# LaTeX2HTML 99.1 release (March 30, 1999)
# Associate images original text with physical files.


$key = q/;c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img78.gif"
 ALT="$ \;c$">|; 

$key = q/;Rightarrow;lambda_1neq0.,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.gif"
 ALT="$ \;\Rightarrow
\;\lambda_1 \neq 0. \,$">|; 

$key = q/,r(mathcal{A})=4.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img53.gif"
 ALT="$ \,r(\mathcal{A})=4.$">|; 

$key = q/;a=sqrt{-bslashlambda_1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img72.gif"
 ALT="$ \;a = \sqrt{-b/ \lambda_1}
\,$">|; 

$key = q/,cneq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img63.gif"
 ALT="$ \,c \neq 0$">|; 

$key = q/,detmathcal{A}_0>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img30.gif"
 ALT="$ \,det \mathcal{A}_0 &gt;0\,$">|; 

$key = q/;lambda_3<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img24.gif"
 ALT="$ \;\lambda_3 &lt;0,$">|; 

$key = q/;a_3=1slashsqrt{-c_3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img34.gif"
 ALT="$ \;a_3 = 1 / \sqrt{-c_3} \,$">|; 

$key = q/;lambda_2>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img46.gif"
 ALT="$ \;\lambda_2&gt;0 \,$">|; 

$key = q/,lambda_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img71.gif"
 ALT="$ \,\lambda_1 \,$">|; 

$key = q/,r(mathcal{A}_0)=1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img67.gif"
 ALT="$ \,r(\mathcal{A}_0) = 1\,$">|; 

$key = q/,lambda_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img74.gif"
 ALT="$ \,\lambda_3 \,$">|; 

$key = q/,r(mathcal{A}_0)=2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img60.gif"
 ALT="$ \,r(\mathcal{A}_0) = 2\,$">|; 

$key = q/,r(mathcal{A}_0)=3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.gif"
 ALT="$ \,r(\mathcal{A}_0) = 3\,$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}-2z=0;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img44.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}-2z=0;
$">|; 

$key = q/;lambda_ineq0;;forall:i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="$ \;\lambda_i
\neq 0 \;\;\forall \:i$">|; 

$key = q/,lambda_1,,lambda_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img41.gif"
 ALT="$ \,\lambda_1,\,\lambda_2 \,$">|; 

$key = q/displaystylex^2,=,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img82.gif"
 ALT="$\displaystyle x^2 \,=\,0 $">|; 

$key = q/;lambda_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img76.gif"
 ALT="$ \;\lambda_2 \,$">|; 

$key = q/;a_i=1slashsqrt{c_i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.gif"
 ALT="$ \;a_i = 1 / \sqrt{c_i}$">|; 

$key = q/,mathcal{Q},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.gif"
 ALT="$ \,\mathcal{Q} \,$">|; 

$key = q/;lambda_3>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img26.gif"
 ALT="$ \;\lambda_3 &gt;0$">|; 

$key = q/,x=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img83.gif"
 ALT="$ \,x=0\,$">|; 

$key = q/,detmathcal{A}_0=lambda_1lambda_2lambda_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.gif"
 ALT="$ \,det \mathcal{A}_0 =
\lambda_1\lambda_2\lambda_3$">|; 

$key = q/,c_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img36.gif"
 ALT="$ \,c_1$">|; 

$key = q/,17,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img2.gif"
 ALT="$ \,17 \,$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}-2z,=,0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img73.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}-2z\,=\,0
$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}-displaystylefrac{z^2}{a_3^2}-1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img35.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}
-\displaystyle\frac{z^2}{a_3^2}-1=0
$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}-displaystylefrac{z^2}{a_3^2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img59.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}
-\displaystyle\frac{z^2}{a_3^2}=0
$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}-displaystylefrac{y^2}{a_2^2}-displaystylefrac{z^2}{a_3^2}-1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img37.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}-\displaystyle\frac{y^2}{a_2^2}
-\displaystyle\frac{z^2}{a_3^2}-1=0
$">|; 

$key = q/,c_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img20.gif"
 ALT="$ \,c_1\,$">|; 

$key = q/;a_1=1slashsqrt{c_1},,a_2=1slashsqrt{c_2},,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img33.gif"
 ALT="$ \;a_1 = 1 / \sqrt{c_1},\, a_2 = 1 / \sqrt{c_2},\,$">|; 

$key = q/,lambda_3=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img61.gif"
 ALT="$ \,\lambda_3=0$">|; 

$key = q/,c_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img21.gif"
 ALT="$ \,c_2\,$">|; 

$key = q/displaystyle-frac{lambda_1}{b}x^2-frac{lambda_2}{b}y^2-2z;=;0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="img39.gif"
 ALT="$\displaystyle -\frac{\lambda_1}{b}x^2-\frac{\lambda_2}{b}y^2-2z\;= \; 0
$">|; 

$key = q/,c_3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img22.gif"
 ALT="$ \,c_3\,$">|; 

$key = q/Leftrightarrow;r(mathcal{A})=4;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.gif"
 ALT="$ \Leftrightarrow \; r(\mathcal{A}) = 4 \;$">|; 

$key = q/:b=sqrt{-displaystylefrac{{detmathcal{A}}}{{lambda_1lambda_2}}};>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="img40.gif"
 ALT="$ \:b=\sqrt{-\displaystyle\frac{{det
\mathcal{A}}}{{\lambda_1\lambda_2}}}\;&gt;0\,$">|; 

$key = q/,lambda_1;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img75.gif"
 ALT="$ \,\lambda_1 \;$">|; 

$key = q/,mathcal{A}_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.gif"
 ALT="$ \,\mathcal{A}_0 \,$">|; 

$key = q/{displaymath}array{{cc}displaystylefrac{x^2}{a_1^2}+1=0<col_mark>hspace{3cm}displaystylefrac{x^2}{a_1^2}-1=0<row_mark>array{{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="53" BORDER="0"
 SRC="img79.gif"
 ALT="\begin{displaymath}
\begin{array}{cc}
\displaystyle\frac{x^2}{a_1^2}+1=0 &amp; \hspace{3cm}
\displaystyle\frac{x^2}{a_1^2}-1=0 \\\\
\end{array}
\end{displaymath}">|; 

$key = q/;c_1=-frac{lambda_1,detmathcal{A}}{detmathcal{A}_0}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img28.gif"
 ALT="$ \;c_1 = -\frac{\lambda_1 \,det \mathcal{A}}{det \mathcal{A}_0} &gt;0$">|; 

$key = q/,lambda_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.gif"
 ALT="$ \,\lambda_i \,$">|; 

$key = q/,detmathcal{A}_0<0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img27.gif"
 ALT="$ \,det
\mathcal{A}_0 &lt;0 \,$">|; 

$key = q/;a_3=1slashsqrt{-lambda_3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img58.gif"
 ALT="$ \;a_3 = 1 / \sqrt{-\lambda_3} \,$">|; 

$key = q/,c=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img52.gif"
 ALT="$ \,c = 0
\,$">|; 

$key = q/{displaymath}array{{cc}displaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}displaystylefrac{y^2}{a_2^2}+1=0<row_mark>array{{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="53" BORDER="0"
 SRC="img64.gif"
 ALT="\begin{displaymath}
\begin{array}{cc}
\displaystyle\frac{x^2}{a_1^2}+\displays...
...}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}+1=0 \\\\
\end{array}
\end{displaymath}">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}-displaystylefrac{y^2}{a_2^2}-2z=0;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img48.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}-\displaystyle\frac{y^2}{a_2^2}-2z=0;
$">|; 

$key = q/;r(mathcal{A}_0)=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="$ \;r(\mathcal{A}_0) = 3$">|; 

$key = q/,lambda_1neq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img68.gif"
 ALT="$ \,\lambda_1\neq 0$">|; 

$key = q/,c_i>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img13.gif"
 ALT="$ \, c_i &gt; 0 \,$">|; 

$key = q/lambda_1x^2=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img81.gif"
 ALT="$ \lambda_1 x^2 = 0 \,$">|; 

$key = q/;a_2=sqrt{-bslashlambda_2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img43.gif"
 ALT="$ \;a_2 = \sqrt{-b/ \lambda_2} \,$">|; 

$key = q/;c_i=-displaystylefrac{{lambda_i,detmathcal{A}_0}}{{detmathcal{A}}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="img6.gif"
 ALT="$ \;c_i = -\displaystyle\frac{{\lambda_i \,det
\mathcal{A}_0}}{{det \mathcal{A}}}$">|; 

$key = q/{displaymath}array{{cc}displaystylefrac{x^2}{a_1^2}+frac{y^2}{a_2^2}=0<col_mark>c{x^2}{a_1^2}-frac{y^2}{a_2^2}=0<row_mark>array{{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="53" BORDER="0"
 SRC="img77.gif"
 ALT="\begin{displaymath}
\begin{array}{cc}
\displaystyle\frac{x^2}{a_1^2}+\frac{y^2...
...ystyle\frac{x^2}{a_1^2}-\frac{y^2}{a_2^2}=0 \\\\
\end{array}
\end{displaymath}">|; 

$key = q/,(2,1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img31.gif"
 ALT="$ \,(2,1)\,$">|; 

$key = q/lambda_1x^2+lambda_2y^2+lambda_3z^2+c;=;0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img51.gif"
 ALT="$ \lambda_1x^2+\lambda_2y^2+\lambda_3z^2+c\;= \; 0 $">|; 

$key = q/:i=1,2,3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img17.gif"
 ALT="$ \:i = 1, 2, 3,$">|; 

$key = q/;a_1=sqrt{-bslashlambda_1},,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img42.gif"
 ALT="$ \;a_1 = \sqrt{-b/ \lambda_1},\,$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}+displaystylefrac{z^2}{a_3^2}+1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img38.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}
+\displaystyle\frac{z^2}{a_3^2}+1=0
$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}+displaystylefrac{z^2}{a_3^2}-1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}
+\displaystyle\frac{z^2}{a_3^2}-1=0
$">|; 

$key = q/1,c_i:;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.gif"
 ALT="$ 1,c_i: $">|; 

$key = q/,c_i,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img19.gif"
 ALT="$ \,c_i\,$">|; 

$key = q/,lambda_1<0;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img45.gif"
 ALT="$ \,\lambda_1&lt;0
\;$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}+displaystylefrac{y^2}{a_2^2}+displaystylefrac{z^2}{a_3^2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img57.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}+\displaystyle\frac{y^2}{a_2^2}
+\displaystyle\frac{z^2}{a_3^2}=0
$">|; 

$key = q/;a_1=1slashsqrt{lambda_1},,a_2=1slashsqrt{lambda_2},,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img55.gif"
 ALT="$ \;a_1 = 1 / \sqrt{\lambda_1},\, a_2 = 1 / \sqrt{\lambda_2},\,$">|; 

$key = q/;a_2=sqrt{bslashlambda_2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="img47.gif"
 ALT="$ \;a_2 = \sqrt{ b/ \lambda_2} \,$">|; 

$key = q/,detmathcal{A}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.gif"
 ALT="$ \, det \mathcal{A} &lt; 0$">|; 

$key = q/,lambda_1<0,;lambda_2<0;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img25.gif"
 ALT="$ \,\lambda_1 &lt; 0,\;\lambda_2
&lt;0 \;$">|; 

$key = q/lambda_1x^2+lambda_2y^2+lambda_3z^2;=;0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img54.gif"
 ALT="$ \lambda_1x^2+\lambda_2y^2+\lambda_3z^2\;= \; 0 $">|; 

$key = q/,detmathcal{A}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img29.gif"
 ALT="$ \,det \mathcal{A} &gt;0$">|; 

$key = q/,lambda_i,detmathcal{A}_0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.gif"
 ALT="$ \,\lambda_i \,det \mathcal{A}_0 \,$">|; 

$key = q/,(1,2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img32.gif"
 ALT="$ \,(1,2)$">|; 

$key = q/,lambda_1>0,;lambda_2>0;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img23.gif"
 ALT="$ \,\lambda_1 &gt; 0,\;\lambda_2 &gt;0
\;$">|; 

$key = q/,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img65.gif"
 ALT="$ \,c$">|; 

$key = q/,r(mathcal{A}_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img49.gif"
 ALT="$ \,r(\mathcal{A}_0)$">|; 

$key = q/displaystyle-frac{lambda_1}{c}x^2-frac{lambda_2}{c}y^2-1;=;0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="img62.gif"
 ALT="$\displaystyle -\frac{\lambda_1}{c}x^2-\frac{\lambda_2}{c}y^2-1\;= \; 0
$">|; 

$key = q/:b>0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img70.gif"
 ALT="$ \:b&gt;0\,$">|; 

$key = q/displaystyle-frac{lambda_1}{b}x^2-2z;=;0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="img69.gif"
 ALT="$\displaystyle -\frac{\lambda_1}{b}x^2-2z\;= \; 0
$">|; 

$key = q/;a_3=1slashsqrt{lambda_3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.gif"
 ALT="$ \;a_3 = 1 / \sqrt{\lambda_3} \,$">|; 

$key = q/;lambda_1,:lambda_2,:lambda_3;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.gif"
 ALT="$ \;\lambda_1,\:\lambda_2,\:\lambda_3 \;$">|; 

$key = q/displaystyledisplaystylefrac{x^2}{a_1^2}-displaystylefrac{y^2}{a_2^2}-1=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="img66.gif"
 ALT="$\displaystyle \displaystyle\frac{x^2}{a_1^2}-\displaystyle\frac{y^2}{a_2^2} -1=0
$">|; 

$key = q/displaystylec_1x^2+c_2y^2+c_3z^2-1=0;;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="$\displaystyle c_1 x^2 + c_2 y^2 + c_3 z^2 - 1 = 0;
$">|; 

1;

