%  \polyalignno {<number_of_variables>}{<lines>}
%  
%  There are 2 * <number_of_variables> + 1 alignment columns that alternate
%  between right & left justified; plus there is a column for the equation
%  number: "<optional labels>".
%
%  <lines> is any number of:
%  <line> = <right math> [ &<left math> &<right math> ] <optional labels> \cr
%  where the contents of [ ] are repeated <number_of_variables> times, and
%  <right math>= math that will be right justified (e.g., 2y)
%  <left math> = math that will be left justified (e.g., +,-,=)
%  <optional labels> are one of:  <nothing>
%                                 \eqno...
%                                 \leqno...
%                                 \eqno...\leqno...
%                                 & explicit numbering
%  The \eqno and \leqno forms will automatically insert just enough &'s
%  to get to the final column, but you can also enter enough &'s to get 
%  to the final column manually.  Note that the equations are centered 
%  on the page, even if they overlap the label.
%  

\edef\ResetCatOfAt{\the\catcode`\@}
\catcode`\@=11

\def\polyleqno{\movetofinalcolumn \kern-\displaywidth
     \hbox to\z@\bgroup\bgroup\aftergroup\polyXleqno$}
\def\polyXleqno{\hss\egroup\kern\displaywidth$\egroup}

\def\p@l@ntry{\toks@=\expandafter{\the\toks@
              &$\@lign\displaystyle{}##{}$\hfil
              &\hfil$\@lign\displaystyle##$}}

\def\movetofinalcolumn{\relax \if@morecolumns 
     \expandafter\movetonextcolumn \fi }

\def\movetonextcolumn{&\movetofinalcolumn}

\def\polyalignno#1#2{%
  \let\if@morecolumns\iftrue
  \toks@{}%
  \looseness#1\relax %  \looseness is a scratch counter
  \loop \p@l@ntry
     \advance\looseness\m@ne \ifnum\looseness>\z@
  \repeat 
  \let\\\cr % can use \\ or \cr
  \let\eqno\movetofinalcolumn \let\leqno\polyleqno 
  \displ@y \tabskip \centering 
  \halign to\displaywidth {\tabskip\z@skip 
  \hfil$\@lign\displaystyle##$%
  \span\the\toks@ \tabskip\centering &\let\if@morecolumns\iffalse
  \tabskip\z@skip\@lign 
  \hbox to\z@\bgroup\hss$##$\egroup\crcr #2\crcr}%
}

\ResetCatOfAt \let\ResetCatOfAt\UnderfIned

%                                 Donald Arseneau
%                                 asnd@reg.triumf.ca

==================================================
%
% --------  Examples  ------------
%

\def\\{$\backslash$}
Test with constant on left, and one label on the left. Note how 
a premature \\cr is OK
$$
\polyalignno3{
99&=& 17x&+& 7y&-&2z\cr
 9&=&117x&-& 2y&-&z\leqno 1.0\cr
33&=&  3x&+&77y\cr}
$$
\\\\ can substitute for \\cr,
and a premature \\eqno works too
$$
\polyalignno3{
99&=& 17x&+& 7y&-&2z\\
 9&=&117x&-& 2y\eqno{\rm WORKS}\\
33&=&  3x&+&77y\\}
$$
Test with constant on left, numbers on right, and a label for the lot.
$$
\polyalignno3{
99&=& 17x&+& 7y&-&2z\eqno 1.1a\\
 9&=&117x&-& 2y&-&z\eqno 1.1b\leqno 1.1\\
33&=&  3x&+&77y \eqno 1.1c}
$$
Try again with constants left justified on the right, and with left equation
numbers.
$$
\polyalignno3{
17x&+&7y&-&2z&=6\leqno{1{-}2a}\\
117x&-&2y&-&z&=-2\leqno{1{-}2b}\\
3x &+&77y&&&=12\leqno{1{-}2c}\\}
$$
And finally with right justified constants on the right.
$$
\polyalignno4{% Extra pair of columns is *unused*
17x&+&7y&-&2z&=&6\eqno(1.3a)\\
117x&-&2y&-&z&=&-2\eqno(1.3b)\\
3x &+&77y&&&=&12\eqno(1.3c)
}
$$
Try with no \\eqno, just \&
$$
\polyalignno4{
0 &=& 4x^3 &-& 6x^2 &+& 3x &+& 1& (1.2.3.4.5.6.7.8.9.a)\cr
0 &=& 5x^3 &+&11x^2 & &    &-& 7& (1.2.3.4.5.6.7.8.9.b)\cr 
}
$$
and finished.
\bye %  - - - - - - - - - - - - - - - - - - - - - - - 

%                                 Donald Arseneau
%                                 asnd@reg.triumf.ca