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\title{
\begin{figure}[htb]
\centering 
   \includegraphics[width=6in]{portada.eps}
\end{figure}
\ \ \\\vspace{.3in} \ \
\newpage
  \ \\\ \\
\vspace{.6in} INSTITUTO POLIT\'ECNICO NACIONAL\\\
{\large ESCUELA
SUPERIOR DE F\'ISICA Y MATEM\'ATICAS }\\\ \\\
\\\
{\large VIBRACIONES EN REDES UNIDIMENSIONALES}\\\ {\large CON}\\
\ {\large INTERACCIONES A PRIMEROS VECINOS}\\\ \\\  \\\ {\large 
T E S I S}\\\ \\\ {\large que presenta}\\\   {\large Isidro
Romero Medina}  \\\ {\large para obtener el t\'{\i}tulo}
\\\ {\large de}
 \\\   {\large Licenciado en F\'{\i}sica y Matem\'aticas} \\
\  \\\  {\large Director de Tesis} \\\ {\large Harold V.
McIntosh}}


\author{}
\date{M\'exico, D. F. \hspace{1.7in}  Septiembre, 1971}



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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath334}%
\begin{displaymath}
Z_{0} =  \left[ \begin{array}{cccc}
x_{k}     \\
x_{k - 1} \\
\vdots    \\
x_{1}     \\
x_{0}     \\
\vdots    \\
x_{-k + 1}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath345}%
\begin{displaymath}
Z_{N} =  \left[ \begin{array}{cccc}
0        \\
0        \\
\vdots    \\
0         \\
x_{N}     \\
\vdots    \\
x_{N - k + 1}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath353}%
\begin{displaymath}
Z_{0} =  \left[ \begin{array}{cccc}
x_{k}       \\
x_{k - 1}   \\
\vdots      \\
x_{1}       \\
0           \\
\vdots      \\
0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath362}%
\begin{displaymath}
\begin{array}{ccc}
\tau_{11}x_{k} + \tau_{12}x_{k-1} + \dots + \tau_{1 k}x_{1} & = & 0   \\
\tau_{21}x_{k} + \tau_{22}x_{k-1} + \dots + \tau_{2 k}x_{1} & = & 0  \\
\vdots   &  \vdots &   \vdots                   \\
\tau_{k-1}x_{k} + \tau_{k2} x_{k-1} + \dots + \tau_{k k} x_{1} & =
& 0
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline578}%
$\tau_{i j} = \tau_{i j} (\lambda) \ \ \ \ \
\ i, j = 1,2, \dots, k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline580}%
$x_{1}, x_{2}, \dots , x_{k}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath390}%
\begin{displaymath}
 \left \vert \begin{array}{cccc}
\tau_{11}      &  \tau_{12}     & \dots  & \tau_{1k}      \\
\tau_{21}      & \tau_{22}      & \dots  & \tau_{2k}      \\
\vdots         & \vdots         &        & \vdots         \\
\tau_{k1}      & \tau_{k2}      &        & \tau_{kk}
\end{array} \right \vert = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline584}%
$x_{k}, x_{k-1}, \dots, x_{1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline588}%
$T$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath409}%
\begin{displaymath}
T Z = \mu Z
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline594}%
$\mu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath414}%
\begin{displaymath}
\mu = e^{\varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline602}%
$\varphi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath420}%
\begin{displaymath}
T = \sum^{2k}_{i = 1} \ \lambda_{i} \ \frac{\arrowvert i
>< i \arrowvert}{< i \arrowvert i >}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath610}%
\begin{displaymath}
\frac{\arrowvert i >< i \arrowvert}{< i \arrowvert i>}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

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

{\newpage\clearpage
\lthtmldisplayA{displaymath431}%
\begin{displaymath}
T^{N} = \sum^{2k}_{i = 1} \ \lambda^{N}_{i} \ \ \frac{\arrowvert i
>< i \arrowvert}{< i \arrowvert i>}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline624}%
$Z$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6189}%
\includegraphics[width=5in]{f02p16.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath652}%
\begin{displaymath}
m_{i} \ddot{x}_{i} = -k_{i i -1} (x_{i} - x_{i-1}) -k_{i i + 1}
(x_{i} - x_{i + 1})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1192}%
\begin{displaymath}
k_{i i} = -k_{i i -1}  -k_{i i + 1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath666}%
\begin{displaymath}
m_{i} \ddot{x}_{i} = k_{i i -1} x_{i-1} + k_{i i} x_{i} + k_{i i +
1} x_{i + 1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath680}%
\begin{displaymath}
 \left [ \begin{array}{c}
m_{1} \ddot{x}_{1}  \\
m_{2} \ddot{x}_{2}  \\
\vdots              \\
m_{N} \ddot{x}_{N}
\end{array}  \right]
=  \left[ \begin{array}{cccc}
k_{11}   &  k_{12}   & \dots  &            \\
k_{21}   &  k_{22}   & \dots  &     0      \\
\ddots   &  \ddots   & \ddots &            \\
0        &           &        &   k_{NN}
\end{array} \right]
 \left[ \begin{array}{c}
x_{1}  \\
x_{2}  \\
\vdots \\
x_{N}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath715}%
\begin{displaymath}
\ddot{Y} = M^{-1/2} \;  K \;  M^{-1/2} \;  Y
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1202}%
$M^{-1/2} \;K \;
M^{-1/2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath722}%
\begin{displaymath}
\frac{k_{i j}}{\sqrt{m_{i} \  m_{j}}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath727}%
\begin{displaymath}
\frac{d^{2}}{d t^{2}} \left [ \begin{array}{c}
y_{1}  \\
y_{2}  \\
\vdots              \\
y_{N}
\end{array}  \right]
=  \left[ \begin{array}{cccc}
a_{11}   &  a_{12}   &        &            \\
a_{21}   &  a_{22}   & a_{23} &     0      \\
\ddots   &  \ddots   & \ddots &            \\
0        &           &        &   a_{NN}
\end{array} \right]
 \left[ \begin{array}{c}
y_{1}  \\
y_{2}  \\
\vdots \\
y_{N}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1206}%
$a_{11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1208}%
$a_{NN}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath754}%
\begin{displaymath}
a_{11} = -a_{12} -a_{10} \ \ a_{NN} = -a_{N,N-1} -a_{N,N+1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath762}%
\begin{displaymath}
a'_{11} = -a_{12} \ \qquad  \ \ a'_{NN} = -a_{N,N-1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1214}%
$Y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath771}%
\begin{displaymath}
A  Y = \lambda Y
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath773}%
\begin{displaymath}
\ddot{Y} = \lambda Y
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath781}%
\begin{displaymath}
\lambda \; y_{i} = a_{i, i -1}  \ y_{i - 1} \; + \; a_{i i}  \
y_{i} \; + \; a_{i, i + 1}  \ y_{i + 1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1224}%
$y_{i + 1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1226}%
\begin{displaymath}
y_{i + 1} = \frac{\lambda - a_{i i}}{a_{i, i + 1}} \ y_{i} \; - \;
\frac{a_{i, i -1}}{a_{i, i + 1}} \  y_{i - 1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath798}%
\begin{displaymath}
y_{i} = y_{i}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath802}%
\begin{displaymath}
 \left [ \begin{array}{c}
y_{i + 1}  \\
y_{i}
\end{array}  \right]
=  \left[ \begin{array}{cccc} \frac{\lambda - a_{i i}}{a_{i, i +
1}}   &  -\frac{a_{i,
i-1}}{a_{i, i + 1}}    \\
1   &  0
\end{array} \right]
 \left[ \begin{array}{c}
y_{i}  \\
y_{i - 1}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath821}%
\begin{displaymath}
T_{i} =  \left[ \begin{array}{cc}
\frac{\lambda - a_{i i}}{a_{i, i + 1}}   &   -\frac{a_{i, i -1}}{a_{i, i + 1}}      \\
1      &  0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1228}%
$y_{i}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1230}%
$y_{i - 1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1236}%
$i+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1240}%
\begin{displaymath}
y_{0} = y_{N + 1} = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar845}%
\begin{eqnarray*}
 Z_{N} &  =  &  T_{N} Z_{N - 1}  \\
Z_{N - 1} &  =  &  T_{N - 1} \ \ \ Z_{N - 2}   \\
  &  \vdots  &                         \\
Z_{1} &  =  &  T_{1} Z_{0}
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath856}%
\begin{displaymath}
 Z_{N} = T_{N} T_{N-1} \dots T_{1} Z_{0} = \tau Z_{0}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1246}%
$\tau_{11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath867}%
\begin{displaymath}
T_{i + 1} =  \left[ \begin{array}{cc}
\frac{\lambda - a_{i + 1, i + 1}}{a_{i + 1, i + 2}}   &   -\frac{a_{i + 1, i}}{a_{i + 1, i + 2}}      \\
       \\
1      &  0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath877}%
\begin{displaymath}
T_{i + 2} =  \left[ \begin{array}{cc}
\frac{\lambda - a_{i + 2, i + 2}}{a_{i + 2, i + 3}}   &   -\frac{a_{i + 2, i + 1}}{a_{i + 2, i + 3}}      \\
       \\
1      &  0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1252}%
$T_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1254}%
$T_{i+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath888}%
\begin{displaymath}
T_{i + 1} T_{i} = \left [ \begin{array}{ccc}
\frac{\lambda - a_{i + 1, i + 1}}{a_{i + 1, i + 2}}  & -\frac{a_{i + 1, i}}{a_{i + 1, i + 2}}  \\
           \\
1     &    0
\end{array}  \right]
 \left[ \begin{array}{ccc}
\frac{\lambda - a_{i i}}{a_{i, i + 1}}  & -\frac{a_{i, i - 1}}{a_{i, i + 1}}       \\
         \\
1   &   0   \\
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath906}%
\begin{displaymath}
=  \left[ \begin{array}{ccc}
 \frac{(\lambda - a_{i + 1, i +
1})(\lambda - a_{i i})}{a_{i + 1, i + 2} a_{i, i + 1}} -
\frac{a_{i + 1, i}}{a_{i + 1, i + 2}}  & -\frac{a_{i, i -
1}}{a_{i, i + 1}}
\frac{(\lambda - a_{i + 1, i + 1})}{a_{i + 1, i + 2}}      \\
      \\
\frac{\lambda - a_{i i}}{a_{i, i + 1}}       &     -\frac{a_{i, i
- 1}}{a_{i, i + 1}}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1256}%
$T_{i+2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath926}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{(\lambda - a_{i + 2, i + 2})(\lambda - a_{i + 1, i
+1})(\lambda - a_{i i})}{a_{i + 2, i + 3} \ \ \ a_{i + 1, i + 2} \
\ \ a_{i, i + 1}}    &    \frac{-a_{i, i + 1}(\lambda - a_{i + 1,
i + 1})(\lambda - a_{i + 2, i + 2})}{a_{i, i + 1} \ \ \
a_{i + 1, i + 2} \ \ \ a_{i + 2, i + 3}}                    \\
    \\
- \frac{a_{i + 1, i}}{a_{i + 1, i + 2}} \frac{(\lambda - a_{i + 2,
i + 2})}{a_{i + 2, i + 3}} -\frac{a_{i + 2, i + 1}(\lambda - a_{i
i})}{a_{i + 2, i + 3} a_{i, i + 1}}     &    + \frac{ a_{i + 2, i
+ 1}}{a_{i + 2, i + 2}}   \frac{a_{i, i - 1}}{a_{i, i + 1}}  \\
    \\
\frac{(\lambda - a_{i + 1, i + 1})(\lambda - a_{i i})} {a_{i + 1,
i+2} \ \ \ a_{i, i+1}} -\frac{a_{i + 1, i}}{a_{i + 1, i + 2}}    &
- \frac{a_{i, i - 1}(\lambda - a_{i + 1, i + 1})}{a_{i, i + 1} \ \
\ a_{i + 1, i + 2}}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1266}%
$Z_{n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath970}%
\begin{displaymath}
 \left [ \begin{array}{ccc}
0       \\
      \\
 y_{N}
\end{array}  \right]
=  \left[ \begin{array}{ccc}
\tau^{(N)}_{11}       &     \tau^{(N - 1)}_{12}        \\
     \\
\tau^{(N - 1)}_{21}   &     \tau^{(N - 2)}_{22}
\end{array} \right]
 \left[ \begin{array}{ccc}
y_{1}      \\
        \\
0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1270}%
$\tau^{(N)}_{11} \  y_{1} = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1272}%
$y_{1}
\neq 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath998}%
\begin{displaymath}
Z_{i} = T_{i} T_{i - 1} \dots T_{1} Z_{0} = \tau^{(i)} Z_{0}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1008}%
\begin{displaymath}
 \left [ \begin{array}{ccc}
y_{i + 1}       \\
       \\
y_{i}
\end{array}  \right]
=  \left[ \begin{array}{ccc}
\tau^{(i)}_{11}       &     \tau^{(i - 1)}_{12}        \\
      \\
\tau^{(i - 1)}_{21}   &     \tau^{(i - 2)}_{22}
\end{array} \right]
 \left[ \begin{array}{ccc}
y_{1}      \\
     \\
0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6304}%
$\displaystyle \begin{array}{ccc}
y_{i + 1}  &  =  &  \tau^{(i)}_{11} \ \ y_{1}             \\
&     &                                        \\
y_{i}      &  =  &  \tau^{(i - 1)}_{21} \ \  y_{1}
\end{array}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1274}%
$y_{1} = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1276}%
$y_{i + 1} = y_{i}
= 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1048}%
\begin{displaymath}
\arrowvert T_{i} \arrowvert = \frac{a_{i, i - 1}}{a_{i, i + 1}}
\neq  0 \ \ \ \ \ \ \ \mbox{para toda} \ i
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1054}%
\begin{displaymath}
Z_{0} = \tau^{(i)^{-1}} Z_{i}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1280}%
$\tau^{(i)^{-1}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1282}%
\begin{displaymath}
 \left [ \begin{array}{ccc}
y_{1}       \\
      \\
0
\end{array}  \right]
=  \left[ \begin{array}{ccc}
\tau^{(i)^{-1}}_{11}       &     \tau^{(i - 1)^{-1}}_{12}        \\
      \\
\tau^{(i - 1)^{-1}}_{21}   &     \tau^{(i - 2)^{-1}}_{22}
\end{array} \right]
 \left[ \begin{array}{ccc}
y_{i + 1}      \\
      \\
y_{i}
\end{array} \right]\eqno{(\mbox{II.24'})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1288}%
$y_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1087}%
\begin{displaymath}
\tau_{11} ^{N}(\lambda) = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1292}%
$\tau_{11} ^{N}(\lambda) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath2875}%
\begin{displaymath}
m_{i} = m    \qquad \qquad i = 1, 2, ..., N
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1348}%
\begin{displaymath}
\qquad
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2877}%
\begin{displaymath}
 k_{i j} = \left\{
\begin{array}{ccc}
  k  &  \qquad & i   \neq  j    \\
-2k  &  \qquad & i   =     j
\end{array} \right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1354}%
\begin{displaymath}
A = \frac{k}{m}  \left[ \begin{array}{cccccc}
a_{1}    &     1     &            &           &          \\
1        &    -2     &   1        &     0     &          \\
         &     1     &   -2       &     1     &          \\
  0      &  \ddots   &  \ddots    &   \ddots  &    1     \\
         &           &            &     1     &   a_{N}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2879}%
$a_{1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2881}%
$a_{N}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1365}%
\begin{displaymath}
\begin{array}{ccccc}
a_{1}  &  =   &  a_{N}  & = & -2  \ \ \ \ \ \mbox{extremos fijos}   \\
    \\
a_{1}  &  =   &  a_{N}  & = & -1  \ \ \ \ \ \mbox{extremos libres}   \\
    \\
    \\
    \\
a_{1} &   =   &  -1 \ \ a_{N}  & = & -2  \ \ \ \ \ \mbox{un extremo libre}  \\
    \\
a_{1} &   =   & -2 \ \ a_{N}   & = & -1  \ \ \ \ \ \mbox{y el otro
confinado}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath1383}%
\begin{displaymath}
T  \left[ \begin{array}{ccc}
\frac{\lambda - a_{0}}{a_{1}}      &     -1        \\
               1                   &     0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6370}%
$\displaystyle a_{0}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6371}%
$\textstyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6372}%
$\displaystyle \frac{-2k}{m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6375}%
$\displaystyle a_{1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay6377}%
$\displaystyle \frac{k}{m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1399}%
\begin{displaymath}
Z_{N} = T^{N} Z_{0}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1404}%
\begin{displaymath}
c = \frac{\lambda - a_{0}}{a_{1}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1409}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
c - \mu    &     -1   \\
 1         &     -\mu
\end{array} \right] = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2887}%
\begin{displaymath}
\mu^{2} - c \mu + 1 = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2889}%
\begin{displaymath}
\mu = \frac{c}{2} \pm \sqrt{\left(\frac{c}{2}\right)^{2} -1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1422}%
\begin{displaymath}
\frac{c}{2} = \mbox{cosh} \: \varphi
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2891}%
\begin{displaymath}
\mu = \mbox{cosh} \: \varphi \pm \sqrt{\mbox{cosh}^2 \: \varphi
-1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2893}%
\begin{displaymath}
\mbox{cosh}^2 \: \varphi - 1 = \mbox{senh}^2 \: \varphi
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2895}%
\begin{displaymath}
\mu = \mbox{cosh} \: \varphi \pm \ \mbox{senh} \: \varphi
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1433}%
\begin{displaymath}
\mu_\pm = e^{\pm \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2897}%
$\frac{\lambda - a_{0}}{2  \ a_{1}} \geq 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2901}%
$\varphi > 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2903}%
$\varphi < 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2905}%
$-1 \leq \frac{\lambda - a_{0}}{2 a_{1}} < 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2907}%
$\varphi = i \beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2909}%
$\frac{\lambda - a_{0}}{2a_{1}} < -1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2911}%
$\varphi = \alpha + i \pi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1446}%
\begin{displaymath}
 \left [ \begin{array}{ccc}
c     &    -1     \\1     &        0
\end{array}  \right]
 \left[ \begin{array}{cccc}
1               \\
u
\end{array} \right]
= \mu_\pm  \left[ \begin{array}{ccc}
1               \\
u
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2915}%
\begin{displaymath}
1 = \mu_\pm \ u
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2917}%
\begin{displaymath}
u_\pm = \frac{1}{\mu_\pm}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1460}%
\begin{displaymath}
u_\pm = e^{\mp \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1463}%
\begin{displaymath}
 \left [ \begin{array}{ccc}
v    &    1
\end{array}  \right]
\left[ \begin{array}{cccc}
c          &      -1     \\
1          &       0
\end{array} \right]
= \mu_\pm  \left[ \begin{array}{ccc} v    &    1
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2919}%
\begin{displaymath}
-v = \mu_\pm
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2921}%
\begin{displaymath}
v = -\mu_\pm
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1474}%
\begin{displaymath}
v_\pm = -e^{\pm \: \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2923}%
\begin{displaymath}
P_\pm = \frac{\arrowvert u_\pm >< v_\pm \arrowvert}{< v_\pm
\arrowvert u_\pm >}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1479}%
\begin{eqnarray*}
\arrowvert u_\pm >< v_\pm \arrowvert &=&  \left [
\begin{array}{c}
1           \\
u_\pm
\end{array}  \right]
 \left[ \begin{array}{ccc}
 v_\pm   &  1
\end{array} \right]                 \\
& = & \left[ \begin{array}{ccc}
v_\pm             &      1    \\
v_\pm \ u_\pm   &     u_\pm
\end{array} \right]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1490}%
\begin{eqnarray*}
< v_\pm \arrowvert u_\pm > &=&  \left [
\begin{array}{ccc}
v_\pm     &    1
\end{array}  \right]
 \left[ \begin{array}{cc}
  1       \\
 u_\pm
\end{array} \right]
\\
& = & v_\pm + \ u_\pm
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1498}%
\begin{displaymath}
 P _{+} = \frac{ \left[ \begin{array}{ccc}
-e^{\varphi}             &      1    \\
-1                       &    e^{-\varphi}
\end{array} \right]}{-e^{\varphi} + e^{-\varphi}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1508}%
\begin{displaymath}
 P _{+} = \frac{ \left[ \begin{array}{ccc}
 e^{\varphi}             &      -1    \\
 1                       &    -e^{-\varphi}
\end{array} \right]}{2 \mbox{senh} \ \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1517}%
\begin{displaymath}
 P _{-} = \frac{ \left[ \begin{array}{ccc}
-e^{-\varphi}            &      1    \\
-1                       &    e^{\varphi}
\end{array} \right]}{-e^{-\varphi} + e^{\varphi}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1527}%
\begin{displaymath}
 P _{-} = \frac{ \left[ \begin{array}{ccc}
-e^{-\varphi}            &      1    \\
-1                       &    e^{\varphi}
\end{array} \right]}{2 \mbox{senh} \ \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1536}%
\begin{displaymath}
T = \frac{e^{\varphi}}{2 \mbox{senh} \varphi}  \left[
\begin{array}{ccc}
e^{\varphi}        &    -1   \\
1                  &    -e^{-\varphi}
\end{array} \right]
+ \frac{e^{-\varphi}}{2 \mbox{senh} \varphi}  \left[
\begin{array}{ccc}
-e^{-\varphi}            &      1    \\
-1                       &    e^{\varphi}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1552}%
\begin{displaymath}
T^{N} = \frac{e^{N \varphi}}{2 \mbox{senh} \varphi}  \left[
\begin{array}{ccc}
e^{\varphi}        &    -1   \\
1                  &    -e^{-\varphi}
\end{array} \right]
+ \frac{e^{-N \varphi}}{2 \mbox{senh} \varphi}  \left[
\begin{array}{ccc}
-e^{-\varphi}            &      1    \\
-1                       &    e^{\varphi}
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1569}%
\begin{displaymath}
= \frac{1}{2 \mbox{senh} \varphi}  \left[
\begin{array}{ccc}
e^{(N + 1)\varphi}   -e^{-(N + 1)\varphi}   & -(e^{N \varphi}   -e^{-N \varphi})       \\
    \\
e^{N \varphi}   -e^{-N \varphi}           & -(e^{(N - 1)\varphi}
-e^{-(N - 1)\varphi})
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1584}%
\begin{displaymath}
T^{N} = \frac{1}{\mbox{senh } \varphi}  \left[
\begin{array}{ccc}
\mbox{senh}(N + 1)\varphi       & - \mbox{senh} \ N \ \varphi       \\
      \\
\mbox{senh} N \ \varphi         & - \mbox{senh} ( N - 1)\varphi
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1596}%
\begin{displaymath}
\frac{\mbox{senh}(N + 1) \varphi}{\mbox{senh} \: \varphi }= 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2925}%
\begin{displaymath}
\mbox{senh}\varphi \neq \infty
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2927}%
\begin{displaymath}
\mbox{senh} (N + 1) \varphi = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2929}%
\begin{displaymath}
(N + 1) \varphi = \ell \pi \ i \ \ \ \ \ \ \ell = 0, \pm 1, \pm 2,
..., \pm (N + 1)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1602}%
\begin{displaymath}
\varphi = \frac{\ell \pi \ i}{N + 1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1606}%
\begin{eqnarray*}
 \mbox{cosh} \varphi & = &  \mbox{cosh} \frac{\ell \pi \ i}{N + 1} \\
 & = & \mbox{cos} \frac{\ell \ \pi}{N + 1}
 \end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2933}%
\begin{displaymath}
\frac{\lambda - a_{0}}{2 a_{1}} = \mbox{cos}\frac{\ell \ \pi}{N +
1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1620}%
\begin{displaymath}
\lambda = a_{0} + 2 a_{1} \mbox{cos}\frac{\ell \ \pi}{N + 1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2939}%
$cos$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2941}%
$\ell = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2943}%
$\ell = N + 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2945}%
$\varphi = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2947}%
$\varphi = \pi \ i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2951}%
$1, 2, ...,N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1629}%
\begin{displaymath}
 \left [ \begin{array}{c}
y_{j + 1}           \\
y_{j}
\end{array}  \right]
= T^{j}  \left[ \begin{array}{cccc}
y_{1}              \\
0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2953}%
$T^{j}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1642}%
\begin{displaymath}
T^{j} = \frac{1}{\mbox{senh} \varphi}  \left[
\begin{array}{ccc}
\mbox{senh}(j + 1)\varphi       &    - \mbox{senh} \ j \ \varphi       \\
\mbox{senh} j \ \varphi         &    - \mbox{senh} (j - 1)\varphi
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2955}%
\begin{displaymath}
y_{j} = \frac{\mbox{senh} \ j \ \varphi}{\mbox{senh} \ \varphi} \:
\ y_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2957}%
\begin{displaymath}
y_{j} = \frac{\mbox{senh} (j \ell \ \pi \ i / N + 1)}
             {\mbox{senh} (\ell \ \pi  \ i / N + 1)}  \:y_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2959}%
\begin{displaymath}
y_{j} = \frac{\mbox{sen}\: j \ell \: \pi \ / N + 1}{\mbox{sen
}\ell \: \pi \: / N + 1}  \:y_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2961}%
\begin{displaymath}
y_{1} = \mbox{sen} (\ell \: \pi / N + 1)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1668}%
\begin{displaymath}
y_{j} = \mbox{sen} (\ j \ \ell \ \pi / N + 1)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath1674}%
\begin{displaymath}
T_{1} = T_{N} =   \left[ \begin{array}{ccc}
\frac{\lambda - a}{a_1}        &    -1       \\
1                              &    0
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2971}%
\begin{displaymath}
a = -\frac{k}{m}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2973}%
$a_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1685}%
\begin{displaymath}
Z_N = T_N \ T^{N - 2} \ T_1 \ Z_0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2975}%
\begin{displaymath}
\begin{array}{ccl}
Z_{N} & = & T_{n} \: T^{-1} \: T \: T^{N-2} \: T \: T^{-1} \: T_{1} \: T_{0}    \\
      &   &                                                \\
      & = & T_{N} \: T^{-1} \: T^{N} \: T^{-1} \: T_{1} \: Z_{0}
\end{array}    \eqno{(\mbox{III.22}')}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1705}%
\begin{displaymath}
T = \left [ \begin{array}{cc}
\frac{\lambda - a_{0}}{a_{1}}     &    -1     \\
      \\
1                                 &    0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2977}%
$T^{-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1714}%
\begin{displaymath}
\left [ \begin{array}{cc}
0     &    1     \\
    \\
-1     &    \frac{\lambda - a_{0}}{a_{1}}
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2981}%
$T^{N}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2983}%
$T^{-1} T_{1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2985}%
$ T_{N} T^{-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1726}%
\begin{eqnarray*}
T^{-1} T_{1} & = & \left [ \begin{array}{cc}
0     &    1     \\
    \\
-1     &    \frac{\lambda - a_{0}}{a_{1}}
\end{array}  \right]
\left[ \begin{array}{cc}
\frac{\lambda - a}{a_{1}}     &    -1     \\
     \\
1                                 &        0
\end{array} \right]
 =  \left[ \begin{array}{cc}
1       &    0             \\
     \\
\frac{a - a_{0}}{a_{1}}    &     1
\end{array} \right]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1745}%
\begin{eqnarray*}
T_{N} T^{-1} & = & \left [ \begin{array}{cc}
\frac{\lambda - a}{a_{1}}      &    -1     \\
   \\
1                             &     0
\end{array}  \right]
\left[ \begin{array}{cc}
0           &       1     \\
    \\
-1          &    \frac{\lambda - a_{0}}{a_{1}}
\end{array} \right]
 =  \left[ \begin{array}{cc}
1       &    -\frac{a - a_{0}}{a_{1}}     \\
    \\
0       &     1
\end{array} \right]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2987}%
$T^{N} T^{-1}
T_{1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1767}%
\begin{eqnarray*}
T^{N} T^{-1} T_{1} & = & \frac{1}{\mbox{senh} \varphi} \left [
\begin{array}{cc}
\mbox{senh}(N + 1)\varphi      &    -\mbox{senh} N \varphi     \\
    \\
\mbox{senh} N \varphi          &    -\mbox{senh}(N - 1)\varphi
\end{array}  \right]
\left[ \begin{array}{cc}
1                         &       0     \\
      \\
\frac{a - a_{0}}{a_{1}}   &    1
\end{array} \right]       \\
&  & \\
 & = & \frac{1}{\mbox {senh} \varphi}  \left[
\begin{array}{cc} \mbox{senh}(N + 1)\varphi - \frac{a -
a_{0}}{a_{1}}
\mbox{senh} N \varphi   &    -\mbox{senh} N \varphi     \\
      \\
\mbox{senh}(N)\varphi - \frac{a - a_{0}}{a_{1}} \mbox{senh}(N -
1)\varphi       &    -\mbox{senh}(N - 1)\varphi
\end{array} \right]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1803}%
\begin{eqnarray*}
\tau & = & \left [\begin{array}{cc}
1      &   - \frac{a - a_{0}}{a_{1}}     \\
     \\
0          &    1
\end{array}  \right]
\left[ \begin{array}{cc} \mbox{senh}(N + 1)\varphi - \frac{a -
a_{0}}{a_{1}}
\mbox{senh} N \varphi   &  -\mbox{senh} N \varphi          \\
      \\
\mbox{senh} N \varphi -[(a - a_{0})/a_{1}]\mbox{senh} (N -
1)\varphi     & -\mbox{senh} (N - 1)\varphi
\end{array} \right] \frac{1}{\mbox{senh} \varphi} \\
                       &                       &                   \\
& = & \frac{1}{\mbox {senh} \varphi}  \left[ \begin{array}{cc}
\mbox{senh}(N + 1)\varphi - \frac{2(a - a_{0})}{a_{1}}
 \mbox{senh} N \varphi       &    -\mbox{senh} N \varphi     \\
     \\
+ \frac{(a - a_{0})^{2}}{a^{2}_{1}} \mbox{senh} (N - 1)\varphi  &
\qquad \quad + \frac{(a - a_{0})}{a_{1}} \mbox{senh}(N - 1)\varphi         \\
     \\
\mbox{senh} N \varphi - \frac{(a - a_{0})}{a_{1}} \mbox{senh} (N -
1) \varphi       & -\mbox{senh} (N - 1)\varphi
\end{array} \right]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2991}%
\begin{displaymath}
\frac{\mbox{senh}(N + 1)\varphi - \frac{2(a - a_{0})}{a_{1}}
\mbox{senh} N \varphi + \frac{(a -
a_{0})^{2}}{a^{2}_{1}}\mbox{senh} (N - 1) \varphi}{\mbox{senh}
\varphi} = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2993}%
$\varphi \neq \infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1858}%
\begin{displaymath}
\mbox{senh} (N + 1)\varphi - \frac{2(a - a_{0})}{a_{1}}
\mbox{senh} N \varphi + \frac{(a - a_{0})^{2}}{a^{2}_{1}}
\mbox{senh} (N - 1) \varphi = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath1873}%
\begin{displaymath}
 \tau = T^{N}  T^{-1}  T_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1879}%
\begin{displaymath}
\tau = \frac{1}{\mbox{senh} \varphi} \left[ \begin{array}{ccc}
\mbox{senh} (N + 1) \ \varphi - \frac{(a - a_{0})}{a_{1}}
\ \mbox{senh} N \varphi  & - \mbox{senh} N \varphi        \\
                         &                                \\
\mbox{senh} N \ \varphi - \frac{(a - a_{0})}{a_{1}} \ \mbox{senh}
(N - 1) \varphi   &   \qquad -\mbox{senh} (N - 1) \varphi
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1896}%
\begin{displaymath}
\mbox{senh} (N + 1) \varphi - \frac{(a - a_{0})}{a_{1}} \
\mbox{senh} N \varphi = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1904}%
\begin{displaymath}
\tau = T_{N}  T^{-1}  T^{N}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar1912}%
\begin{eqnarray*}
\tau & = & \left[ \begin{array}{ccc}
1         &        \frac{-(a - a_{0})}{a_{1}}     \\
    \\
0         &           1
\end{array}  \right]
 \left[ \begin{array}{ccccccc}
\mbox{senh} (N + 1) \varphi     &   -\mbox{senh} \ \ \ N \varphi           \\
     \\
\mbox{senh} \ N \varphi     &   -\mbox{senh} (N - 1) \varphi
\end{array}  \right] \frac{1}{\mbox{senh} \ \varphi} \\
       &      &    \\
& = & \frac{1}{\mbox{senh} \varphi}  \left[ \begin{array}{ccc}
\mbox{senh} (N + 1) \varphi  \frac{-(a - a_{0})}{a_{1}} \
\mbox{senh} \ N \varphi   & \qquad
-\mbox{senh} \ N \varphi + \frac{(a - a_{0})}{a_{1}} \ \mbox{senh}(N - 1) \varphi      \\
     \\
\mbox{senh} \ \ N \varphi      &    -\mbox{senh} (N - 1) \varphi
\end{array}  \right]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3009}%
$m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1956}%
\begin{displaymath}
a = \frac{k}{M}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1961}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{\lambda + 2a}{a}    &    -1     \\
      \\
1                        &     0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3021}%
$L$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3023}%
$N-L-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6550}%
\includegraphics[width=5in]{f03p43.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1975}%
\begin{displaymath}
\tau \left[ \begin{array}{ccc}
\frac{\lambda - a_{0}}{a_{1}}      &     -1       \\
      \\
1                          &      0
\end{array}  \right]^{N - L - 1}
 \left[ \begin{array}{ccc}
\frac{\lambda + 2a}{a}     &       -1        \\
     \\
1                          &        0
\end{array}  \right]
 \left[ \begin{array}{ccc}
\frac{\lambda - a_{0}}{a_{1}}     &     -1     \\
     \\
1                                 &      0
\end{array}  \right]^{L}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1994}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{\lambda + 2a}{a}      &      -1       \\
     \\
1         &       0
\end{array}  \right]
=  \left[ \begin{array}{cc}
\frac{\lambda - a_{0}}{a_{1}}       &       -1          \\
     \\
1         &       0
\end{array}  \right]
+  \left[ \begin{array}{cc}
\varepsilon       &       0        \\
     \\
0                 &       0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2009}%
\begin{displaymath}
\varepsilon   =  \lambda(\frac{1}{a} - \frac{1}{a_{1}})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2015}%
\begin{displaymath}
\tau =  \left[ \begin{array}{cc}
\frac{\lambda - a_0}{a_1}    &     -1       \\
                    &                   \\
1                            &      0
\end{array}  \right]^{N}
+ \varepsilon  \left[ \begin{array}{cc}
\frac{\lambda - a_0}{a_1}     &    -1             \\
                    &                   \\
1                             &     0
\end{array}  \right]^{N - L - 1}
 \left[ \begin{array}{cc}
1                   &     0             \\
                    &                   \\
0                   &     0
\end{array}  \right]
\left[ \begin{array}{cc}
\frac{\lambda - a_0}{a_1}      &     -1   \\
                    &                   \\
           1                   &     0
\end{array}  \right]^{L}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3025}%
\begin{displaymath}
\frac{\lambda - a_{0}}{2a_{1}} = \mbox{cosh} \ \varphi
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2041}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{\lambda - a_0}{a_1}        &       -1       \\
               &                \\
1                   &        0
\end{array}  \right]^{N - L - 1}
= \frac{1}{\mbox{senh} \ \varphi}  \left[ \begin{array}{ccc}
\mbox{senh} \ (N - L) \varphi \qquad      &    -\mbox{senh} \ (N - L - 1) \varphi        \\
                    &                 \\
\mbox{senh} \ (N - L - 1) \varphi  &    -\mbox{senh} \ (N - L - 2)
\varphi
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2058}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{\lambda - a_0}{a_1}         &        -1      \\
               &                \\
1                   &        0
\end{array}  \right]^{L}
= \frac{1}{\mbox{senh} \ \varphi}  \left[ \begin{array}{ccc}
\mbox{senh} \ (L + 1) \varphi      &    -\mbox{senh} \ L \ \varphi        \\
                    &                 \\
\mbox{senh} \ L \ \varphi  &    -\mbox{senh} \ (L - 1)\varphi
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2075}%
\begin{displaymath}
\tau = \frac{1}{\mbox{senh}} \ \varphi \left[ \begin{array}{cc}
\mbox{senh} \ (N + 1) \varphi    &     -\mbox{senh} \ N \ \varphi     \\
     \\
\mbox{senh} \ N \ \varphi      &     -\mbox{senh} \ (N + 1)
\varphi
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2086}%
\begin{displaymath}
\frac{+ \epsilon}{\mbox{senh}^{2} \ \varphi}  \left[
\begin{array}{cc}
\mbox{senh} (N-L)\varphi \ \mbox{senh}(L+1)\varphi  &  -\mbox{senh}(N-L) \ \varphi \ \mbox{senh} L \ \varphi  \\
     \\
\mbox{senh} (N-L-1)\varphi \ \mbox{senh}(L+1)\varphi   &
-\mbox{senh}(N-L-1) \ \varphi \ \mbox{senh} L \ \varphi
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2102}%
\begin{displaymath}
\frac{\mbox{senh} (N+1)\varphi}{\mbox{senh} \ \varphi} +
\frac{\epsilon}{\mbox{senh}^{2} \ \varphi} \
\mbox{senh}(N-L)\varphi \ \mbox{senh}(L+1) \varphi = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2112}%
\begin{displaymath}
\frac{\mbox{senh}(N+1)\varphi}{\frac{\mbox{senh}(N-L)\varphi \
\mbox{senh}(L+1)\varphi}{\mbox{senh}   \varphi}} = - \epsilon
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3031}%
$a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3035}%
\begin{displaymath}
\frac{\mbox{senh}(N+1)\varphi}{\frac{\lambda \
\mbox{senh}(N-L)\varphi \  \mbox{senh}(L+1)\varphi}{\mbox{senh} \
\varphi}} = - \frac{M-m}{k}  \eqno{(\mbox{III.33}')}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3041}%
$'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6582}%
\includegraphics[width=5in]{f04p47.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3049}%
$M \rightarrow \infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3051}%
$N - 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3053}%
$M = m$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3055}%
$M \rightarrow 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3057}%
$M = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3059}%
$m/k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3061}%
$k_{i}=
k/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2143}%
\begin{displaymath}
 \left[ \begin{array}{cc}
\frac{M}{k}\lambda + 2      &   -1       \\
     \\
1                            &    0
\end{array}  \right]
 \left[ \begin{array}{cc}
\frac{M}{k}\lambda + 2      &  -1          \\
     \\
1                           &   0
\end{array}  \right]
 \left[ \begin{array}{cc}
\frac{m}{k}\lambda + 2      &    -1        \\
     \\
1                           &     0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2161}%
\begin{displaymath}
 \left[ \begin{array}{cc}
\frac{m}{k}\ \ \lambda+ \frac{1}{2}  +  1      &   -\frac{1}{2}       \\
    \\
1                            &    0
\end{array}  \right]
 \left[ \begin{array}{cc}
\frac{2m}{k}\lambda + 2 +1      &     -2          \\
     \\
1                           &   0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3065}%
$k/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6601}%
\includegraphics[width=4in,angle=-90]{f05p50.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3071}%
$x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3073}%
$z$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6607}%
\includegraphics[width=3.7in]{f06p52.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3075}%
$K\rightarrow 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3077}%
$K\rightarrow
\infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2209}%
\begin{displaymath}
             K = k (1 + \epsilon)
      \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2213}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{\frac{\lambda-a_{0}}{a_{1}} + \epsilon}
{1+\epsilon}          &       - \frac{1}{1 + \epsilon}      \\
                             &        \\
1                 &         0
\end{array}  \right] \qquad \mbox{para la  } i
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2224}%
\begin{displaymath}
 \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}} + \epsilon         &       -(1+\epsilon)     \\
                   &        \\
1                 &         0
\end{array}  \right] \qquad \mbox{para  } i + 1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2234}%
\begin{displaymath}
T_{i} = \frac{1}{1+\epsilon}   \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}}      &       -1     \\
                   &        \\
1                 &         0
\end{array}  \right]
+  \frac{\varepsilon}{1 + \epsilon} \left[ \begin{array}{ccc}
1      &       0    \\
1      &       0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2249}%
\begin{displaymath}
T_{i + 1} =  \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}}      &       -1     \\
                   &        \\
1                 &         0
\end{array}  \right]
+ \: \epsilon  \left[ \begin{array}{ccc}
1      &       -1    \\
       &             \\
 0      &       0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2263}%
\begin{displaymath}
T_{i + 1} \ \ T_{i} =  \left(\left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}}      &       -1     \\
                   &        \\
1                 &         0
\end{array}  \right]
+ \epsilon  \left[ \begin{array}{ccc}
1      &       -1    \\
       &             \\
 0      &       0
\end{array}  \right]\right) \cdot
\left( \frac{1}{1+\epsilon} \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}}      &       -1     \\
                   &        \\
1                 &         0
\end{array}  \right]
+ \frac{\epsilon}{1+\epsilon}  \left[ \begin{array}{ccc}
1      &       0    \\
       &             \\
 1      &       0
\end{array}  \right]\right)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2287}%
\begin{displaymath}
\quad =  \frac{1}{1+\epsilon} \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}}      &       -1     \\
                   &        \\
1                 &         0
\end{array}  \right]^{2}
+ \frac{\epsilon}{1+\epsilon}  \left[ \begin{array}{ccc}
\frac{\lambda - a_{0}}{a_{1}} -1      &       0    \\
       &             \\
 1      &       0
\end{array}  \right]
+  \frac{\epsilon}{1+\epsilon} \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}} -1       &       -1     \\
                   &        \\
0                 &         0
\end{array}  \right]
+ \frac{\epsilon^{2}}{1+\epsilon}  \left[ \begin{array}{ccc}
0      &       0    \\
       &             \\
 0      &       0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2316}%
\begin{displaymath}
T_{i+1} \ T_{i} = \frac{1}{1+\epsilon} \left[ \begin{array}{ccc}
\frac{\lambda-a_{0}}{a_{1}}     &       -1     \\
                   &        \\
1                &         0
\end{array}  \right]
+ \frac{\epsilon}{1+\epsilon}  \left[ \begin{array}{ccc}
\frac{2(\lambda-a_{0})}{a_{1}}-2     &       -1    \\
       &             \\
 1      &       0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3091}%
$T^{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3093}%
$Q$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3095}%
\begin{displaymath}
T_{i+1} \: T_{i} = \frac{1}{1 + \epsilon} (T^{2} + \epsilon  Q)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3099}%
\begin{displaymath}
\tau = T^{N-i-1} \left[\frac{1}{1+\epsilon}(T^{2} + \epsilon
Q)\right]
 T^{i-1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2346}%
\begin{displaymath}
\tau = \frac{1}{1+\epsilon} \ T^{N} + \frac{\epsilon}{1+\epsilon}
\ T^{N-i-1} Q \ T^{i-1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2355}%
\begin{displaymath}
\frac{T_{11}^{N}}{Q_{11}} = -\epsilon
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3101}%
$T^{N}_{11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3105}%
$Q_{11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3107}%
$T^{N-i-1} \ Q \ T^{i-1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3115}%
$\epsilon$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6649}%
\includegraphics[width=4.4in]{f07p57.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3117}%
$\epsilon
\rightarrow \infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3121}%
$\epsilon = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3123}%
$\epsilon \rightarrow -1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6657}%
\includegraphics[width=4in,angle=-90]{f08p59.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3127}%
$\epsilon\rightarrow 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6664}%
\includegraphics[width=3.8in]{f09p61.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6666}%
\includegraphics[width=4.7in]{f10p62.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2398}%
\begin{displaymath}
\left[ \begin{array}{cccc}
\frac{-2k}{M}        &  \frac{k}{\sqrt{Mm}}   &    0                    \\
                     &                        &                         \\
\frac{k}{\sqrt{Mm}}  &  \frac{-2k}{m}         &  \frac{k}{\sqrt{Mm}}    \\
                     &                        &                         \\
0                    &        \ddots          &     \ddots
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2415}%
\begin{displaymath}
\begin{array}{ccc}
\lambda x_{2i} & = &  \frac{k}{\sqrt{Mm}} x_{2i-1} -\frac{2k}{m}
x_{2i} + \frac{k}{\sqrt{Mm}} x_{2i+1}   \\
   \\
\lambda x_{2i + 1} & = &  \frac{k}{\sqrt{Mm}} x_{2i} -\frac{2k}{M}
x_{2i+1} + \frac{k}{\sqrt{Mm}} x_{2i + 2}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2440}%
\begin{displaymath}
\left[ \begin{array}{cccc}
(\lambda + \frac{2k}{m}) \frac{\sqrt{Mm}}{k}      &   -1     \\
                                                 &          \\
1                                                &    0
\end{array}  \right] \qquad \mbox{part\'{\i}culas \ pares}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2450}%
\begin{displaymath}
\; \; \; \left[ \begin{array}{cccc}
(\lambda + \frac{2k}{m}) \frac{\sqrt{Mm}}{k}      &   -1     \\
                                                 &          \\
1                                                &    0
\end{array}  \right] \qquad  \mbox{part\'{\i}culas \ impares}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2461}%
\begin{displaymath}
\xi = \frac{\lambda \sqrt{Mm}}{2k} \ \ \ \ \ \ \ \ \rho =
\frac{\sqrt{M}}{\sqrt{m}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3131}%
$\xi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3133}%
$\rho^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2467}%
\begin{displaymath}
\alpha = \left(\lambda + \frac{2k}{m}\right) \frac{\sqrt{Mm}}{k} \
\qquad \qquad \beta = \left(\lambda + \frac{2k}{M} \right)
\frac{\sqrt{Mm}}{k}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3137}%
$\rho$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3139}%
\begin{displaymath}
\alpha = 2(\xi + \rho) \qquad \qquad \beta = 2(\xi +
\frac{1}{\rho})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2479}%
\begin{displaymath}
T_{D} =  \left[ \begin{array}{cc}
\alpha       &      -1    \\
             &            \\
1            &       0
\end{array}  \right]
 \left[ \begin{array}{cc}
\beta    & -1        \\
         &           \\
 1        &  0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2488}%
\begin{displaymath}
T_{D} =  \left[ \begin{array}{cc}
\alpha  \beta - 1      &      -\alpha    \\
           &                     \\
\beta            &       -1
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3141}%
$T_D$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2495}%
\begin{displaymath}
\left[ \begin{array}{cc}
\alpha  \beta - 1  -\mu    &      -\alpha    \\
                 &                  \\
\beta            &       -1 -\mu
\end{array}  \right] = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3143}%
\begin{displaymath}
(\mu +1)[(\mu + 1) - \alpha \beta] + \alpha \beta = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3145}%
\begin{displaymath}
(\mu + 1)^{2} - \alpha  \beta (\mu + 1) + \alpha \beta = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3147}%
$\alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3149}%
$\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2502}%
\begin{displaymath}
\mu^{2} + (2 - \alpha  \beta)\mu + 1 = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3159}%
$\alpha  \beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2505}%
\begin{displaymath}
\alpha  \beta = 2(\xi + \rho) \cdot 2(\xi + \frac{1}{\rho}) =
4\left[\xi^{2} + \xi(\rho + \frac{1}{\rho}) + 1\right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2512}%
\begin{displaymath}
P = \frac{1}{2}\left(\rho + \frac{1}{\rho}\right)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3163}%
$P$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3165}%
\begin{displaymath}
P = \frac{1}{2} \left(\frac{\sqrt{M}}{\sqrt{m}} +
\frac{\sqrt{m}}{\sqrt{M}}\right) = \frac{1}{2} \ \frac{M +
m}{\sqrt{m}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3173}%
\begin{displaymath}
\mu^{2} + (2 - 4 \xi^{2} - 8 P \xi - 4) \mu + 1 = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3175}%
\begin{displaymath}
\mu^{2} - (4 \xi^{2} + 8 P \xi + 2)\mu + 1 = 0
\eqno{(\mbox{III.42'})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3177}%
\begin{displaymath}
\mu = \frac{\alpha  \beta - 2}{2} \pm \frac{\sqrt{(\alpha  \beta -
2)^{2} - 4}}{2}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3179}%
\begin{displaymath}
\mu = \frac{\alpha  \beta - 2}{2} \pm \frac{\sqrt{\alpha
\beta(\alpha  \beta - 4)}}{2}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3181}%
\begin{displaymath}
\mu = \mu (\xi, P)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3185}%
\begin{displaymath}
\alpha  \beta (\alpha  \beta - 4) = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3191}%
\begin{displaymath}
[4(\xi^{2} + 2 \xi  P + 1)][4 \xi^{2} + 8 \xi  P + 4 - 4] = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3193}%
\begin{displaymath}
(\xi^{2} + 2 P \xi + 1)(\xi^{2} + 2 P \xi) = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath2546}%
\begin{displaymath}
 \xi^{2} + 2 P \xi + 1 = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3195}%
\begin{displaymath}
\xi (\xi + 2 P) = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3197}%
\begin{displaymath}
\xi = 0      \eqno{(\mbox{III.46a})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3199}%
\begin{displaymath}
\xi + 2P = 0    \eqno{(\mbox{III.46b})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\addtocounter{equation}{1}
{\newpage\clearpage
\lthtmldisplayA{displaymath2554}%
\begin{displaymath}
P = \frac{\mu^{2} - (4 \xi^{2} + 2) \mu + 1}{8 \xi  \mu}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3211}%
$\xi = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3213}%
\begin{displaymath}
\mu^{2} - 4 \mu  \xi^{2} - 2 \mu + 1 = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3215}%
\begin{displaymath}
\xi = \pm \frac{\mu - 1}{2\sqrt{\mu}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6752}%
\includegraphics[width=4.8in]{f11p69.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3231}%
$(P, \xi)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3233}%
\begin{displaymath}
P = 1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3249}%
$\mu < 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3251}%
$\xi = -2 P$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3253}%
$\mu > 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6771}%
\includegraphics[width=4in,angle=-90]{f12p71.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3259}%
$1/2.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3261}%
$2.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6778}%
\includegraphics[width=3.7in]{f13p73.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3263}%
$3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3265}%
$6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6786}%
\includegraphics[width=4in,angle=-90]{f14p75.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6790}%
\includegraphics[width=3.7in]{f15p76.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6794}%
\includegraphics[width=4in,angle=-90]{f16p78.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6798}%
\includegraphics[width=3.7in]{f17p79.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath3271}%
\begin{displaymath}
m_{i} = m_{0} + i_{\epsilon}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6804}%
\includegraphics[width=4in,angle=-90]{f18p81.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6808}%
\includegraphics[width=3.7in]{f19p82.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath3273}%
\begin{displaymath}
\frac{m_{i}}{m_i + 1} = \epsilon
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6814}%
\includegraphics[width=4in,angle=-90]{f20p84.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6818}%
\includegraphics[width=3.7in]{f21p85.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6823}%
\includegraphics[width=4in,angle=-90]{f22p87.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6827}%
\includegraphics[width=3.7in]{f23p88.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{chapter}
\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath4565}%
\begin{displaymath} \arrowvert T_{i}
\arrowvert = \frac{a_{i, i-1}}{a_{i, i+1}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3385}%
\begin{displaymath}
                 \arrowvert T_{i} \arrowvert = 1
        \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3388}%
\begin{displaymath}
T =  \left[ \begin{array}{cc}
a        &     b     \\
c        &     d
\end{array} \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3393}%
\begin{displaymath}
\left \arrowvert \begin{array}{cc}
a        &     b     \\
c        &     d
\end{array}  \right \arrowvert = ad - bc = 1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3399}%
\begin{displaymath}
\left \arrowvert \begin{array}{cc}
a - \mu      &     b     \\
c        &     d - \mu
\end{array}  \right \arrowvert = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4569}%
\begin{displaymath}
\mu^{2} - (a + d) \mu + (ad - bc) = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4571}%
\begin{displaymath}
\mu^{2} - (a + d)\mu + 1 = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4573}%
\begin{displaymath}
\mu = \frac{a + d}{2} \pm \sqrt{\left(\frac{a + d}{2}\right)^{2} -
1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4575}%
\begin{displaymath}
\frac{a + d}{2} = \mbox{cosh} \ \varphi
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3420}%
\begin{displaymath}
 \left[ \begin{array}{cc}
a      &    b  \\
c      &    d
\end{array}  \right]
 \left[ \begin{array}{cc}
1           \\
\nu
\end{array}  \right] = \mu_\pm
 \left[ \begin{array}{c}
1         \\
\nu
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4577}%
\begin{displaymath}
\nu_\pm = \frac{\mu_\pm - a}{b}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3433}%
\begin{displaymath}
 \left[ \begin{array}{cc}
\upsilon   &   1
\end{array}  \right]
 \left[ \begin{array}{ccccccc}
a        &    b          \\
c        &    d
\end{array}  \right]
= \mu_\pm  \left[ \begin{array}{cc} \upsilon  &  1
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4579}%
\begin{displaymath}
\upsilon_\pm = \frac{\mu_\pm - \alpha}{b}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3448}%
\begin{displaymath}
P_{\pm} = \frac{\left[ \begin{array}{ccc}
\upsilon_{\pm}                  &        1     \\
\nu_{\pm} \upsilon_{\pm}     &        \nu_{\pm}
\end{array}  \right]}{\nu_{\pm} + \upsilon_{\pm}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4583}%
\begin{displaymath}
\nu_{\pm} + \upsilon_{\pm} = \frac{\mu_{\pm} - a}{b} +
\frac{\mu_{\pm} - d}{b} = \frac{2\mu_{\pm} - (a + d)}{b}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4585}%
\begin{displaymath}
a + d = e^{\varphi} + e^{-\varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4587}%
\begin{displaymath}
\nu_{\pm} + \upsilon_{\pm} = \frac{2e^{\pm \varphi} - e^{\varphi}
- e^{-\varphi}}{b}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4589}%
\begin{displaymath}
\nu_{\pm} + \upsilon_{\pm} = \pm \frac{2 \mbox{senh} \varphi}{b}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4591}%
\begin{displaymath}
\nu_{\pm} \upsilon_{\pm} = \frac{(\mu_{\pm} - a)(\mu_{\pm} -
d)}{b^{2}} = \frac{\mu_{\pm}^{2} - (a + d) \mu_{\pm} + ad}{b^{2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4593}%
\begin{displaymath}
\qquad = \frac{e^{\pm 2\varphi} - e^{\varphi} e^{\pm\varphi} -
e^{-\varphi} e^{\pm\varphi} + ad}{b^{2}} = \frac{ad - 1}{b^{2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4597}%
\begin{displaymath}
\nu_{\pm} \upsilon_{\pm} = \frac{ad - ad + bc}{b^{2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4599}%
\begin{displaymath}
\nu_{\pm} \upsilon_{\pm} = \frac{c}{b}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3505}%
\begin{displaymath}
P_{\pm} = \pm \frac{b}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\frac{\mu_{\pm} - d}{b}     &     1    \\
      \\
\frac{c}{b}                 &     \frac{\mu_{\pm}- a}{b}
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3519}%
\begin{displaymath}
P_{\pm} = \pm \frac{1}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\mu_{\pm} - d     &     b    \\
c                 &     \mu_{\pm}- a
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3530}%
\begin{displaymath}
T = \frac{\mu_{+}}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\mu_{+} - d     &     b    \\
     \\
c               &     \mu_{+}- a
\end{array}  \right]
- \frac{\mu_{-}}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\mu_{-} - d     &     b    \\
    \\
c               &     \mu_{-}- a
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4601}%
$LnT$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3546}%
\begin{displaymath}
L_{n}  T = \frac{\varphi}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\mu_{+} - d     &     b    \\
c               &     \mu_{+} - a
\end{array}  \right]
+ \frac{\varphi}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\mu_{-} - d     &     b    \\
c               &     \mu_{-} - a
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3563}%
\begin{displaymath}
L_{n}  T = \frac{\varphi}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\mu_{+} + \mu_{-} - 2d     &     2b    \\
2c               &     \mu_{+} + \mu_{-} - 2a
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4603}%
\begin{displaymath}
\mu_{+} + \mu_{-} = a + d
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3577}%
\begin{displaymath}
L_{n} \ T = \frac{\varphi}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
a - d     &     2b    \\
2c               &     d - a
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3585}%
\begin{displaymath}
L_{n} \ T = \frac{\varphi}{2\mbox{senh} \ \varphi}  \left[
\begin{array}{ccc}
\frac{a - d}{2}     &     b    \\
    \\
c               &     \frac{d - a}{2}
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3597}%
\begin{displaymath}
I =  \left[\begin{array}{ccc}
1     &     0    \\
0     &     1
\end{array}  \right]
\ \ \ \ \ \ \ J =  \left[\begin{array}{ccc}
0     &     1    \\
1     &     0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3605}%
\begin{displaymath}
K =  \left[\begin{array}{ccc}
1     &     0    \\
0     &     -1
\end{array}  \right]
\ \ \ \ \ \ \ L =  \left[\begin{array}{ccc}
0     &     1    \\
-1     &     0
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3613}%
\begin{displaymath}
L_{n}  \left[\begin{array}{ccc}
a     &     b    \\
c     &     d
\end{array}  \right] = \frac{\varphi}{\mbox{senh} \ \varphi} \left[\begin{array}{ccc}
\frac{(b + c)}{2} J + \frac{(a - d)}{2} K + \frac{(b - c)}{2} L
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3630}%
\begin{displaymath}
U = \frac{(b + c)}{2} \ J + \frac{(a - d)}{2} \ K + \frac{(b -
c)}{2} \ L = \left[\begin{array}{ccc}
\frac{a - d}{2}     &     b    \\
     \\
c     &     \frac{d - a}{2}
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4605}%
$Tr (U) = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4607}%
$\mbox{cosh} \ \varphi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4609}%
\begin{displaymath}
\mbox{senh} \ \varphi = \sqrt{\left(\frac{a + d}{2}\right)^{2} -
1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4611}%
\begin{displaymath}
\mbox{senh} \ \varphi = \sqrt{\left(\frac{a + d}{2}\right)^{2} -
ad + bc}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3655}%
\begin{displaymath}
\mbox{senh} \ \varphi = \sqrt{\left(\frac{b + c}{2}\right)^{2} +
\left(\frac{a - d}{2}\right)^{2} - \left(\frac{b -
c}{2}\right)^{2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4613}%
$U$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3667}%
\begin{displaymath}
\widehat{U} = \frac{U}{\mbox{senh}  \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3672}%
\begin{displaymath}
\left[ \begin{array}{ccc}
 a   &   b     \\
 c   &   d
\end{array}  \right] = e^{\varphi  \widehat{U}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4615}%
\begin{displaymath}
\left[ \begin{array}{ccc}
 a   &   b     \\
 c   &   d
\end{array}  \right] = \mbox{cosh}  \varphi I + \mbox{senh}  \varphi  \widehat{U}
\eqno{(\mbox{IV.10'})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4617}%
$I, J, K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4621}%
\begin{displaymath}
\begin{tabular}{|c|c|c|c|c|} \hline
$\bullet$ &  \mbox{I} & \mbox{J} & \mbox{K}  & \mbox{L} \\\hline
\mbox{I}&  \mbox{I} & \mbox{L} & \mbox{K}  & \mbox{L} \\\hline
\mbox{J}&  \mbox{J} & \mbox{I} & \mbox{-L} & \mbox{-K} \\\hline
\mbox{K}&  \mbox{K} & \mbox{L} & \mbox{I}  & \mbox{J}  \\\hline
\mbox{L}& \mbox{L} & \mbox{K} & \mbox{-J} & \mbox{-I}    \\\hline
\end{tabular}  \eqno{(4.11)}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\addtocounter{equation}{1}
{\newpage\clearpage
\lthtmldisplayA{displaymath3715}%
\begin{displaymath}
\left[\begin{array}{ccc}
1     &     0    \\
0     &     0
\end{array}  \right] \qquad
 \left[\begin{array}{ccc}
0     &     0    \\
1     &     0
\end{array}  \right] \qquad
 \left[\begin{array}{ccc}
0    &     1    \\
0     &    0
\end{array}  \right] \qquad
 \left[\begin{array}{ccc}
0     &     0    \\
0     &     1
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath4623}%
\begin{displaymath}
A = a_{0} I + a_{1} J + a_{2} K + a_{3} L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4625}%
\begin{displaymath}
B = b_{0} I + b_{1} J + b_{2} K + b_{3} L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4627}%
\begin{displaymath}
AB = (a_{0} I + a_{1} J + a_{2} K + a_{3} L)(b_{0} I + b_{1} J +
b_{2} K + b_{3} L)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar3746}%
\begin{eqnarray*}
AB & = & (a_{0} b_{0} + a_{1} b_{1} + a_{2} b_{2} - a_{3} b_{3}) I
+
     (a_{0} b_{1} + a_{1} b_{0} + a_{2} b_{3} - a_{3} b_{2}) J +
     \\& & \\
& & (a_{0} b_{2} + a_{2} b_{0} + a_{3} b_{1} - a_{1} b_{3}) K +
(a_{0} b_{3} + a_{3} b_{0} + a_{2} b_{1} - a_{1} b_{2}) L
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4631}%
$B$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3780}%
\begin{displaymath}
\begin{array}{ccc}
A & = & a_{0}  I + \widehat{a}  \\
B & = & b_{0}  I + \widehat{b}
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4633}%
$I$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4635}%
$\widehat{a}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4637}%
$\widehat{b}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3792}%
\begin{displaymath}
(\widehat{a}, \widehat{b}) = a_{1} b_{1} + a_{2} b_{2} - a_{3}
b_{3}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3805}%
\begin{displaymath}
\widehat{a} \times  \widehat{b} \: = \: (a_{2}b_{3} - a_{3}b_{2})
J + (a_{3} b_{1} - a_{1} b_{3})  K + (a_{2} b_{1} - a_{1} b_{2}) L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4645}%
$2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar3821}%
\begin{eqnarray*}
e^{\varphi \ \widehat{u}} & = & \mbox{cosh} \ \varphi \ ( I +
\mbox{tanh} \ \varphi \ \widehat{U})   \\
e^{\psi \ \widehat{V}} & = & \mbox{cosh} \ \psi \ (I + \mbox{tan}
\psi \widehat{V})
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar3831}%
\begin{eqnarray*}
e^{\psi \ \widehat{V}} e^{\psi \ \widehat{V}} & = &[\mbox{cosh} (I
+ \mbox{tanh} \ \varphi \ \widehat{U})][\mbox{cosh} \psi \ (I +
\mbox{tanh} \ \psi \ \widehat{V})] \\
& = & \mbox{cosh} \ \varphi \ \mbox{cosh} \ \psi \ (I +
\mbox{tanh} \
\varphi \ \widehat{U})(I + \mbox{tanh} \ \psi \ \widehat{V})  \\
& = & \mbox{cosh} \ \varphi \ \mbox{cosh} \ \psi \ [I +
\mbox{tanh} \ \varphi \widehat{U} + \mbox{tanh} \ \psi \
\widehat{V} + \mbox{tanh} \ \varphi \ \mbox{tanh} \ \psi \
\widehat{U} \ \widehat{V}]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4647}%
$\widehat{U}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4649}%
$\widehat{V}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3859}%
\begin{displaymath}
\widehat{U} \ \widehat{V} = I (\widehat{U}, \widehat{V}) +
\widehat{U} \times \widehat{V}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar3867}%
\begin{eqnarray*}
 e^{\varphi \ \widehat{U}} \ e^{\psi \ \widehat{V}} &=&
I[\mbox{cos}h \ \varphi \ \mbox{cos}h \ \psi + \ \mbox{senh} \
\varphi \ \mbox{senh} \ \psi \ (\widehat{U}, \ \widehat{V})] \\
& \ \ & + \ \mbox{cosh} \ \varphi \ \mbox{cosh} \ \psi \
[\mbox{tanh} \ \varphi \ \widehat{U} + \mbox{tanh} \ \psi \
\widehat{V} +  \ \mbox{tanh} \ \varphi \ \mbox{tanh} \ \psi \
(\widehat{U} \times \widehat{V})]
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar3887}%
\begin{eqnarray*}
e^{\varphi \ \widehat{U}} \ e^{\psi \ \widehat{V}} &=& e^{\theta \
\widehat{W}}\\
& = &\mbox{cosh} \ \theta \ I + \mbox{cosh} \ \theta \ \mbox{tanh}
\ \theta \ \widehat{W}
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3896}%
\begin{displaymath}
\mbox{cosh} \ \theta = \mbox{cosh} \ \varphi \ \mbox{cosh} \ \psi
+ \mbox{senh} \ \varphi \ \mbox{senh} \ \psi \ (\widehat{U}, \
\widehat{V})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4653}%
$cosh$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3908}%
\begin{displaymath} \mbox{tanh} \ \theta \ \widehat{W} =
\frac{\mbox{cosh} \ \varphi \ \mbox{cosh} \ \psi}{\mbox{cosh} \
\theta} [\mbox{tanh} \ \varphi \ \widehat{U} + \mbox{tanh} \ \psi
\ \widehat{V} + \mbox{tanh} \ \varphi \ \mbox{tanh} \ \psi \
(\widehat{U} \times \widehat{V})]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4655}%
\begin{displaymath}
\widetilde{W} = \mbox{tanh} \ \theta \ \widehat{W}
\eqno{(\mbox{IV.18a})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4657}%
$W$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4659}%
\begin{displaymath}
\widetilde{W} = \frac{W}{\mbox{cosh} \ \theta}
\eqno{(\mbox{IV.18b})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\addtocounter{equation}{1}
{\newpage\clearpage
\lthtmldisplayA{displaymath3933}%
\begin{displaymath}
\widetilde{U} = \mbox{tanh} \ \varphi \ \widehat{U} \qquad \qquad
\widetilde{V} = \mbox{tanh} \ \psi \ \widehat{V}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4661}%
\begin{displaymath}
\widetilde{W} = \frac{\mbox{cosh} \ \varphi  \ \mbox{cosh} \
\psi}{\mbox{cosh} \ \varphi \  \mbox{cosh} \ \psi + \mbox{senh} \
\varphi \  \mbox{senh} \ \psi} (\widetilde{U} + \widetilde{V} +
\widetilde{U} + \widetilde{V})
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3952}%
\begin{displaymath}
\widetilde{W} = \frac{\widetilde{U} + \widetilde{V} +
\widetilde{U} \times \widetilde{V}}{1 + (\widetilde{U},
\widetilde{V})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4663}%
$J,K,L$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4667}%
\begin{displaymath}
\widetilde{U} = \mbox{tanh} \ \varphi \ \widehat{U}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4671}%
$\widetilde{U}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4673}%
$\mbox{tanh} \ \varphi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4679}%
\begin{displaymath}
\widetilde{U} = \frac{U}{\mbox{cosh} \ \varphi}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4685}%
\begin{displaymath}
\widehat{U} = u_{1}  \ J + u_{2}  \ K + u_{3}  \ L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3978}%
\begin{displaymath}
(U, \ U) = u_{1}^{2} + u_{2}^{2} - u_{3}^{2} = 1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmldisplayA{displaymath3988}%
\begin{displaymath}
     R = S_{2} \ S_{1}
    \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4687}%
$S_{2}^{2} = S_{1}^{2} = I$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4689}%
\begin{displaymath}
R_{1} = S_{2} \: S_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4691}%
\begin{displaymath}
R_{2} = S_{3} \: S_{2}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4693}%
\begin{displaymath}
R = R_{2} \: R_{1} = S_{3} \: S_{2} \: S_{2} \: S_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4695}%
\begin{displaymath}
S_{2}  \: S_{2} = S_{2}^{2}  = I
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4697}%
\begin{displaymath}
R = S_{3} \: S_{1}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap6983}%
\includegraphics[width=4.2in]{f24p106.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4701}%
\begin{displaymath}
\widehat{V} = v_{1} \ J + v_{2} \ K + v_{3} \ L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4703}%
\begin{displaymath}
(\widehat{U}, \ \widehat{U}) = 1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4705}%
\begin{displaymath}
(\widehat{V}, \ \widehat{V}) = 1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4707}%
$\widehat{U} \ \ \widehat{U}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4709}%
$\widehat{V} \ \ \widehat{V}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4711}%
\begin{displaymath}
\widehat{U} \  \widehat{U} = (\widehat{U}, \ \widehat{U}) \ I +
\widehat{U} \times \widehat{U}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4713}%
\begin{displaymath}
\widehat{V} \  \widehat{V} = (\widehat{V}, \ \widehat{V}) \ I +
\widehat{V} \times \widehat{V}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4715}%
\begin{displaymath}
\widehat{U} \times \widehat{U} = (u_{2}  u_{3} - u_{3}  u_{2}) \:
J \ + \ (u_{3}  u_{1} - u_{1}  u_{3}) \: K \ + \  (u_{2}  u_{1} -
u_{1}  u_{2}) \: L = \widehat{0}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4717}%
\begin{displaymath}
\widehat{V} \times \widehat{V} = \widehat{0}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4066}%
\begin{displaymath}
\begin{array}{ccc}
 \widehat{U}  \ \widehat{U} & = & I  \\
 \widehat{V}  \ \widehat{V} & = & I
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4077}%
\begin{eqnarray*}
(\widehat{U} \times \widehat{V},  \widehat{U} \times \widehat{V})
& = & ([(u_{2}  v_{3} - u_{3}  v_{2}) \ J + (u_{3}
v_{1} - u_{1}  v_{3}) \ K +    \\
 & & \ (u_{2}  v_{1} - u_{1}  v_{2}) \ L], [(u_{2}  v_{3} - u_{3}
v_{2}) \ J +       \\
 & & \ (u_{3}  v_{1} - u_{1}  v_{3}) \ K + (u_{2}  v_{1} - u_{1}
 v_{2}) \ L])
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{eqnarraystar4107}%
\begin{eqnarray*}
(\widehat{U} \times \widehat{V}, \ \ \widehat{U} \times
\widehat{V}) & = & (u_{2}  v_{3} - u_{3}  v_{2})^{2} + (u_{3}
v_{1} - u_{1}  v_{3})^{2} - (u_{2}  v_{1} - u_{1}  v_{2})^{2} \\
& = & u_{2}^{2}  v_{3}^{2} - 2u_{2}  u_{3}  v_{2}  v_{3} +
u_{3}^{2}  v_{2}^{2}  + u_{3}^{2}  v_{1}^{2} - 2u_{1}  u_{3}
v_{1}  v_{3} +
u_{1}^{2}  v_{3}^{2}   \\
& \ \ &  - u_{2}^{2}  v_{1}^{2} + 2u_{1}  u_{2}  v_{1}  v_{2} -
u_{1}^{2}  v_{2}^{2}   \\
& = & v_{1}^{2} (-u_{1}^{2} - u_{2}^{2} + u_{3}^{2}) + u_{1}^{2}
v_{1}^{2} + v_{2}^{2} (-u_{1}^{2} - u_{2}^{2} + u_{3}^{2}) +
u_{2}^{2}  v_{2}^{2}   \\
& \ \ & + v_{3}^{2} (u_{1}^{2} + u_{2}^{2} - u_{3}^{2}) -
u_{3}^{2}  v_{3}^{2} + 2u_{1}  u_{2}  v_{1}  v_{2} - 2u_{1} u_{3}
v_{1}  v_{3}  \\
&  & - 2u_{2}  u_{3}  v_{2}  v_{3}  \\
& = & - v_{1}^{2} - v_{2}^{2} + v_{3}^{2} + (u_{1}  v_{1} + u_{2}
v_{2} - u_{3}  v_{3})^{2}     \\
& = & (u_{1}  v_{1} + u_{2}  v_{2} - u_{3} \ v_{3})^{2} - 1
\end{eqnarray*}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4232}%
\begin{displaymath}
 u_1 v_1 + u_2 v_2   - u_3 v_3 = \mbox{cosh} \ \theta
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4723}%
$\mbox{cosh} \ \theta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4729}%
\begin{displaymath}
(\widehat{U}  \times  \widehat{V}, \widehat{U}  \times
\widehat{V}) = \mbox{cosh}^{2} \ \theta  -  1
 =  \mbox{senh}^{2} \ \theta
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4248}%
\begin{displaymath}
\widehat{U} \times \widehat{V} = \mbox{senh}  \: \theta  \:
\widehat{W}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4731}%
$\widehat{W}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4737}%
$\mbox{senh} \
\theta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4739}%
$\theta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4745}%
\begin{displaymath}
\widehat{U} \widehat{V} = (\widehat{U}, \widehat{V}) I +
\widehat{U} \times \widehat{V}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4266}%
\begin{displaymath}
\widehat{U} \ \widehat{V} = \mbox{cosh} \ \theta \ I + \mbox{senh}
\ \theta \ \widehat{W}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4747}%
\begin{displaymath}
\widehat{U} \ \widehat{V} = e^{\theta  \widehat{W}}
\eqno{(\mbox{IV.26'})}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4749}%
$e^{\theta \widehat{W}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4761}%
\begin{displaymath}
 (\widehat{U}, \widehat{W}) = u_1  w_1 + u_2  w_2 - u_3  w_3
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4285}%
\begin{displaymath}
(\widehat{U}, \widehat{W}) = [\widehat{U}, N \widehat{W}] = [u_1,
u_2, u_3]  \left[ \begin{array}{cccc}
1      &    0   &    0            \\
0      &    1   &    0            \\
0      &    0   &    -1
\end{array}  \right]
\left[ \begin{array}{c}
 w_1    \\
 w_2    \\
 w_3
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4297}%
\begin{displaymath}
N =  \left[ \begin{array}{cccc}
1      &    0   &    0            \\
0      &    1   &    0            \\
0      &    0   &    -1
\end{array}  \right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4763}%
\begin{displaymath}
 [\widehat{X}, N \ \widehat{W}]  =  0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4305}%
\begin{displaymath}
w_1  x_1  +  w_2  x_2  -  w_3   x_3  =  0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4307}%
\begin{displaymath}
 x^{2}_{1} + x^{2}_{2}  -  x^{2}_{3}  =  1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4765}%
$a \ x_3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4767}%
\begin{displaymath}
x_3  =  \frac{w_1  x_1 + w_2  x_2}{w_{3}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4769}%
\begin{displaymath}
x^{2}_{3}  =  \frac {w^{2}_{1}  x^{2}_{1} + w^{2}_{2} x^{2}_{2} +
2 w_{1}  w_{2}  x_{1}  x_{2}}{w^{2}_{3}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4771}%
\begin{displaymath}
x^{2}_{1} + x^{2}_{2} - \frac{w^{2}_{1}  x^{2}_{1} + w^{2}_{2}
x^{2}_{2} + 2 w_{1}  w_{2}  x_{1}  x_{2}}{w^{2}_{3}}  =  1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4351}%
\begin{displaymath}
x^{2}_{1} (w^{2}_{3} + w^{2}_{1}) + x^{2}_{2} (w^{2}_{3} -
w^{2}_{2}) - 2 w_{1}  w_{2}  x_{1}  x_{2} - w^{2}_{3} = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4773}%
$x_{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4372}%
\begin{displaymath}
 x^{2}_{1} (w^{2}_{1}+w^{2}_{2}) - x^{2}_{3} (w^{2}_{2} -
w^{2}_{3}) - 2 w_{1} w_{3} x_{1}x_{3} - w^{2}_{2} = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4777}%
$x_{3}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4403}%
\begin{displaymath}
[\widehat{U}, N \ \widehat{V}] = \mbox{cosh} \ \theta
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4793}%
$(x_{1}, x_{2})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4795}%
$x_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7071}%
\includegraphics[width=3.2in]{f25p113.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap7075}%
\includegraphics[width=3.2in]{f26p114.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4801}%
$\widetilde{W}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4803}%
\begin{displaymath}
\widetilde{W} \: = \: \mbox{tanh} \: \theta \:  \widehat{W}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4426}%
\begin{displaymath}
\left[ \begin{array}{cc}
\tau_{11}       &    \tau_{12}     \\
\tau_{21}       &    \tau_{22}
\end{array} \right] =
\mbox{cosh} \ \theta \ I + \mbox{senh} \ \theta \ \widehat{W}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4807}%
\begin{displaymath}
\widehat{W} = w_{1} \: J + w_{2} \: K + w_{3} \: L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4444}%
\begin{displaymath}
\left[ \begin{array}{cc}
\tau_{11}       &    \tau_{12}     \\
\tau_{21}       &    \tau_{22}
\end{array} \right] =
\mbox{cosh} \: \theta \:I + (w_{1} \:J + w_{2}\: K + w_{3} \ L)\:
\mbox{senh} \: \theta
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4811}%
$\tau_{11} = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4813}%
\begin{displaymath}
\tau_{11} = \mbox{cosh} \ \theta + \mbox{senh} \ \theta \ w_{2} =
0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4463}%
\begin{displaymath}
\mbox{tanh} \ \theta = -\frac{1}{w_{2}}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4817}%
\begin{displaymath}
\widetilde{W} = - \frac{1}{w_{2}}(w_{1}\:J + w_{2}\:K + w_{3} \:L)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4476}%
\begin{displaymath}
\widetilde{W} = - \frac{w_{1}}{w_{2}}\:J - 1 -
\frac{w_{3}}{w_{2}}\:L
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4483}%
\begin{displaymath}
W_{2} = -1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4821}%
$J, \ K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}
