<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%------------------------------------------------------------------------
% This is equations.sty (August 1988).  Written by
%   Charles Karney
%   Plasma Physics Laboratory   Phone:   +1 609 243 2607
%   Princeton University        MFEnet:  Karney@PPC.MFEnet
%   PO Box 451                  ARPAnet: Karney%PPC.MFEnet@NMFECC.ARPA
%   Princeton, NJ 08543-0451    Bitnet:  Karney%PPC.MFEnet@ANLVMS.Bitnet
% with some ideas and macros borrowed from John Hobby and Stephen Gildea.
 
% --------------------USER DOCUMENTATION--------------------
 
% Use this as a style option, e.g.,
%     \documentstyle[equations]{article}
% These probably don't work in conjunction with the leqno option.
 
% This implements the following:
 
% (1) \yesnumber turns on an equation number in an eqnarray* environment
% (just as \nonumber turns it off in an eqnarray environment).  E.g.,
%     \begin{eqnarray*}
%        a &amp;=&amp; b \\
%          &amp; &amp; + c \\
%          &amp; &amp; + d \\
%          &amp; &amp; + e \\
%          &amp; &amp; + f \\
%          &amp; &amp; + g \yesnumber
%     \end{eqnarray*}
 
% (2) The eqalign environment is just like Plain TeX's \eqalign.  E.g.,
%     \begin{equation}
%           \begin{eqalign}
%              a &amp;= b, \\
%              c &amp;= d.
%           \end{eqalign}
%     \end{equation}
% Note that \begin{equation} \end{equation} or equivalent is needed.
 
% (3) The eqalignno environment is just like Plain TeX's \eqalignno.  E.g.,
%     \begin{eqalignno}
%        a &amp;= b, \label{foo}\\
%        c &amp;= d. \label{bar}
%     \end{eqalignno}
% Note the absence of \begin{equation} \end{equation}.  \nonumber can be used
% to suppress the equation number.  eqalignno* is the same except that the
% equation numbers are suppressed (unless a \yesnumber appears).
 
% (4) The eqaligntwo environment is a two-equation per line equivalent of
% eqalignno.  E.g.,
%     \begin{eqaligntwo}
%        a &amp;= b, &amp; x &amp;= y, \label{foo} \\
%        c &amp;= d, &amp; z &amp;= w. \label{bar}
%     \end{eqalign}
% eqaligntwo* is defined similarly.
 
% (5) The cases environment is just like Plain TeX's \cases.  E.g.,
%     \begin{equation}
%        u(x) =
%           \begin{cases}
%             0, &amp; for $x &lt; 0$,
%             1, &amp; for $x \ge 0$.
%           \end{cases}
%     \end{equation}
% Note the first column is treated as math, the second column as text.
 
% (6) I've borrowed John Hobby's modifications to the eqnarray environment
% (to fix up the spacing around the operator), and to the \big, \bigg, etc.
% operators (to make them scale with the point size).
 
% (7) I've borrowed Stephen Gildea's subequations environment, and fixed it
% so that it ignores spaces after the environment and so that you can refer
% both to the overall set of equations and to individual subequations.  E.g.,
%     \begin{subequations} \label{foo}
%       \begin{eqalignno}
%          a &amp;= b, \label{foo-a} \\
%          c &amp;= d, \label{foo-b}
%       \end{eqalignno}
%       text text text text ...
%       \begin{equation}
%          e = f. \label{foo-c}
%       \end{equation}
%     \end{subequations}
% Now \ref{foo}, \ref{foo-a}, \ref{foo-b}, \ref{foo-c}, produce 1, 1a, 1b,
% 1c.
 
% --------------------END OF USER DOCUMENTATION--------------------
 
\typeout{Document Substyle `EQUATION', modified by MPI, Version 1.0, 09.03.89}
 
 
% Fix up eqnarray* so that \yesnumber and \nonumber do the obvious things
\newif\if@defeqnsw \@defeqnswtrue
 
% This is John Hobby's (hobby@research.att.com) version to fix up the spacing.
\def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation
\if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi
%\global\@eqnswtrue       inactivated march 89 G. Kruljac
\tabskip\@centering\let\\=\@eqncr
$$\halign to \displaywidth\bgroup\hfil\global\@eqcnt\z@
  $\displaystyle\tabskip\z@{##}$&amp;\global\@eqcnt\@ne
  \hfil$\displaystyle{{}##{}}$\hfil
  &amp;\global\@eqcnt\tw@ $\displaystyle{##}$\hfil
  \tabskip\@centering&amp;\llap{##}\tabskip\z@\cr}
 
% Here's the eqnarray* environment:
 
%\let\@seqncr=\@eqncr
\@namedef{eqnarray*}{\@defeqnswfalse\eqnarray}
\@namedef{endeqnarray*}{\endeqnarray}
 
\def\yesnumber{\global\@eqnswtrue}
 
\def\@@eqncr{\let\@tempa\relax\global\advance\@eqcnt by \@ne
    \ifcase\@eqcnt \def\@tempa{&amp; &amp; &amp; &amp;}\or \def\@tempa{&amp; &amp; &amp;}\or
     \def\@tempa{&amp; &amp;}\or \def\@tempa{&amp;}\else\fi
     \@tempa \if@eqnsw\@eqnnum\stepcounter{equation}\fi
     \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi
     \global\@eqcnt\z@\cr}
 
% Several formulas like \eqalign (to go inside \begin{equation}
% \end{equation} or $$ $$.
 
\def\@eqnacr{{\ifnum0=`}\fi\@ifstar{\@yeqnacr}{\@yeqnacr}}
 
\def\@yeqnacr{\@ifnextchar [{\@xeqnacr}{\@xeqnacr[\z@]}}
 
\def\@xeqnacr[#1]{\ifnum0=`{\fi}\cr \noalign{\vskip\jot\vskip #1\relax}}
 
\def\eqalign{\null\,\vcenter\bgroup\openup1\jot \m@th \let\\=\@eqnacr
\ialign\bgroup\strut
\hfil$\displaystyle{##}$&amp;$\displaystyle{{}##}$\hfil\crcr}
\def\endeqalign{\crcr\egroup\egroup\,}
 
% \cases
 
\def\cases{\left\{\,\vcenter\bgroup\normalbaselines\m@th \let\\=\@eqnacr
    \ialign\bgroup$##\hfil$&amp;\quad##\hfil\crcr}
\def\endcases{\crcr\egroup\egroup\right.}
 
% Several formulas like \eqalignno (doesn't need the $$ $$)
 
\def\eqalignno{\stepcounter{equation}\let\@currentlabel=\theequation
\if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi
\let\\=\@eqncr
$$\displ@y \tabskip\@centering \halign to \displaywidth\bgroup
  \global\@eqcnt\@ne\hfil
  $\@lign\displaystyle{##}$\tabskip\z@skip&amp;\global\@eqcnt\tw@
  $\@lign\displaystyle{{}##}$\hfil\tabskip\@centering&amp;
  \llap{\@lign##}\tabskip\z@skip\crcr}
 
\def\endeqalignno{\@@eqncr\egroup
      \global\advance\c@equation\m@ne$$\global\@ignoretrue}
 
 
% Here's the eqalignno* environment:
 
\@namedef{eqalignno*}{\@defeqnswfalse\eqalignno}
\@namedef{endeqalignno*}{\endeqalignno}
 
% Two formulas like \eqalignno (doesn't need the $$ $$)
 
\def\eqaligntwo{\stepcounter{equation}\let\@currentlabel=\theequation
\if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi
\let\\=\@eqncr
$$\displ@y \tabskip\@centering \halign to \displaywidth\bgroup
  \global\@eqcnt\m@ne\hfil
  $\@lign\displaystyle{##}$\tabskip\z@skip&amp;\global\@eqcnt\z@
  $\@lign\displaystyle{{}##}$\hfil\qquad&amp;\global\@eqcnt\@ne
  \hfil$\@lign\displaystyle{##}$&amp;\global\@eqcnt\tw@
  $\@lign\displaystyle{{}##}$\hfil\tabskip\@centering&amp;
  \llap{\@lign##}\tabskip\z@skip\crcr}
 
\def\endeqaligntwo{\@@eqncr\egroup
      \global\advance\c@equation\m@ne$$\global\@ignoretrue}
 
\@namedef{eqaligntwo*}{\@defeqnswfalse\eqaligntwo}
\@namedef{endeqaligntwo*}{\endeqaligntwo}
 
% subequations
%%% File: subeqn.sty
%%% The subequations environment %%%
%
% Within the subequations environment, the only change is that
% equations are labeled differently.  The number stays the same,
% and lower case letters are appended.  For example, if after doing
% three equations, numbered 1, 2, and 3, you start a subequations
% environmment and do three more equations, they will be numbered
% 4a, 4b, and 4c.  After you end the subequations environment, the
% next equation will be numbered 5.
%
% Both text and equations can be put inside the subequations environment.
%
% If you make any improvements, I'd like to hear about them.
%
% Stephen Gildea
% MIT Earth Resources Lab
% Cambridge, Mass. 02139
% mit-erl!gildea
% gildea@erl.mit.edu
%
\newtoks\@stequation
 
\def\subequations{\refstepcounter{equation}%
  \edef\@savedequation{\the\c@equation}%
  \@stequation=\expandafter{\theequation}%   %only want \theequation
  \edef\@savedtheequation{\the\@stequation}% %expanded once
  \edef\oldtheequation{\theequation}%
  \setcounter{equation}{0}%
  \def\theequation{\oldtheequation\alph{equation}}}
 
\def\endsubequations{%
  \setcounter{equation}{\@savedequation}%
  \@stequation=\expandafter{\@savedtheequation}%
  \edef\theequation{\the\@stequation}%
  \global\@ignoretrue}
 
% More from John Hobby
% Fix Plain's \bigl, \Bigl, etc. macros so that they try to scale with
% LaTeX size changes.  This uses the fact that \@setsize sets \ht\strutbox
% to be 70% of the normal unstretched baselineskip.
 
\def\big#1{{\hbox{$\left#1\vcenter to1.428\ht\strutbox{}\right.\n@space$}}}
\def\Big#1{{\hbox{$\left#1\vcenter to2.142\ht\strutbox{}\right.\n@space$}}}
\def\bigg#1{{\hbox{$\left#1\vcenter to2.857\ht\strutbox{}\right.\n@space$}}}
\def\Bigg#1{{\hbox{$\left#1\vcenter to3.571\ht\strutbox{}\right.\n@space$}}}
 
% More from Schaefer-Siebert / Kruljac
%                              kruljac@ds0mpi11
 
\def\@leqncr{{\ifnum0=`}\fi\@ifstar{\global\@eqpen\@M
    \@yeqncr}{\global\@eqpen\interdisplaylinepenalty \@lyeqncr}}
 
\def\@lyeqncr{\@ifnextchar [{\@lxeqncr}{\@lxeqncr[\z@]}}
 
\def\@lxeqncr[#1]{\ifnum0=`{\fi}\@@leqncr
   \noalign{\penalty\@eqpen\vskip\jot\vskip #1\relax}}
 
 
\def\@@leqncr{\let\@tempa\relax\global\advance\@eqcnt by \@ne
    \ifcase\@eqcnt \def\@tempa{&amp; &amp; &amp; &amp;}\or \def\@tempa{&amp; &amp; &amp;}\or
     \def\@tempa{&amp; &amp;}\or \def\@tempa{&amp;}\else\fi
     \@tempa \if@eqnsw\@leqnnum\stepcounter{equation}\fi
     \if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi
     \global\@eqcnt\z@\cr}
 
 \def\@leqnnum{{\hspace{\fill}\hbox{\normalsize (\theequation)}}}
 
\def\lequation#1{\bgroup#1$$ % $$ BRACE MATCHING HACK
\refstepcounter{equation}}
 
\def\endlequation{\eqno \@leqnnum % $$ BRACE MATCHING HACK
$$\egroup\global\@ignoretrue}
 
\def\leqnarray#1{\stepcounter{equation}\let\@currentlabel=\theequation
\if@defeqnsw\global\@eqnswtrue\else\global\@eqnswfalse\fi
\tabskip\@centering\let\\=\@leqncr
\bgroup#1
$$\halign to \displaywidth\bgroup\hfil\global\@eqcnt\z@
  $\displaystyle\tabskip\z@{##}$&amp;\global\@eqcnt\@ne
  \hfil$\displaystyle{{}##{}}$\hfil
  &amp;\global\@eqcnt\tw@ $\displaystyle{##}$\hfil
  \tabskip\@centering&amp;\llap{##}\tabskip\z@\cr}
 
\def\endleqnarray{\@@leqncr\egroup
      \global\advance\c@equation\m@ne$$\egroup\global\@ignoretrue}
 
% Here's the leqnarray* environment:
 
\@namedef{leqnarray*}{\@defeqnswfalse\leqnarray}
\@namedef{endleqnarray*}{\endleqnarray}
 
\endinput
 
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