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+\title{Maths in \protect\LaTeX: Part~5, Getting started on arrays}
+%%%
+\author[R.~A.~Bailey]{R.~.A.~Bailey\\
+Queen Mary and Westfield College\\University of London}
+%%%
+%%% here are extra environments that I have used in this article.
+%%% I hope that they do not conflict with anything of the editor's.
+%%%
+%%%
+\newcommand{\writer}[1]{{\sc #1}:}
+\newcommand{\book}[1]{{\it #1},}
+\newcommand{\publish}[2]{{\rm #1, #2,}}
+\newcommand{\byear}[1]{{\rm (#1).}}
+\newtheorem{preqn}{Exercise}
+\newenvironment{qn}{\preqn\rm}{\endpreqn}
+%%% Dear Mr Editor, I should like the content of exercises
+%%% to come out in what all copy-editors call ROMAN, not in what
+%%% Frank Mittelbach calls roman. And this should be irrespective
+%%% of the surrounding text.
+%%%
+\newcommand{\latexword}[1]{{\tt #1}}
+%%% and the above should be in typewriter-ordinary IRRESPECTIVE of the
+%%% surrounding fonts
+%%%
+%%%
+%%% editor: I have been as generic as I can, but of course you can't
+%%% put \verb inside a \newcommand. I have consistently used + as the
+%%% delimiter for \verb, except when I needed the + in Maths!
+%%%
+\newcommand{\lamport}{{\it \LaTeX: A Document Preparation System\/} by Leslie 
+Lamport}
+\newcommand{\shortlamp}{{\it The Manual}}
+\let\torl\AllTeX
+\newcommand{\phz}{\phantom{0}}
+\newcommand{\dfrac}[2]{\displaystyle\frac{#1}{#2}}
+\newcommand{\CS}{\mathop{{\rm CS}}\nolimits}
+\newcommand{\normof}[1]{{\left\Vert#1\right\Vert}^2}
+\newcommand{\littlehead}[1]{\textit{#1}}
+
+\begin{Article}
+\section*{Recall}
+This is the fifth part of a sequence of tutorials on typesetting Mathematics in 
+\LaTeX. The first four appeared in issues~4.4, 4.5, 5.1 and~5.2 of \BV. 
+The series
+includes some things which can be found in \cite{leslie}, but I am 
+working in more things which, while straightforward and necessary for 
+Mathematical work, are  not in \cite{leslie} or \cite{newleslie}. In this 
+tutorial and the next
+I cover arrays: matrices, tables of data, aligned equations, and other
+items with a two-dimensional layout.
+ 
+I do not cover the extra array goodies 
+provided by packages such as Frank Mittelbach's \latexword{array.sty}, David 
+Carlisle's plethora of array add-ons or the new packages from the American 
+Mathematical Society. I hope that someone else will write a follow-on article 
+describing some or all of those, particularly if they can give easier methods 
+of doing some of the more cumbersome items in this tutorial.
+
+In case you missed the previous
+tutorials, I remind you that
+I expect you, the reader, to do some work. Every so 
+often comes a group of exercises, which you are supposed to do. Use \LaTeX\ to 
+typeset everything in the exercise except sentences in italics, which are 
+instructions. If you are not satisfied that you can do the exercise, then tell 
+me. Either write to me at
+\begin{verse}
+School of Mathematical Sciences\\
+Queen Mary and Westfield College\\
+Mile End Road\\
+London E1 4NS
+\end{verse}
+with hard copy of your input and output, 
+or email me at \mbox{\tt r.a.bailey@qmw.ac.uk}
+with a copy of the 
+smallest possible piece of \LaTeX\ input file that contains your
+attempt at the answer.
+In either case
+I will include a solution in the following issue of \BV: you will remain 
+anonymous if you wish.
+
+\addtocounter{section}{7}
+\section{Mathematical arrays}
+\subsection{Basics}
+The \latexword{array} environment is used to set all manner of aligned 
+Mathematics. A simple example is:
+\[
+\begin{array}{l}
+\verb+\begin{array}{lrrr}+\\
+\verb+x & 1 & 2 & 5\\+\\
+\verb+x^2 & 1 & 4 & 25 \\+\\
+\verb+\end{array}+
+\end{array}
+\quad \qquad
+\begin{array}{lrrr}
+x & 1 & 2 & 5\\
+x^2 & 1 & 4 & 25 
+\end{array}
+\]
+This array has four columns. The alignment of these columns is shown in the 
+argument \latexword{lrrr} of the environment. The first column is left-aligned;
+the other three are right-aligned. It is also possible to have a centred 
+column, shown by a~\latexword{c}. Typically columns of Mathematical symbols are
+centred, integers are right-aligned and powers of a single symbol are 
+left-aligned.
+>From now on I shall refer to the argument such as \latexword{lrrr} as the
+{\em columns specifier}.
+
+Entries in each row of the \latexword{array} are separated by ampersands. 
+The end of a row 
+(except the last one) is shown by \verb+\\+. If the last few entries in a row 
+are blank there is (usually) no need to put in all the ampersands.
+
+The \latexword{array} environment can be used only in Maths mode. All its 
+entries are automatically in Maths mode, in \verb+\textstyle+. 
+If you have fractions or summations in an array you may want to precede them by
+\verb+\displaystyle+.
+
+Each entry is in a separate `box', 
+which is typeset before the array is built up. Thus switches such as 
+\verb+\bf+, \verb+\displaystyle+ and \verb+\raggedright+ extend no further than
+the end of the current entry. If a binary relation or operator, such as~$=$ 
+or $+$, occurs at the beginning or end of an entry  \torl\ cannot give it the 
+proper spacing relative to the previous or succeeding entry.
+
+\subsection{Matrices and vectors}
+The most obvious use for \latexword{array} is for a matrix. Use \verb+\left+ 
+and \verb+\right+ to surround the matrix with fences of the right size.
+For example, the matrix in Figure~\ref{mv} begins
+\begin{verbatim}
+     \left(
+     \begin{array}{rrr}
+\end{verbatim}
+
+\begin{figure*}
+\begin{minipage}{.5\textwidth}
+\[
+\left(
+\begin{array}{rrr}
+2 & -1 & -1\\
+-1 & 3 & -1\\
+-1 & -1 & 4
+\end{array}
+\right)
+\left(
+\begin{array}{r}
+5\\-3\\8
+\end{array}
+\right)
+ = 
+\left(
+\begin{array}{r}
+5\\-22\\30
+\end{array}
+\right)
+\]
+\caption{A matrix equation}
+\label{mv}
+\end{minipage}
+\begin{minipage}{.5\textwidth}
+\[
+\renewcommand{\arraystretch}{2}
+\det \left [
+\begin{array}{cccc}
+\dfrac{\partial h_1}{\partial y_1} & \dfrac{\partial h_1}{\partial y_2} & 
+    \cdots & \dfrac{\partial h_1}{\partial y_n}\\
+\dfrac{\partial h_2}{\partial y_1} & \dfrac{\partial h_2}{\partial y_2} & 
+   \cdots &  \dfrac{\partial h_2}{\partial y_n}\\
+\vdots &  \vdots &\ddots & \vdots\\
+\dfrac{\partial h_n}{\partial y_1} & \cdots & \cdots &
+   \dfrac{\partial h_n}{\partial y_n}
+\end{array}
+ \right]
+\]
+\caption{The Jacobian of~$h$}
+\label{jacob}
+\end{minipage}
+\end{figure*}
+
+The commands \verb+\vdots+ and \verb+\ddots+ are useful in matrices. They are 
+both used in Figure~\ref{jacob}.
+
+
+An array can have a single column, as in the column vectors in Figure~\ref{mv},
+or a single row, which is useful for a permutation or a row vector.
+
+\subsection{Arrays of arrays}
+Try to think logically, rather than visually, about the contents of an array.
+For example, if you have six $5\times 5$ Latin squares set out in a $2 \times 
+3$ rectangle then you might think that you have $10~$rows and $15$~columns. 
+However, it is more logical to do \verb+\begin{array}{ccc}+ and then make each 
+entry in this array a new array with five centred columns. One advantage of 
+this approach is that it makes editing
+the file much easier when you decide to 
+change the order of the Latin squares, or to lay them out in a $3\times 2$ 
+rectangle.
+
+\subsection{Changing the row spacing}
+The default spacing between the rows of an \latexword{array} is fine for simple
+matrices with numbers in, but is often insufficient for large entries, 
+particularly if you use \verb+\displaystyle+. To adjust the amount of space 
+after the current row, put an explicit length in square brackets after the 
+\verb+\\+: for example, \verb+\\[20pt]+. 
+
+I find it useful to work in multiples 
+of \verb+\jot+: this length is the usual extra space between lines of displayed
+Mathematics. Curiously, \verb+\jot+ is equal to $3$~points in all of \LaTeX's 
+three point sizes, so an adjustment in a $10$~pt document may not be correct if 
+you change the document to $11$~pt. If you work in multiples of 
+the length \verb+\baselineskip+
+then no extra adjustment is needed if the point size of the 
+document is changed.
+
+Because of the complicated way in which the row-spacing in an array is adjusted
+to fit the contents, the length given in \verb+[  ]+ is not exactly an {\em 
+addition}. You have to experiment to find out what works best: doubling the 
+length given does not usually double the space between the rows.
+
+The existence of this optional length parameter to \verb+\\+ implies that no row
+of an array may start with \verb+[+. If you need to do this, 
+precede it with an emtpy pair of braces: \verb+{}[...+.
+
+Changing the inter-row spacing for the whole array is much simpler than 
+changing it for a single row. This spacing is governed by the value of the
+number
+\verb+\arraystretch+, whose default setting is~$1$. 
+Between the rows of an array, the length \verb+\baselineskip+ is multiplied by
+\verb+\arraystretch+.
+In Figure~\ref{jacob}, 
+the rows have been spread out by putting
+\begin{verbatim}
+     \renewcommand{\arraystretch}{2}
+\end{verbatim}
+before the \verb+\begin{array}+. However, this command was given {\em after\/} 
+the start of the displayed Maths, so that it remained local to that particular 
+display. If you experiment with other values of \verb+\arraystretch+ in this 
+example you will find that there is virtually no separation between the first 
+two rows  when \verb+\arraystretch+ is smaller than~$2$.
+Figure~\ref{ima} also shows an array in which \verb+\arraystretch+
+has been altered.
+
+Arrays inside arrays can have different values of \verb+\arraystretch+.
+For example, in
+\begin{verbatim}
+     \[
+     \renewcommand{\arraystretch}{2}
+     \begin{array}{ccc} 
+     A & B & 
+        \renewcommand{\arraystretch}{1.2}
+        \begin{array}{lr}
+        ...
+\end{verbatim}
+the outer array has \verb+\arraystretch+ equal to~$2$, while the inner 
+array which is the third entry in the first row 
+has \verb+\arraystretch+ equal to~$1.2$. 
+
+\begin{figure*}
+\[
+\renewcommand{\arraystretch}{2}
+\addtolength{\arraycolsep}{1.5\arraycolsep}
+\begin{array}{cccp{1.5in}c}
+W_{1_G} & V_0 & 1 & one stratum\\
+W_\psi &\left(V_{\rm rows} + V_{\rm cols} \right) \cap V_0^\perp &
+2(q^2 + q) & 
+\raggedright
+direct sum of two
+unidentifiable strata
+&
+\\
+W_\eta &\left(V_{\rm rows} + V_{\rm cols} \right) ^\perp & q^3 & one stratum
+\end{array}
+\]
+\caption{A table of mathematical information}
+\label{ima}
+%This gives four candidates for automorphisms:
+\[
+\begin{array}{l@{\colon p + q\omega + r \omega^2 + 
+s\omega^3 + t\omega^4 \mapsto p + 
+{}}r@{\omega + {}}r@{\omega^2 + {}}r@{\omega^3 + {}}r@{\omega^4}}
+\alpha_1 & q & r & s & t\\
+\alpha_2 & s & q & t & r\\
+\alpha_3 & r & t & q & s\\
+\alpha_4 & t & s & r & q 
+\end{array}
+\]
+\caption{Four automorphisms of a splitting field}
+\label{galois}
+\end{figure*}
+
+\subsection{Changing the column spacing}
+The default inter-column spacing in an array is one quad. To change the spacing
+between a pair of adjacent columns, use \verb+@{...}+ between their alignment 
+codes in the columns specifier, putting  inside the braces
+the actual horizontal space 
+required. For example,
+\verb+\begin{array}{cc@{\qquad}c@{}c}+ 
+produces an array with four centred columns in which the inter-column spaces 
+are, in order, one quad, two quads and zero.
+
+At the start and end of the array there is usually a space of width half the 
+inter-column space. These can be suppressed by putting \verb+@{}+ at the start 
+and end of the columns specifier.
+
+The overall inter-column spacing is controlled by the value of the length
+\verb+\arraycolsep+, which is half the default inter-column space. 
+This can be changed by using \verb+\addtolength+ in the 
+usual way. I like to alter \verb+\arraycolsep+ by multiples of itself.
+%, so that 
+%the changes are correct no matter what the point size of the document. 
+For example, to treble the size of \verb+\arraycolsep+ do
+\begin{verbatim}
+  \addtolength{\arraycolsep}{2\arraycolsep}
+\end{verbatim}
+while to halve it make the last parameter \verb+-0.5\arraycolsep+. 
+In Figure~\ref{ima}, \verb+\arraycolsep+ has been multiplied by~$2.5$.
+
+Finally, the command \verb+\extracolsep+ inside an \verb+@{...}+
+can be used to insert extra space to the left of all subsequent columns:
+for example, \verb+@{\extracolsep{1em}}+. This is not suppressed by a 
+subsequent \verb+@{...}+.
+
+My remarks about quads above are not strictly true unless your current text font
+is Computer Modern roman in $10$~pt. A quad length is defined to be $1$~em, 
+whose size depends on the current font, whereas \verb+\arraycolsep+ is defined 
+to be $5$~pt, irrespective of the point size of the document. I find it very 
+odd that \LaTeX, which has been so carefully set up so that the user can 
+apparently change painlessly
+between $10$~pt, $11$~pt and $12$~pt, does not change the
+size of Maths lengths like \verb+\jot+ and \verb+\arraycolsep+ in the different
+point sizes.
+
+\subsection{Words in arrays}
+The odd word or two in an array can be put in by using \verb+\mbox+. But what 
+should you do when you have a verbal phrase extending over more than one line, 
+as in the final column of Figure~\ref{ima}?
+
+Most people's first thought is to have a separate line of the array for each 
+line of text. As usual, this visual approach is inferior to the logical one 
+when it comes to revising your file. Hard experience has shown me that you have
+to keep altering which words go on which lines as you alter other parts of the 
+array. It is better to use the fourth type of column code~\verb+p+, which 
+creates a paragraph box of a specified width. Thus the 
+columns specifier in Figure~\ref{ima} is actually
+\verb+cccp{1.5in}+. Just guess a length to put inside 
+\verb+p{...}+ at first: changing it later is much simpler than shuffling words 
+around.
+
+Figure~\ref{ima} demonstrates another good reason to use the \verb+p+ code.
+The rows are far apart, because \verb+\arraystretch+ has been magnified. But we
+{\em don't\/} want the lines of text at the end of the second row to be that 
+far apart. We could make an inner array with a smaller value of
+\verb+\arraystretch+, but it's simpler to use~\verb+p+.
+
+The only problem with the naked \verb+p+ columns is that they are set 
+justified on both sides, which is absurd in narrow columns. I usually put 
+\verb+\centering+ or \verb+\raggedright+ at the beginning of every entry in 
+such a column. Unfortunately, you cannot do this in the final column of an
+array, because \LaTeX\ gets confused about whether \verb+\\+ refers to the row 
+of the array or the line in the paragraph. If necessary, I get round this by 
+adding a dummy unused column at the end of the array. Thus I told a lie above:
+the columns specifier in Figure~\ref{ima} is really
+\verb+cccp{1.5in}c+. 
+
+\subsection{Inter-column constants}
+The code \verb+@{...}+ can be used for putting anything between two columns 
+that must appear in every row, not just spaces. This is convenient for any 
+repeated items. For example, the columns specifier in 
+Figure~\ref{galois} is
+\begin{verbatim}
+  {l@{\colon p + q\omega + r \omega^2 + 
+  s\omega^3 + t\omega^4 \mapsto p + 
+  {}}r@{\omega + {}}r@{\omega^2 + 
+  {}}r@{\omega^3 + {}}r@{\omega^4}}
+\end{verbatim}
+while the first row is just 
+\begin{verbatim}
+     \alpha_1 &q & r & s & t
+\end{verbatim}
+Notice the pairs \verb+{}+ that have been inserted to make $+$ behave 
+as a binary operator.
+
+
+\subsection{Positioning}
+By default, each array is positioned so that it is centred vertically. In 
+Figure~\ref{mv}, the equals sign is aligned with the centre of each of the 
+three arrays. To align an array on its top row, start it with
+\verb+\begin{array}[t]{...}+;
+to align it on its bottom row, use \verb+[b]+ in place of \verb+[t]+.
+
+These positionings work well for simple tables of numbers. However, the 
+logical interaction between them and \verb+\left+ and \verb+\right+ fences 
+produces bizarre results (try it!). If you want matrices aligned on their top 
+rows, as in Figure~\ref{topmat}, you have to do something quite complicated, 
+which I shall come back to later.
+
+\begin{figure*}
+\newcommand{\topthing}[1]{\begin{array}{@{}c@{}}#1\\\phz\end{array}}
+\newcommand{\vecmu}{\mbox{\boldmath$\mu$}}
+\newcommand{\vecx}{{\bf x}}
+\[
+\topthing{3(\vecx - \vecmu)^{\rm T} \Sigma^{-1} (\vecx - \vecmu) = (x-1,y+2)} 
+\left[\begin{array}{rr}
+4&1\\1&1
+\end{array}
+\right] 
+\left(\begin{array}{c}
+x-1\\y+2
+\end{array}
+\right)
+\]
+\caption{Matrices aligned on their top rows}
+\label{topmat}
+\end{figure*}
+
+\subsection{Cases}
+We often write equations where the value on the right-hand side depends on some
+condition, as in
+%The $(\omega,t)$-entry of~$X$ is defined by
+\begin{equation}
+X(\omega,t) = \cases{1 & if $\phi(\omega) = t$\cr
+     0 & otherwise.}
+\label{eqncase}
+\end{equation}
+You can do the right-hand side of this using an \latexword{array}, but it is
+simpler to use the direct plain \TeX\ command \verb+\cases+. The right-hand 
+side of Equation~(\ref{eqncase}) was produced with
+\begin{verbatim}
+    \cases{1 & if $\phi(\omega) = t$\cr
+           0 & otherwise.}
+\end{verbatim}
+The argument of \verb+\cases+ is like a two-column array, with its rows 
+separated by \verb+\cr+ instead of \verb+\\+. The first entry in each row is in
+Maths mode; the second is in non-Maths mode. Both
+columns are left-aligned.
+
+\subsection{Binomial coefficients}
+If you write your binomial coefficients as
+$6 \choose 2$ rather than ${}^6{\rm C}_2$ 
+you may be tempted to build them with an
+\latexword{array}. Don't. Use the plain \TeX\ infix command \verb+\choose+, as 
+follows.
+\[
+\verb+6 \choose 2+ \quad \qquad {6 \choose 2}
+\]
+It comes out rather differently in \verb+\textstyle+ and in 
+\verb+\displaystyle+.
+
+Because \verb+\choose+ does not have its two arguments in braces, it is
+safest if you enclose the whole expression in grouping braces, as in
+\verb+{6 \choose 2}+, to ensure that \verb+\choose+ does not search too far 
+back or too far forward for its arguments.
+If you need to use this a lot, make a \LaTeX-style macro such as
+\begin{verbatim}
+     \newcommand{\binom}[2]{{#1\choose#2}}
+\end{verbatim}
+
+
+The command \verb+\atop+ works similarly, but omits the parentheses.
+
+\section*{Exercises}
+\addtocounter{preqn}{54}
+\begin{qn}
+The matrix $\left[\begin{array}{cc}
+1 & 3\\
+3 & 5
+\end{array}
+\right]$
+is symmetric.
+\end{qn}
+
+\begin{qn}
+The determinant is given by
+\[
+\det A = \left |
+\begin{array}{ccc}
+a_{11} & \ldots & a_{1n}\\
+\vdots & \ddots & \vdots\\
+a_{n1} & \ldots & a_{nn}
+\end{array}
+\right|
+\]
+\end{qn}
+
+\begin{qn}
+For a rotation,
+\[
+\left(\begin{array}{rr}
+\cos \theta & \sin \theta\\
+-\sin \theta & \cos \theta
+\end{array}
+\right)
+\left(\begin{array}{c}x \\ y\end{array}\right)
+ = 
+\left(\begin{array}{c}
+x\cos \theta + y\sin \theta\\
+y\cos\theta - x\sin\theta
+\end{array}
+\right).
+\]
+\end{qn}
+
+\begin{qn}
+Here is a pair of mutually orthogonal Latin squares:
+\[
+\begin{array}{cc}
+\begin{array}{ccc}
+A & B & C\\
+B & C & A\\
+C & A & B\\
+\end{array}
+&
+\begin{array}{ccc}
+\alpha & \beta & \gamma\\
+\gamma & \alpha & \beta\\
+\beta & \gamma & \alpha
+\end{array}
+\\ \\
+\mbox{Square 1} & \mbox{Square 2}
+\end{array}
+\]
+\end{qn}
+
+\begin{qn}
+A few real functions:
+\[
+\begin{array}{lp{2in}c}
+x \mapsto x^2 & \raggedright differentiable everywhere & \\
+x \mapsto \left| x\right | & \raggedright 
+differentiable everywhere except at the origin &\\
+x \mapsto \lfloor x \rfloor & \raggedright 
+continuous everywhere except at the integers
+\end{array}
+\]
+\end{qn}
+
+\begin{qn}
+\it Typeset the table in Figure~\ref{ima}.
+\end{qn}
+
+\begin{qn}
+\addtolength{\arraycolsep}{-0.5\arraycolsep}
+Pascal's Triangle begins
+\[
+\begin{array}{*{13}{c}}
+ & & & & & & 1 & & & & & & \\
+ & & & & & 1 & & 1 & & & & & \\
+ & & & & 1 & & 2 & & 1 & & & &\\
+ & & & 1 & & 3 & & 3 & & 1 & & &\\
+ & & 1 & & 4 & & 6 & & 4 & & 1 & &\\
+ & 1 & & 5 & & 10 & & 10 & & 5 & & 1 &\\
+ 1 & & 6 & & 15 & & 20 & & 15 & & 6 & & 1
+\end{array}
+\]
+\end{qn}
+
+\begin{qn}
+The Stirling numbers of the second kind are defined by
+\[
+S(n,k) =  \frac{1}{k!} \sum_{j=1}^{k} (-1)^{k-j} {k \choose j} j^n.
+\]
+\end{qn}
+
+\begin{qn}
+If the random variable $X$ is symmetric about~$0$ and if $Y = X^2$ then
+\[
+f_Y(y) = \cases{ 0 & if $y \leq 0$ \cr
+          \dfrac{f_X(\sqrt{y})}{\sqrt{y}} & if $y > 0$.}
+\]
+\end{qn}
+
+\begin{qn}
+\[
+\left[
+\begin{array}{cccc}
+\sigma_{11} & \sigma_{12} &\cdots & \sigma_{1n}\\
+\sigma_{21} & \sigma_{22} &\cdots & \sigma_{2n}\\
+ \vdots & \cdots & \ddots& \vdots\\
+\sigma_{1n} & \cdots & \cdots & \sigma_{nn}
+\end{array}
+\right]
+\]
+\end{qn}
+
+\begin{qn}
+\[
+\begin{array}{ccccccccc}
+M_0 & \subseteq & M_1 & \subseteq & M_2 & \subseteq & \cdots & \subseteq & 
+M_m\\
+\uparrow && \uparrow && \uparrow\\
+\mbox{null} && \mbox{linear} && \mbox{quadratic}
+\end{array}
+\]
+\end{qn}
+
+\section{Tables of data}
+\subsection{General}
+The extra topics 
+that I cover in this section are most pertinent to tables of data, but do 
+sometimes arise in Mathematical arrays. Likewise,
+many considerations about Mathematical arrays also apply to tables of data.
+For example, the table in Figure~\ref{eel} does not really have eight columns:
+it has two long columns but has been broken down into four parts to fit on the 
+page nicely. So it is set as an array with one row and four 
+columns, each of whose entries is an array with two columns. 
+
+\LaTeX\ has another environment, called \latexword{tabular}, which is very 
+similar to \latexword{array}. It may be used inside or outside Maths mode. By 
+default, each of its entries is set in ordinary mode. However, any
+extra space on either side of the ampersands has no effect.  
+The length \verb+\arraycolsep+ is replaced by the length 
+\verb+\tabcolsep+.
+Tables of data often have
+a mix of words and symbols, and you have to decide which is less trouble: using 
+an \latexword{array} and putting the words in \verb+mbox+es or 
+\verb+p+~columns, or using a \latexword{tabular} and putting the symbols in 
+\verb+$...$+. If the latter, don't forget to put numbers in \verb+$...$+ if 
+they involve a minus sign.
+
+\begin{figure*}
+\[
+\begin{tabular}{c|c|c|c}
+\begin{tabular}{cc}
+Year 1 & Year 2\\
+\hline
+269 & 466\\
+100 & 219\\
+197 & 421\\
+216 & 708\\
+\end{tabular}
+&
+\begin{tabular}{cc}
+Year 1 & Year 2\\
+\hline
+134 & 590\\
+\phz 74 & 137\\
+\phz 88 & 356\\
+\phz 25 & 212
+\end{tabular}
+&
+\begin{tabular}{cc}
+Year 1 & Year 2\\
+\hline
+211 & 505 \\
+102 & 363\\
+191 & 563\\
+\phz 67 & 338\\
+\end{tabular}
+&
+\begin{tabular}{cc}
+Year 1 & Year 2\\
+\hline
+209 & 352\\
+\phz 29 & 254\\
+\phz 19 & 106\\
+\phz 44 & 268
+\end{tabular}
+\end{tabular}
+\]
+\caption{Number of eelworms in two successive years}
+\label{eel}
+\end{figure*}
+
+\subsection{Horizontal rules}
+
+\begin{figure*}[hbtp]
+\[
+\begin{tabular}{l@{\qquad}l@{\qquad}lrrrrrrrrr}
+\hline
+ & & & \multicolumn{3}{c}{1961} & \multicolumn{3}{c}{1971} & 
+\multicolumn{3}{c}{1981}\\
+\hline
+\multicolumn{3}{l}{Billion passenger kilometres travelled by:}\\
+ & \multicolumn{2}{l}{Air} && 1 &&& 2 &&& 3&\\
+ & \multicolumn{2}{l}{Rail} && 39 &&& 36 &&& 34&\\
+ & \multicolumn{2}{l}{Road}\\
+ & & Buses and coaches && 67 &&& 51 &&& 42&\\
+ & & Cars, taxis and two-wheeled motor vehicles && 142 &&& 284 &&& 380&\\
+ & & Bicycles && 10 &&& 4 &&& 4&\\
+\cline{4-12}
+ & \multicolumn{2}{l}{Total} && 259 &&& 377 &&& 406&\\
+\hline
+\end{tabular}
+\]
+\caption{Passenger transport (from \protect\cite{keyd})}
+\label{cars}
+\end{figure*}
+
+To obtain a horizontal rule stretching the full width of the array, use 
+\verb+\hrule+. Don't forget to finish the previous row with \verb+\\+, even if
+it is the last row of the array, as at the end of Figure~\ref{cars}. Two or 
+more \verb+\hrule+s can follow each other with no intervening \verb+\\+. 
+A little vertical space is put between two such horizontal rules.
+However, any vertical rules in the array (see below) will be suppressed between
+the successive horizontal rules.
+
+For a horizontal rule spanning only columns \mbox{$2$--$4$}, say, use 
+\verb+\cline{2-4}+. Again, finish the preceding row with \verb+\\+. 
+Figure~\ref{cars} shows an example of this.
+Successive 
+\verb+\cline+s will be interpreted as being in the same row, as in 
+\verb+\cline{2-4}\cline{7-7}+, for example, so that
+\verb+\cline{2-4}\cline{2-4}+ does {\em not\/} produce two horizontal lines 
+with a little space between them. Likewise, there is no space between an 
+\verb+\hline+ and an immediately preceding or succeeding \verb+\cline+.
+
+\subsection{Vertical rules}
+
+A vertical rule is obtained by putting \verb+|+ at the appropriate place in the
+columns specifier. For example, the columns specifier for the 
+\latexword{tabular} in Figure~\ref{travel} is
+\begin{verbatim}
+     |r@{ Month}l@{\quad\pounds}r|
+\end{verbatim}
+Two or more \verb+|+s may come next to each other.
+
+\begin{figure*}
+\begin{minipage}{.5\textwidth}
+\[
+\begin{tabular}{|r@{ Month}l@{\quad\pounds}r|}
+\hline
+\multicolumn{3}{|c|}{Premiums}\\
+\hline
+1 & & 18\\
+3 & s & 48\\
+6 & s & 95 \\
+9 & s & 136\\
+\hline
+\end{tabular}
+\]
+\caption{Travel insurance}
+\label{travel}
+\end{minipage}
+\begin{minipage}{.5\textwidth}
+
+\[
+\begin{tabular}{lr@{}l}
+Bootstrap & 0.301\\
+Jackknife & 0.314\\
+Delta method & 0.244& *\\
+Half-samples & 0.364 & \dag\\
+Random subsampling & 0.423 & \ddag\\
+Normal theory & 0.302
+\end{tabular}
+\]
+\caption{Sticking out to the right (from \protect\cite{brad})}
+\label{efron}
+\end{minipage}
+\end{figure*}
+
+For a vertical rule in a single item, stretching from the top to bottom of its 
+row, use \verb+\vline+. This cannot be used in a \verb+p+~column, but may be 
+put inside \verb+@{...}+.
+
+If you omit the trailing ampersands in a row with few items the trailing 
+vertical rules will also be omitted. This can be a nuisance in a sparse table.
+If you want a vertical rule at each side of a sparse table, consider doing 
+this:
+\begin{verbatim}
+     \begin{array}{|c|}
+     \begin{array}{@{}...@{}}
+     ...
+     \end{array}
+     \end{array}
+\end{verbatim}
+
+The length \verb+\arrayrulewidth+ controls the thickness of  all the horizontal
+and vertical rules just described. The width of the space between the vertical 
+rules created by \verb+||+ in a the columns specifier is controlled by the 
+length \verb+\doublerulesep+, as is the vertical space between two successive 
+\verb+\hline+s.
+
+\subsection{Spanning several columns}
+The \verb+\multicolumn+ command is used to create an entry spanning several 
+columns. It takes three arguments. The first is the number of columns to be 
+spanned; the second is the code for the type of column to be used;
+the third is the text of the entry. 
+\begin{verbatim}
+     \begin{tabular}{crrrr}
+     Type & \multicolumn{4}{c}{Grade}\\
+     \cline{2-5}
+      & 12 & 16 & 24 & 30\\
+     \hline
+     $A$ & 0 & 1 & 21 & 24\\
+     $B$ & 1 & 6 & 24 & 13
+     \end{tabular}
+\end{verbatim}
+\[
+\begin{tabular}{crrrr}
+Type & \multicolumn{4}{c}{Grade}\\
+\cline{2-5}
+ & 12 & 16 & 24 & 30\\
+\hline
+$A$ & 0 & 1 & 21 & 24\\
+ $B$ & 1 & 6 & 24 & 13
+\end{tabular}
+\]
+
+Any \verb+|+ in the columns specifier is 
+regarded as belonging to its preceding column, and may be overwritten by
+\verb+\multicolumn+ unless it is included in the second argument. The exception
+is any \verb+|+ to the left of the first column, which is regarded as part of 
+the first column.
+For example, the first row in Figure~\ref{travel} is
+\begin{verbatim}
+     \multicolumn{3}{|c|}{Premiums}\\
+\end{verbatim}
+Expressions \verb+@{...}+ in the columns specifier are similarly assigned to 
+columns and similarly overwritten.
+
+Indented row labels for subheadings can also be conveniently obtained by using 
+\verb+\multicolumn+. The \latexword{tabular} in Figure~\ref{cars} begins with 
+three \verb+l+~columns. The first row after the second horizontal rule is a 
+\verb+\multicolumn+ spanning three columns; the next begins with \verb+&+ 
+followed by a 
+\verb+\multicolumn+ spanning two columns.
+
+Unfortunately, if the width of a multicolumn entry spanning three or more 
+columns is more than the total width of the spanned columns, \LaTeX\ cannot 
+adjust the spacing between those columns to make it equal: it puts all the 
+extra space to the left of the last of the spanned columns. The only way that
+I know
+around this is to use \verb+@{...}+ to put the same amount of extra space 
+between each adjacent pair of the spanned columns.
+
+\subsection{Changing the type of an entry}
+Although its name does not suggest it, the command
+\verb+\multicolumn+ is also used to 
+change the type of a single entry. This is most often used to give a centred 
+heading to a column of right-aligned data.
+\begin{verbatim}
+     \begin{array}{rrr}
+     \multicolumn{1}{c}{n} & 
+     \multicolumn{1}{c}{n^2} &
+     \multicolumn{1}{c}{\phi(n)}\\
+     9 & 81 & 6\\
+     10 & 100 & 4\\
+     11 & 121 & 10
+     \end{array}
+\end{verbatim}
+\[
+\begin{array}{rrr}
+\multicolumn{1}{c}{n} & \multicolumn{1}{c}{n^2} 
+& \multicolumn{1}{c}{\phi(n)}\\
+9 & 81 & 6\\
+10 & 100 & 4\\
+11 & 121 & 10
+\end{array}
+\]
+As this example shows, this works fine when the centred heading is narrower 
+than the widest other entry in the column. However, when it is wider then all 
+the other entries in the column are aligned with its right-hand edge, when 
+probably what you wanted was to have them right-aligned among themselves and 
+then all centred under the heading.
+How can this be fixed?
+
+If most of the numbers are the same length, a reasonable solution is to make the
+whole column centred and put in \verb+\phantom{0}+s to make the shorter numbers
+as long as the longest. This works because all digits have the same width.
+ That is what I did in Figure~\ref{eel}.
+
+I adopted a different solution in Figure~\ref{cars}, where I decided that the 
+year headings, not being numbers to be added to other numbers in their columns,
+should be centred. This time I used the columns specifier \verb+rrr+ for each 
+column of data. In the headings I used \verb+\multicolumn{3}{c}{...}+ and in 
+the data rows I left one blank entry either side of each data entry. The result
+is rather widely spaced: space could be saved by using \verb+@{...}+ between 
+each data column and its blank neighbours.
+
+The command
+\verb+\multicolumn{1}+ can also be used to remove a vertical rule or the 
+contents of an \verb+@{...}+ from one particular row. For example, if the 
+columns specifier is \verb+l|r+ then the vertical rule is missing from any row 
+whose first entry is made with \verb+\multicolumn{1}{l}+.
+
+\subsection{Decimal tabulation}
+Sometimes a column of numbers with decimal points should be aligned
+on those decimal points. If they all have the same number of digits after the 
+point, there is no problem: simply use an \verb+r+ column. If only a few numbers
+are missing a few digits after the point, fill in the spaces with 
+\verb+\phantom{0}+ and still use an \verb+r+ column. Only if you have a long 
+column of numbers with very variable numbers of digits after the point is it 
+worth going to the trouble of using the columns specifier \verb+r@{.}l+ and 
+entering a number such as 3.142 as \verb+3 & 142+.
+
+\subsection{Things that stick out}
+Sometimes you have a column of numbers that should be right-aligned but some of
+the entries have accretions that should stick out to the right, as in 
+Figure~\ref{efron}, or a left-aligned column with pieces sticking out to the 
+left. There are three ways of dealing with this, each appropriate in different 
+circumstances.
+
+
+If the pieces that stick out are all digits, use \verb+\phantom{0}+ as 
+described above.
+
+Sometimes the piece that sticks out is very small and occurs in only one row. 
+If a displayed Mathematical array ends the sentence 
+then sometimes the full stop needs to
+come outside the array, vertically centred (with \verb+\end{array}.\]+); but 
+sometimes the final row of the array
+needs a full stop at the end of it without upsetting the column alignment.
+Then the trick is to add the full stop in such a way that \torl\ thinks that it
+doesn't take up any space. According to \cite[page~98]{leslie}, you do this 
+with
+\verb+\makebox[0pt][l]{.}+, but the plain \TeX\ \verb+\rlap{.}+ is shorter and 
+achieves the same effect. For a small item sticking out on the left use
+\verb+\makebox[0pt][r]{ }+ or \verb+\llap{ }+. Of course, the content of 
+\verb+\makebox{ }+ or \verb+\rlap+ or \verb+\llap+ is not in Maths mode.
+
+Statisticians sometimes put a question mark to indicate dubious data, and 
+scientists often decorate numbers with stars to show statistical significance.
+In such cases it is best to use the columns specifier \verb+r@{}l+ for the 
+column: the \verb+r+ part is used for the numbers and the \verb+l+ part is used
+for any decorations. 
+In Figure~\ref{efron} the columns specifier is \verb+lr@{}l+ and
+the fourth line is
+\begin{verbatim}
+     Half-samples & 0.364 & \dag\\
+\end{verbatim}
+
+To see the difference between these last two solutions, compare the following.
+\begin{center}
+$
+\begin{array}{ll}
+\begin{array}{l}
+\verb+\begin{array}{|r@{}l|}+\\
+\verb!20 & ++!\\
+\verb+\end{array}+
+\end{array}
+&
+\begin{array}{|r@{}l|}
+20 & ++
+\end{array}
+\\ \\
+\begin{array}{l}
+\verb+\begin{array}{|r|}+\\
+\verb!20 \rlap{$++$}!\\
+\verb+\end{array}+
+\end{array}
+&
+\begin{array}{|r|}
+20 \rlap{$++$}
+\end{array}
+\end{array}
+$
+\end{center}
+
+\subsection{Columns specifiers}
+We have seen that a columns specifier is a string of the following objects:
+\verb+r+, \verb+l+, \verb+c+, \verb+p{...}+, \verb+@{...}+ and \verb+|+. It is 
+easy to lose track when setting up the column specifier for a complicated 
+array. However, repeated elements may be simplified by using \verb+*+. Thus an
+array with $15$~centred columns can be specified with \verb+*{15}{c}+. This 
+shorthand is really advantageous for something complicated like
+\verb+l|*{4}{r@{}lrr|}+. Moreover, \verb+*+~expressions can be nested within 
+each other.
+
+\section*{Exercises}
+\begin{qn}
+\renewcommand{\arraystretch}{1.6}
+\[
+\begin{array}{c|ccccc}
+\multicolumn{6}{c}{\mbox{Values of $\nu_{\gamma\chi}$ for the design}}\\
+\hline
+\gamma & \chi_0 & \chi_1 & \chi_2 &\chi_3 & \chi_4\\
+\hline
+\mu & 1 & 0 & 0 & 0 & 0 \\
+\sigma & 1 & 0 & 0 & 0 & 0\\
+\rho & 1 & 0 & 0 & \frac{1}{4} & 0\\
+\kappa & 1 & \displaystyle\frac{3 + e_2 - e_1}{36} & 
+\displaystyle\frac{3 + e_4 - e_2}{36} & 0 &
+\displaystyle\frac{3 + e_1 - e_4}{36}\\
+\varepsilon & 1 & 1 & 1 & 1 & 1\\
+\hline
+\end{array}
+\]
+\end{qn}
+
+\begin{qn}
+Table of results:
+\[
+\begin{array}{l|cc|r}
+ & B_1 & B_2\\
+\hline 
+A_1 & 37 & \phz53 & 90\\
+A_2 & 41 & \phz48 & 89\\
+\hline
+ & 78 & 101 & 179
+\end{array}
+\]
+\end{qn}
+
+\begin{qn}
+\it Typeset the table in Figure~\ref{eel}.
+\end{qn}
+
+\begin{qn}
+\[
+\begin{array}{lrrrr@{.}l}
+\mbox{Source} & \mbox{df} & \multicolumn{1}{c}{\mbox{SS}} & 
+\multicolumn{1}{c}{\mbox{MS}} & \multicolumn{2}{c}{\mbox{VR}}\\
+\hline
+x_1 & 1 & 104474 & 104474 & 1220 & 5\\
+x_2 \mid x_1 & 1 & 2284 & 2284 & 26 & 68\\
+\cline{1-3}
+\mbox{regression} & 2 & 106758\\
+\mbox{residual} & 15 & 1284 & 85.6\\
+\cline{1-3}
+\mbox{`total'} & 17 & 108042
+\end{array}
+\]
+\end{qn}
+
+\begin{qn}
+\it Typeset the table in Figure~\ref{cars}.
+\end{qn}
+
+\begin{qn}
+\[
+\begin{tabular}%
+{||r@{ }l|r@{}l|cr@{}lc|@{}c@{\hspace{\arraycolsep}\pounds}r||}
+\multicolumn{2}{||c|}{Date} & \multicolumn{2}{c|}{Miles} & 
+\multicolumn{4}{c|}{Gallons} & \multicolumn{2}{c||}{Cost}\\
+\hline
+27 & December & 46674 & && 4 & .18 &&& 5.56\\
+3 & January & 46038 & & &3 & .93  && &5.00\\
+6 & January & 47956 & ? && 7 & .1 && &10.50\\
+10 & January & 47292 & && 7 & .89&& & 10.50\\
+17 & January & 47464 &&& 4 & &&&5.00\\
+\multicolumn{10}{||c||}{\dotfill}\\
+3 & February & 48112 &&& 6 & .88 &&& 9.00
+\end{tabular}
+\]
+\end{qn}
+
+\begin{thebibliography}{9}
+\bibitem{keyd}
+\writer{Central Statistical Office}
+\book{Key Data 1986}
+\publish{Her Majesty's Stationery Office}{London}
+\byear{1986}
+
+\bibitem{brad}
+\writer{Efron,~B.}
+\book{The Jackknife, the Bootstrap and Other Resampling Plans}
+\publish{Society for Industrial and Applied Mathematics}{Philadelphia}
+\byear{1982}
+
+\bibitem{leslie}
+\writer{Lamport,~L.}
+\book{\LaTeX: A Document Preparation System}
+first edition,
+\publish{Addison-Wesley}{Reading, Mass.}
+\byear{1986}
+
+\bibitem{newleslie}
+\writer{Lamport,~L.}
+\book{\LaTeX: A Document Preparation System}
+second edition, 
+\publish{Addison-Wesley}{Reading, Mass.}
+\byear{1994}
+
+\end{thebibliography}
+\end{Article}
+
+
-- 
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