  \nu\left(\Dlapla + \frac{2}{a^2}\right)\Dlapla\psi. 
  \end{eqnarray*}
  %
  ã‚ˆã£ã¦
  %
  \begin{equation}
    \Deqlab{æ¸¦åº¦æ–¹ç¨‹å¼}
    \DP{\Dlapla\psi}{t} 
      + \frac{1}{a^2}J(\psi,\Dlapla\psi)
      + \frac{2\Omega}{a^2}\DP{\psi}{\lambda}
    = \nu \left(\Dlapla + \frac{2}{a^2}\right)\Dlapla\psi.
  \end{equation}
  %
  ãŸã&nbsp;ã— $J(f,g)=(\partial_\lambda f)(\partial_\mu g)
                -(\partial_\lambda g)(\partial_\mu f)$ ã¯
  ãƒ¤ã‚³ãƒ“ã‚¢ãƒ³ã§ã‚ã‚‹. 
  ãƒ©ãƒ—ãƒ©ã‚·ã‚¢ãƒ³ã‚’ $\mu $ ã§è¡¨ã—ã¦ãŠãã¨
  \begin{equation}
     \Dlapla= \frac{1}{a^2\cos^2\varphi}\DP[2]{}{\lambda}
             +\frac{1}{a^2\cos\varphi}\DP{}{\varphi}
                            \cos\varphi\DP{}{\varphi}
            = \frac{1}{a^2(1-\mu^2)}\DP[2]{}{\lambda}
             +\frac{1}{a^2}\DP{}{\mu}(1-\mu^2)\DP{}{\mu}.
  \end{equation}


\section{è§’é‹å‹•é‡ä¿å­˜å‰‡}

  $\psi$ ã®çƒé¢èª¿å’Œå‡½æ•° $Y_n^m(\mu,\lambda)$ ã§å±•é–‹ã—ãŸæˆåˆ†ã®ã†ã¡, 
  $n=0$ ã¯å‰›ä½“å›žè»¢æµã‚’è¡¨ã‚ã™. 
  $(n,m)=(1,0)$ ã¯æ¥µæ–¹å‘ã‚’å›žè»¢è»¸ã¨ã™ã‚‹å‰›ä½“å›žè»¢æµ, 
  $(n,m)=(1,1),(1,-1)$ ã¯èµ¤é“é¢ä¸Šã®è»¸ã‚’å›žè»¢è»¸ã¨ã™ã‚‹å‰›ä½“å›žè»¢æµã§ã‚ã‚‹. 
  ã—ãŸãŒã£ã¦ã“ã‚Œã‚‰ã®æˆåˆ†ã¯å„ã€…ã®è»¸ã«å¯¾ã™ã‚‹è§’é‹å‹•é‡ã¨ãªã£ã¦ãŠã‚Š, 
  éžå›žè»¢ç³»ã§ã¯å…¨ã¦, å›žè»¢ç³»ã§ã¯ $(n,m)=(1,0)$ æˆåˆ†ã&nbsp;ã‘ä¿å­˜ã—ã¦ã„ã‚‹ã¯ãšã§ã‚ã‚‹. 
  ãã®ã“ã¨ã‚’ç¢ºã‹ã‚ã¦ã¿ã‚ˆã†. 

  ã„ã¾æµç·šé–¢æ•°ãŒçƒé¢èª¿å’Œå‡½æ•°ã®è¶³ã—ã‚ã‚ã›ã§
  %
  \begin{equation}
    \Deqlab{çƒé¢èª¿å’Œå‡½æ•°è¡¨ç¾}
    \psi(\lambda,\mu,t) = \sum_{n=1}^\infty\sum_{m=-n}^n 
                           \tilde{\psi}_{n,m} Y_n^m(\lambda,\mu)
                        = \sum_{n=1}^\infty\sum_{m=-n}^n 
                           \tilde{\psi}_{n,m} P_n^m(\lambda)e^{i\mu\lambda}
  \end{equation}
  %
  ã¨è¡¨ã‚ã•ã‚Œã¦ã„ã‚‹ã¨ã™ã‚‹. 
  ã“ã®ã¨ã $\psi$ ãŒå®Ÿæ•°ã§ã‚ã‚‹ã“ã¨ã‹ã‚‰ä¿‚æ•°ã«ã¯
  %
  \begin{equation}
    \tilde{\psi}_{n,-m} = \tilde{\psi}_{n,m}^*
  \end{equation}
  %
  ãŒæˆã‚Šç«‹ã¤. 

  ã¾ãš $(n,m)=(1,0)$ æˆåˆ†ã«ã¤ã„ã¦è€ƒãˆã‚ˆã†. 
  æ¸¦åº¦æ–¹ç¨‹å¼\Deqref{æ¸¦åº¦æ–¹ç¨‹å¼}ã«ãŠã„ã¦, ç²˜æ€§é&nbsp;…ã® 1,0 æˆåˆ†ã¯
  \begin{equation}
    \left(\Dlapla + \frac{2}{a^2}\right)\Dlapla 
        \tilde{\psi}_{1,0} Y_1^0(\lambda,\mu)
   =  \left( \frac{-1\times 2}{a^2} + \frac{2}{a^2}\right) 
         \frac{-1\times 2}{a^2}\tilde{\psi}_{1,0} Y_1^0(\lambda,\mu) 
   = 0. 
  \end{equation}
  %
  \Deqref{æ¸¦åº¦æ–¹ç¨‹å¼}ã®$\beta$ é&nbsp;…ã® 1,0 æˆåˆ†ã¯
  \begin{equation}
      \frac{2\Omega}{a^2}\DP{}{\lambda}
        \tilde{\psi}_{1,0} Y_1^0(\lambda,\mu) = 0. 
  \end{equation}
  %
  éžç·šå½¢é&nbsp;…ã‹ã‚‰ã®å¯„ä¸Žã¯, \Deqref{çƒé¢èª¿å’Œå‡½æ•°è¡¨ç¾}ã‚’ãƒ¤ã‚³ãƒ“ã‚¢ãƒ³ã«ä»£å…¥ã—ã¦
  $Y_1^0(\lambda,\mu)=\mu$ ã‚’ã‹ã‘ã¦çƒé¢ä¸Šã§ç©åˆ†ã™ã‚‹ã“ã¨ã«ã‚ˆã‚Šå¾—ã‚‰ã‚Œã‚‹. 
  %
  \begin{eqnarray*}
    &amp; &amp;
    \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu
       \frac{1}{a^2}J(\psi,\Dlapla\psi)Y_1^0(\lambda,\mu)\\
    &amp;=&amp; \frac{1}{a^2} \sum_{n,m}\sum_{n',m'} 
          \int_0^{2\pi}\int_{-1}^1 
             \left[
               im \tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
                     \mu Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
                     \mu \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] d\lambda d\mu \\
    &amp;=&amp; \frac{1}{a^2} \sum_{n,m}\sum_{n',m'} 
          \tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
          \int_0^{2\pi}\int_{-1}^1 
             \left[
                im \mu Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \mu \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right]d\lambda d\mu 
  \end{eqnarray*}
  %
  ã“ã“ã§ $m=-m'$ æˆåˆ†ã®ã¿ 0 ã§ãªã„ã“ã¨ã«æ³¨æ„ã™ã‚‹ã¨
  %
  \begin{eqnarray*}
    &amp; &amp;  \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                im \mu Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \mu \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] \\
    &amp;=&amp;  \int_0^{2\pi}\left[ im \mu Y_n^m Y_{n'}^{m'}\right]_{-1}^1 d\lambda
       -\int_0^{2\pi}\int_{-1}^1 
             im Y_n^m Y_{n'}^{m'}d\lambda d\mu 
       -\int_0^{2\pi}\int_{-1}^1 
             im \mu \DP{Y_n^m}{\mu}Y_{n'}^{m'} d\lambda d\mu  \\
    &amp; &amp; +\int_0^{2\pi}\int_{-1}^1
             \left[
               -im' \mu \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] d\lambda d\mu  \\
    &amp;=&amp;  \int_0^{2\pi}\left[ im \mu Y_n^m Y_{n'}^{-m}\right]_{-1}^1 d\lambda
       -\int_0^{2\pi}\int_{-1}^1 
             im Y_n^m Y_{n'}^{-m}d\lambda d\mu 
       -\int_0^{2\pi}\int_{-1}^1 
             im \mu \DP{Y_n^m}{\mu}Y_{n'}^{-m} d\lambda d\mu  \\
    &amp; &amp; +\int_0^{2\pi}\int_{-1}^1 
             im \mu \DP{Y_n^m}{\mu}Y_{n'}^{-m} d\lambda d\mu \\
    &amp;=&amp;  \int_0^{2\pi}\left[ im \mu Y_n^m Y_{n'}^{-m}\right]_{-1}^1 d\lambda
        -\int_0^{2\pi}\int_{-1}^1 im Y_n^m Y_{n'}^{-m} d\lambda d\mu\\
    &amp;=&amp;  \int_0^{2\pi}\left[ im \mu Y_n^m Y_{n'}^{-m}\right]_{-1}^1 d\lambda
        -\int_0^{2\pi}\int_{-1}^1 im Y_n^m Y_{n}^{-m} d\lambda d\mu
  \end{eqnarray*}
  %
  æœ€å¾Œã« $Y_n^m$ ã®ç›´äº¤é–¢ä¿‚ã‚’ç”¨ã„ã¦ã„ã‚‹. 
  ç¬¬ 1 é&nbsp;…ç›®ã®å€¤ã‚’è©•ä¾¡ã™ã‚‹. 
  $Y_n^m$ ã® $\mu=\pm 1$ ã§ã®å€¤ã¯ $m\neq 0$ ã®ã¨ã 0 ã§ã‚ã‚‹. 
  $m=0$ ã®ã¨ãã¯ä¿‚æ•° $m$ ãŒã‹ã‹ã£ã¦ã„ã‚‹ã®ã§çµå±€ã“ã®é&nbsp;…ã¯ 0 ã¨ãªã‚‹. 
  ã—ãŸãŒã£ã¦, $n,m$, $n',m'$ æˆåˆ†ã‹ã‚‰ã¯
  %
  \begin{equation}
    -\tilde{\psi}_{n,m}\tilde{\psi}_{n,-m}
        \int_0^{2\pi}\int_{-1}^1 im Y_n^m Y_{n}^{-m}d\lambda d\mu 
   = -|\tilde{\psi}_{n,m}|^2
        \int_0^{2\pi}\int_{-1}^1 im Y_n^m Y_{n}^{-m} d\lambda d\mu 
  \end{equation}
  %
  ä¸€æ–¹, $n,-m$ ã‹ã‚‰
  %
  \begin{equation}
    -|\tilde{\psi}_{n,-m}|^2
        \int_0^{2\pi}\int_{-1}^1 (-im Y_n^{-m} Y_{n}^{m})d\lambda d\mu
    =|\tilde{\psi}_{n,m}^*|^2
        \int_0^{2\pi}\int_{-1}^1 im Y_n^{-m} Y_n^m d\lambda d\mu 
  \end{equation}
  %
  ã—ãŸãŒã£ã¦ã“ã®é&nbsp;…ã¯äº’ã„ã«ã‚­ãƒ£ãƒ³ã‚»ãƒ«ã™ã‚‹. 
  çµå±€éžç·šå½¢é&nbsp;…ã‹ã‚‰ã® $(n,m)=(1,0)$ ã¸ã®å¯„ä¸Žã¯ 0 ã¨ãªã‚‹. 
  æœ€å¾Œã«æ™‚é–“å¤‰åŒ–é&nbsp;…ã¯
  %
  \begin{equation}
    \DP{}{t}\Dlapla\tilde{\psi}_{1,0}(t)Y_1^0 
    = -\frac{1(1+1)}{a^2}\DD{\tilde{\psi}_{1,0}}{t}Y_1^0
  \end{equation}
  ã‚ˆã£ã¦
  %
  \begin{equation}
    \DD{\tilde{\psi}_{1,0}}{t} = 0
  \end{equation}
  %
  ã¨ãªã‚Šæ¥µæ–¹å‘ã‚’è»¸ã¨ã™ã‚‹è§’é‹å‹•é‡ã¯ä¿å­˜ã™ã‚‹. 

  æ¬¡ã« $(n,m)=(1,1)$ æˆåˆ†ã«ã¤ã„ã¦è€ƒãˆã‚ˆã†. 
  ç²˜æ€§é&nbsp;…ã® $(1,1)$ æˆåˆ†ã¯ $(n,m)=(1,0)$ ã¨åŒæ§˜ã«ã—ã¦ 0 ã§ã‚ã‚‹. 
  \Deqref{æ¸¦åº¦æ–¹ç¨‹å¼}ã®$\beta$ é&nbsp;…ã® 1,1 æˆåˆ†ã¯
  \begin{equation}
      \frac{2\Omega}{a^2}\DP{}{\lambda}
        \tilde{\psi}_{1,1} Y_1^1(\lambda,\mu) = 
     = i\frac{2\Omega}{a^2}\tilde{\psi}_{1,1} Y_1^1(\lambda,\mu). 
  \end{equation}
  %
  éžç·šå½¢é&nbsp;…ã‹ã‚‰ã®å¯„ä¸Žã¯, \Deqref{çƒé¢èª¿å’Œå‡½æ•°è¡¨ç¾}ã‚’ãƒ¤ã‚³ãƒ“ã‚¢ãƒ³ã«ä»£å…¥ã—ã¦
  $Y_1^1(\lambda,\mu)=\sqrt{1-\mu^2}e^{i\lambda}$ ã‚’ã‹ã‘ã¦
  çƒé¢ä¸Šã§ç©åˆ†ã™ã‚‹ã“ã¨ã«ã‚ˆã‚Šå¾—ã‚‰ã‚Œã‚‹. 
  %
  \begin{eqnarray*}
    &amp; &amp;
    \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu
       \frac{1}{a^2}J(\psi,\Dlapla\psi)Y_1^1(\lambda,\mu)\\
    &amp;=&amp; \frac{1}{a^2} \sum_{n,m}\sum_{n',m'} 
          \int_0^{2\pi}\int_{-1}^1 
             \left[
               im \tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
                   \sqrt{1-\mu^2}e^{i\lambda} Y_n^m \DP{Y_{n'}^{m'}}{\mu}
             \right.\\
    &amp; &amp; \hspace{50mm}\left.
               -im' \tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
                   \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] d\lambda d\mu \\
    &amp;=&amp; \frac{1}{a^2} \sum_{n,m}\sum_{n',m'} 
          \tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
          \int_0^{2\pi}\int_{-1}^1 
             \left[
                im \sqrt{1-\mu^2}e^{i\lambda} Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \sqrt{1-\mu^2}e^{i\lambda}\DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right]d\lambda d\mu 
  \end{eqnarray*}
  %
  ã“ã“ã§ $m'=-1-m$ æˆåˆ†ã®ã¿ 0 ã§ãªã„ã“ã¨ã«æ³¨æ„ã™ã‚‹ã¨, 
  $m=n$ ã®ã¨ãã¯ 0 ã«ãªã‚Š, $0&lt; m &lt; n+1$ ãªã‚‹ $n,m$ ã«å¯¾ã—ã¦
  %
  \begin{eqnarray*}
    &amp; &amp;  \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                im \sqrt{1-\mu^2}e^{i\lambda} Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] \\
    &amp;=&amp; \int_0^{2\pi}\left[ im\sqrt{1-\mu^2}e^{i\lambda}Y_n^m Y_{n'}^{m'}
                      \right]_{-1}^1 d\lambda
       -\int_0^{2\pi}\int_{-1}^1 
             -im \frac{\mu}{\sqrt{1-\mu^2}}e^{i\lambda}
                 Y_n^m Y_{n'}^{m'}d\lambda d\mu \\
    &amp; &amp;-\int_0^{2\pi}\int_{-1}^1 
             im \sqrt{1-\mu^2}e^{i\lambda}\DP{Y_n^m}{\mu}Y_{n'}^{m'} 
                 d\lambda d\mu 
       +\int_0^{2\pi}\int_{-1}^1
             \left[
               -im' \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] d\lambda d\mu  \\
    &amp;=&amp; \int_0^{2\pi}\int_{-1}^1 
             im \frac{\mu}{\sqrt{1-\mu^2}}e^{i\lambda}
                 Y_n^m Y_{n'}^{m'}d\lambda d\mu 
       -i(m+m')\int_0^{2\pi}\int_{-1}^1 
                 \sqrt{1-\mu^2}e^{i\lambda}\DP{Y_n^m}{\mu}Y_{n'}^{m'} 
                 d\lambda d\mu  \\
    &amp;=&amp; \int_0^{2\pi}\int_{-1}^1 
             im \frac{\mu}{\sqrt{1-\mu^2}}e^{i\lambda}
                 Y_n^m Y_{n'}^{m'}d\lambda d\mu 
        +i\int_0^{2\pi}\int_{-1}^1 
                 \sqrt{1-\mu^2}e^{i\lambda}\DP{Y_n^m}{\mu}Y_{n'}^{m'} 
                 d\lambda d\mu  \\
    &amp;=&amp; i\int_0^{2\pi}\int_{-1}^1 \frac{1}{\sqrt{1-\mu^2}}e^{i\lambda}
           \left[ m\mu Y_n^m + (1-\mu^2)\DP{Y_n^m}{\mu} \right]Y_{n'}^{m'} 
                 d\lambda d\mu \\
  \end{eqnarray*}
  %
  ã“ã“ã§ãƒ«ã‚¸ãƒ£ãƒ³ãƒ‰ãƒ«å‡½æ•°ã®æ¼¸åŒ–å¼
  \begin{equation}
    (1-\mu^2)\DP{P_n^m}{\mu} = \sqrt{1-\mu^2}P_n^{m+1}-m\mu P_n^m
  \end{equation}
  %
  ã‚ˆã‚Š
  \begin{eqnarray*}
        e^{i\lambda} \left[ m\mu Y_n^m + (1-\mu^2)\DP{Y_n^m}{\mu}\right]
    &amp;=&amp; e^{i(m+1)\lambda} \left[ m\mu P_n^m + (1-\mu^2)\DP{P_n^m}{\mu}\right]\\
    &amp;=&amp; \sqrt{1-\mu^2}P_n^{m+1}e^{i(m+1)\lambda}
     = \sqrt{1-\mu^2}Y_n^{m+1}.
  \end{eqnarray*}
  %
  ã—ãŸãŒã£ã¦, $n,m$ æˆåˆ†ã‹ã‚‰ã®å¯„ä¸Žã¯
  %
  \begin{eqnarray*}
    &amp; &amp;  \frac{1}{a^2}\tilde{\psi}_{n,m}\tilde{\psi}_{n',m'}
         \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                im \sqrt{1-\mu^2}e^{i\lambda} Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] \\
    &amp;=&amp;  i \frac{1}{a^2}\tilde{\psi}_{n,m}\tilde{\psi}_{n',-m-1}
            \int_0^{2\pi}\int_{-1}^1 
            Y_n^{m+1}Y_{n'}^{m'} d\lambda d\mu d\lambda d\mu \\
    &amp;=&amp;  i \frac{1}{a^2}\tilde{\psi}_{n,m}\tilde{\psi}_{n,-m-1}
           \int_0^{2\pi}\int_{-1}^1 
            Y_n^{m+1}Y_{n}^{-m-1} d\lambda d\mu d\lambda d\mu 
  \end{eqnarray*}
  %
  æœ€å¾Œã«ç›´äº¤é–¢ä¿‚ã‚’ä½¿ã£ã¦ $n'=n$ ã¨ã—ãŸ. 
  
  ä¸€æ–¹ $m&lt;0$ ã®æˆåˆ†ã«å¯¾ã—ã¦, $m \rightarrow -m-1 &gt; -n-1$ ã‚’é¸ã‚“ã§è¨ˆç®—ã™ã‚‹ã¨
  $m'=m$ ã®æ™‚ã®ã¿ 0 ã§ãªã, 
  %
  \begin{eqnarray*}
    &amp; &amp;  \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                im \sqrt{1-\mu^2}e^{i\lambda} Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] \\
    &amp;=&amp;  \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                i(-m-1)\sqrt{1-\mu^2}e^{i\lambda} Y_n^{-m-1} \DP{Y_{n'}^{m'}}{\mu}
               -im' \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^{-m-1}}{\mu}Y_{n'}^{m'} 
             \right] \\
    &amp;=&amp;  \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                i(-m-1)\sqrt{1-\mu^2}e^{i\lambda} Y_n^{-m-1} \DP{Y_{n'}^{m}}{\mu}
               -im \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^{-m-1}}{\mu}Y_{n'}^{m} 
             \right] \\
    &amp;=&amp; -\int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
               im \sqrt{1-\mu^2}e^{i\lambda} Y_{n'}^{m} \DP{Y_n^{-m-1}}{\mu}
               -i(-m-1)\sqrt{1-\mu^2}e^{i\lambda}\DP{Y_{n'}^{m}}{\mu}Y_n^{-m-1} 
             \right] \\
    &amp;=&amp; -i \int_0^{2\pi}\int_{-1}^1 
            Y_{n'}^{m+1}Y_{n}^{-m-1} d\lambda d\mu d\lambda d\mu \\
    &amp;=&amp; -i \int_0^{2\pi}\int_{-1}^1 
            Y_{n}^{m+1}Y_{n}^{-m-1} d\lambda d\mu d\lambda d\mu.
  \end{eqnarray*}
  %
  æœ€å¾Œã®è¨ˆç®—ã¯ $n,m$ ã®æ™‚ã®å¼å¤‰å½¢ã¨åŒæ§˜ã«è¡Œã£ãŸ. 
  ã—ãŸãŒã£ã¦, $n,-m-1$ æˆåˆ†ã‹ã‚‰ã®å¯„ä¸Žã¯
  %
  \begin{eqnarray*}
    &amp; &amp;  \frac{1}{a^2}\tilde{\psi}_{n,-m-1}\tilde{\psi}_{n',m'}
         \int_0^{2\pi}\int_{-1}^1 d\lambda d\mu 
             \left[
                im \sqrt{1-\mu^2}e^{i\lambda} Y_n^m \DP{Y_{n'}^{m'}}{\mu}
               -im' \sqrt{1-\mu^2}e^{i\lambda} \DP{Y_n^m}{\mu}Y_{n'}^{m'} 
             \right] \\
    &amp;=&amp; -i \frac{1}{a^2}\tilde{\psi}_{n,-m-1}\tilde{\psi}_{n',m}
            \int_0^{2\pi}\int_{-1}^1 
            Y_n^{-m-1}Y_{n'}^{m} d\lambda d\mu d\lambda d\mu \\
    &amp;=&amp; -i \frac{1}{a^2}\tilde{\psi}_{n,-m-1}\tilde{\psi}_{n,m}
           \int_0^{2\pi}\int_{-1}^1 
            Y_n^{-m-1}Y_{n}^{m} d\lambda d\mu d\lambda d\mu 
  \end{eqnarray*}
  %
  ã‚ˆã£ã¦ $n,m$ æˆåˆ†ã¨ $n,-m-1$ æˆåˆ†ã‹ã‚‰ã®å¯„ä¸ŽãŒæ‰“ã¡æ¶ˆã—ã‚ã£ã¦
  éžç·šå½¢é&nbsp;…ã¯ 0 ã¨ãªã‚‹. 

  æœ€å¾Œã«æ™‚é–“å¤‰åŒ–é&nbsp;…ã¯ $(1,0)$ æˆåˆ†ã¨åŒã˜ã§ã‚ã‚Š
  %
  \begin{equation}
    \DP{}{t}\Dlapla\tilde{\psi}_{1,1}(t)Y_1^1
    = -\frac{1(1+1)}{a^2}\DD{\tilde{\psi}_{1,1}}{t}Y_1^1
  \end{equation}
  %
  ã‚ˆã£ã¦ $(1,1)$ æˆåˆ†ã®å¼ã¯
  %
  \begin{equation}
    -\frac{2}{a^2}\DD{\tilde{\psi}_{1,1}}{t} 
       + i \frac{2\Omega}{a^2}\tilde{\psi}_{1,1}=0,
    \qquad \tilde{\psi}_1^1 = A e^{i \Omega t}.
  \end{equation}
  %
  $A$ ã¯ä»»æ„å®šæ•°ã§ã‚ã‚ŠåˆæœŸå€¤ã§å®šã¾ã‚‹. 
  ã—ãŸãŒã£ã¦ $(1,1)$ æˆåˆ†ã®æŒ¯èˆžã¯
  %
  \begin{equation}
    \psi(\mu,\lambda,t) = A e^{i(\lambda+\Omega t)}P_1^1(\mu)
  \end{equation}
  %
  ç·¯åº¦æ–¹å‘ã«é€Ÿåº¦ $\Omega$ ã§é€†å‘ãã«ä¼æ’­ã™ã‚‹. 
  ã“ã®è§£ã¯å›žè»¢ç³»ã§è¦‹ã‚Œã°æ³¢æ•° $(1,1)$ ã®ãƒ­ã‚¹ãƒ“ãƒ¼æ³¢ã§ã‚ã‚‹. 
  æ…£æ€§ç³»ã‹ã‚‰è¦‹ã¦ä¿å­˜ã—ã¦ã„ã‚‹æ¥µã‚’ã¨ãŠã‚‹å‰›ä½“å›žè»¢æµã‚’, å›žè»¢ç³»ã‹ã‚‰è¦‹ã‚‹ã¨
  $\Omega$ ã§é€†ã¾ã‚ã‚Šã—ã¦ã„ã‚‹ã‚ˆã†ã«è¦‹ãˆã‚‹ã®ãŒã“ã®è§£ã§ã‚ã‚‹. 

  $(1,-1)$ æˆåˆ†ã¯ $(1,1)$ æˆåˆ†ã¨åŒæ§˜ã«ç²˜æ€§é&nbsp;…, éžç·šå½¢é&nbsp;…ã®å¯„ä¸ŽãŒ 0 ã¨ãªã‚‹.
  %
  \begin{equation}
    -\frac{2}{a^2}\DD{\tilde{\psi}_{1,-1}}{t} 
    - i \frac{2\Omega}{a^2}\tilde{\psi}_{1,-1}=0,
    \qquad \tilde{\psi}_1^1 = A e^{-i \Omega t}.
  \end{equation}
  %
  ã—ãŸãŒã£ã¦ $(1,1)$ æˆåˆ†ã®æŒ¯èˆžã¯
  %
  \begin{equation}
    \psi(\mu,\lambda,t) = A e^{-i(\lambda+\Omega t)}P_1^1(\mu)
  \end{equation}
  %
  ç·¯åº¦æ–¹å‘ã«é€Ÿåº¦ $\Omega$ ã§é€†å‘ãã«ä¼æ’­ã™ã‚‹. 
  ç‰¹ã«å›žè»¢ãŒãªã‘ã‚Œã°æ™‚é–“å¾®åˆ†ãŒ 0 ã¨ãªã‚Šä¿å­˜ã™ã‚‹. 


\begin{Dreference}
  \item åœ°çƒæµä½“é›»è„³å€¶æ¥½éƒ¨ç†è«–ãƒžãƒ‹ãƒ¥ã‚¢ãƒ«ã€Œãƒ­ã‚¹ãƒ“ãƒ¼æ³¢(2 æ¬¡å…ƒéžç™ºæ•£çƒé¢)ã€
    https://www.gfd-dennou.org/GFD\_Dennou\_Club/dc-arch/zz1998/gfd-note/waveli/ros2dnds/index.htm
\end{Dreference}
\end{document}

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