\documentclass[12pt,a4paper]{article}
\usepackage{amssymb,amsmath,amsxtra,amsthm}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{question}{Q.}
\newtheorem*{question*}{Q}
\newtheorem{exercise}[question]{Exercise}
\begin{document}
\title{Thoughts on the Essay Question}
\author{T.~W.~K\"{o}rner}
\maketitle
\begin{footnotesize}
\noindent
{\bf Small print} The opinions expressed in this
note are the author's own. Even the best advice
(and there is no reason to suppose that the advice
here is the best advice) does not apply to all
people and in all circumstances\footnote{`Two days
wrong!' sighed the Hatter `I told you butter wouldn't
suit the works!' he added, looking angrily at the
March Hare.

`It was the \emph{best} butter,' the March Hare
meekly replied.

`Yes, but some crumbs must have got in as well,'
the Hatter grumbled: `you shouldn't have put it in with
the breadknife.' [Alice in Wonderland]}. The advice
given is intended for students taking 1A, 1B and,
particularly, Part II of the Cambridge mathematics tripos.
It does not apply to Part~III which is a very different
course with different objectives.

I should {\bf very much} appreciate being told
of any corrections or possible improvements. This document
is written in \LaTeX2e and stored in the file labelled
\verb+~twk/FTP/Excess.tex+ on emu in (I hope) read permitted form.
It also available via my web home page.
My e-mail address is \verb+twk@dpmms+.
\end{footnotesize}
\section{What is an essay question?} What students
dislike most about essay questions is that they
have to make choices for themselves about what
to include and how to treat it. Since this dislike
is often shared by examiners (giving students
genuine choices makes marking much more difficult)
many essay questions are simply disguised bookwork.
Consider the following essay question.

\begin{question}\label{Steinitz} 
Write an essay on the Steinitz exchange
lemma and its use in establishing the notion of dimension.
\end{question}

\noindent A little thought shows that it could be rewritten
as follows.

\begin{question*}{\bf\ref{Steinitz}$\mathbf '$.}
Define the terms `spanning set',
`linearly independent set' and `basis'.

State and prove the Steinitz exchange lemma.

Show that any vector space $V$ with a finite
spanning set has a basis. Use the Steinitz
exchange lemma to show that all bases of
$V$ have the same number of elements
and so we can define the dimension of $V$.
\end{question*}

Now consider a possible essay question on
numerical analysis.

\begin{question}\label{Gauss} Suppose that we wish to
find 
\[\int_{-1}^{1}f(x)\,{\mathrm d}x\]
but that it is expensive to obtain 
values of $f$. Discuss the use of orthogonal
polynomials in evaluating the integral.
\end{question}

In principle, this is a more open question
than Question~\ref{Steinitz} but in practice
it is limited by the fact that the essay
must be on a subject that you have covered.
If the only method of integration discussed
in the course is Gaussian quadrature then
the question can only be on Gaussian quadrature
and can be restated as follows.

\begin{question*}{\bf\ref{Gauss}$\mathbf '$.}
Explain the method of Gaussian quadrature.
\end{question*}

A little reflection converts this into a bookwork question.
\begin{question*}{\bf\ref{Gauss}$\mathbf ' '$.}
If $x_{1},\ x_{2},\ \dots,\ x_{n}$ are distinct
points of $[-1,1]$ show that we can find
$\lambda_{1},\ \lambda_{2},\ \dots,\ \lambda_{n}$
such the formula
\begin{equation*}
\int_{-1}^{1}f(x)\,{\mathrm d}x
=\sum_{j=1}^{n}\lambda_{j}f(x_{j}) \tag*{$\bigstar$}
\end{equation*}
is exact for all polynomials of degree $n-1$
or less.

If the $x_{1},\ x_{2},\ \dots,\ x_{n}$ are chosen to be 
the roots of the Legendre polynomial of degree
$n$ show that the formula $\bigstar$
is exact for all polynomials of degree $2n-1$
or less. Conclude that the method of approximate
integration given by $\bigstar$ is likely to be
improved by this choice of $x_{j}$.
\end{question*}

There is substantially more that can be said on Gaussian 
quadrature but (supposing the course to say no more)
the rules of the game say that you are not expected
to know more than the syllabus demands.

The next example of an essay question is, in my opinion,
rather more demanding.

\begin{question}\label{Power}
Assuming any results about $\exp$
and $\log$ that you need, define $x^{\alpha}$
for $x>0$ and $\alpha>0$ and establish
the basic results concerning the function
$x \mapsto x^{\alpha}$.
\end{question}

Here the problem is to decide which are the
basic properties. Perhaps the reader should make
her own list before looking at mine.
 
\begin{question*}{\bf\ref{Power}$\mathbf '$.} Set
\[x^{\alpha}=\exp(\alpha\log x).\]

Stating carefully any results about $\exp$
and $\log$ that you need prove the following results.
(Here $x,y>0$, $\alpha$ and $\beta$ are real and
$n$ is strictly positive integer.)

(i) $(xy)^{\alpha}=x^{\alpha}y^{\alpha}$.

(ii) $x^{({\alpha}+{\beta})}=x^{\alpha}x^{\beta}$.

(iii) $x^{\alpha\beta}=(x^{\alpha})^{\beta}$.

(iv) ${\displaystyle \frac{dx^{\alpha}}{dx}=\alpha x^{\alpha -1}.}$

(v) ${\displaystyle x^{n}=\overbrace{xx\ldots x}^{n}.}$
\end{question*}

Of course there are other ways to define $x^{\alpha}$
and the examiner would accept these\footnote{However,
the other ways that I can think of involve
quite a lot of hard work.}. If I was marking 
Question~\ref{Power}, I think that I would give full marks
to an essay which included a satisfactory treatment
of (v) (it is essential that our new definition
of $x^{n}$ should coincide with the old) together with
three out of the four points (i), (ii), (iii) and (iv).

Perhaps you disagree with my list (remember that I only
ask for three out of the first four points). Do you disagree
that points~(i) to~(v) are basic or do you feel that
that I should have included more points? Some people
might feel that I should have included the fact that
$x^{0}=1$ (though this follows from (ii)) or shown
explicitly that our new and old definitions
of $x^{\alpha}$ coincide when $\alpha$ is rational
(although this follows from (v), I would certainly
do this in any lecture course). Others might
wish to include the fact that $x^{\alpha}\rightarrow\infty$
as $x\rightarrow\infty$ if $\alpha>0$. 
On the other hand, most people
would agree that the Taylor series for $(1+x)^{\alpha}$
is not a basic result in this context. 
There is room for disagreement
but I hope that, after reflection, most mathematicians
will agree that my list was reasonable (particularly
if my marking scheme was flexible).

My final example is a fully fledged essay.
\begin{question}~\label{Moebius}
Write an essay on M\"{o}bius maps.
\end{question}
I shall use this as my typical example and I shall
assume only the material contained in the first year
course `Algebra and Geometry'.

Question~\ref{Moebius} is a full fledged essay because
it involves organising material in a different way
from the course (results on M\"{o}bius maps
are scattered throughout the lectures), there is
too much material to cover in detail
in a short essay so we must select which topics
to treat and, even after making a choice of topics,
it is not possible to give all the proofs in full
so we must choose what to prove, what to sketch
and what to state.
\section{How is the essay marked?} If examiners have to
mark a large number of mathematics essays 
quickly and reasonably fairly, it seems to me that
they have have very little choice but to prepare
a mark scheme in advance. It will not be quite
as crude as `give 2 marks for each fact mentioned'
but will consist of instructions like
`give 2 marks for mention of either $A$ or $B$
and 3 marks if both mentioned'. (Although the
examiner will try to cover all possibilities
in advance the mark scheme will almost
inevitably need to be modified to cover
unexpected paths taken by examinees. If the
changes are substantial the examiner may
need to run through all the scripts a second time
to maintain consistency).

If the essay allows genuine choice then the
total marks available will exceed the maximum
permitted mark for a question and the examiner
will need to apply a scaling formula. Often
the scaling formula is the crude
\[\text{final mark}=\min(\text{total marks, maximum mark}).\]

It is generally accepted that the distribution
of marks for essay questions looks very different
from the distribution of marks for ordinary
`problem' or `bookwork' questions\footnote{Of course,
any distribution can be made to resemble any
other by `grading to the curve' but, in my view,
such drastic rescaling would be ridiculous. More to
the point, no such rescaling is attempted in
Cambridge mathematics exams.}. Since you can either
do a piece of mathematics or not, marks for
`problem' questions tend to be either very high
or very low and the same is true to a lesser
extent for `bookwork' questions. However,
it is harder to produce a very good or very bad
essay and so it is harder to get very high or
very low marks for an essay question.

Most directors of study would therefor advise you
do do a non-essay question in preference to an essay
question \emph{if you can}. The sting is, of course,
in the words \emph{if you can}. A glance around the
room two hours into
an examination establishes that very few students
can find enough questions that they can do
to occupy the full examination period. You are
certainly better off writing an essay than staring at the
ceiling. I therefore tell my
students that they should normally
expect to tackle any appropriate essay.

Here is some further specific advice.
\begin{itemize}
\item \emph{Tackle non-essay questions first.} If you
get stuck there is longer for your subconscious
to work at the problem. Return to the question
at intervals to see if your subconscious has come up
with anything.

\item \emph{Do the essay questions last.} The difficulty
with essay questions is getting the organisation 
right rather than solving mathematical problems.
Visit the question from time to time to make
notes and to let your subconscious consider
the matter. When you have definitely finished
with the non-essay questions look at your notes,
pick up your pen and write like hell.

\item \emph{Allocate roughly equal time to all
the essay questions you tackle.} If your essay is good
it will already have gained most of the marks available
and extra work can not gain many marks. If your essay is
poor any improvement is likely to gain marks.
\end{itemize}
 
\section{Planning an essay for supervision}
Since most of the marks for an essay are
for specific points rather than for presentation
and since any examination question must be
done in a hurry you neither can nor should
aim very high in the actual examination.

There are, however, various reasons for
trying to do substantially better when preparing
the same essay for supervision.

(1) If a similar essay turns up in the exam you
will have a good model to guide you.

(2) If your supervision standards are high you
can relax them in the exam. If they are low you
will find it hard to improve them under the
stress of examination conditions.

(3) Writing a good essay is a good form of revision.
It helps you to see how different parts of the course
hang together.

(4) Remember, however, that in a few years you may
be defending a PhD, proposing unpopular changes
to top management, giving a talk to a selection
board or even lecturing to your own students.
In such circumstances, it matters not merely
what you say but how you say it. You will need
to marshal your arguments and present them attractively.
Treat the essay as training for such circumstances.

If an essay forms part of a week's work for
a supervision you should start thinking about it at the beginning
of the week, jotting down ideas as they occur
to them. It is a good idea to run through the
essay in your head several times during the week
before putting pen to paper. (Compare my
advice for doing the essay in the exam.)

Now imagine that you have to give a talk to other
mathematicians who know nothing about the subject.
Part of your job is to do some hard mathematics
but you must do more than this. You must also try
to give some idea of where your topic comes
from and where it is going. You also wish
to convince your audience that the topic is interesting.

One problem that faces you at once is that there
is not time to prove all the results that you wish to
discuss. This should not be treated as an invitation
to prove nothing. Instead you must be selective.
Your imaginary audience wants to see the important
things proved properly but is happy to take
the less important things on trust. Thus
your first business is to choose the central
theorem of your essay which will be proved fully.
Next you need to consider the lemmas you will
need in your proof. Some of these you may
decide to prove, for some you may decide to give
a sketch proof, others you will decide to merely state.
In the same way, your central theorem may have
associated corollaries, and counter examples
and you will need to decide how much detail
to give for each.

Here is a possible check list. Not all the
items will be appropriate to each essay
and those that do appear may occur in
a different order. (For example,
the preliminary definitions may mixed
up with the preliminary lemmas.


(1) History. [If you decide to include some history,
keep it down to a couple of sentences. This is a mathematics
exam.]

(2) Preliminary definitions. [You may decide to give
some illustrative examples.]

(3) Preliminary lemmas. [Proved, sketched or stated.]

(4) Main theorem. [Proved]

(5) Corollaries. [Proved, sketched or stated.]

(6) Counter-examples showing that the results
can not be extended. [I would tend to sketch rather
than prove or state.]

(7) In what directions does the result lead?
Why is it useful?

You do not have to write your essay in linear order.
Since most people tend to spend too much time on the
preliminaries it may be a good idea to start by
writing out the main theorem and then add the preliminaries.
If you find that your essay is too short
you can expand it by adding further proofs
at the end with brief directions towards
the proofs in the text. It is a good idea to 
write on alternate lines to leave room
for such alterations.

Every supervisor knows that students can be divided
into a large class who write too little for
supervisions and a small class who write too much.
If you belong to the class which writes too
much, remember that condensation and selection
are key skills which the essay is intended to
encourage. If you can not prevent yourself from
producing fifteen pages of beautifully neat
handwriting covering every aspect of the subject
in complete detail, write your masterpiece
and then set yourself the task of producing
a three page summary\footnote{You may on other hand,
be one of those undergraduates
who spent the first part of the week rehearsing
for the Boat Club's production of Aida, but had genuinely
left lots of time but your pet hamster died
and then you lost your room key on a ten mile
hike in aid of charity
and then you had to take tea with the Master
and then there was so much to do that you were
too depressed to do anything, particularly
because of your cold. In such a case remember
that producing a skeleton plan of your essay
is substantially better than producing nothing.}.
                                                 
All this advice may have a contrary effect
to that intended and make the task of
writing an essay seem dauntingly complicated.
It is not. Try it and see. Your first essay
will be hard to write but not as hard as you
thought. Your second essay will be easier to
write and read better than the first and
your third essay will be easier to write than
your second. Thereafter essay questions will be simply
part of the day's work.

Up to now I have stressed the similarities
between an essay and a talk. Let me finish this section
by pointing out one important difference. At best,
your essay will be a sort of condensed talk
which would need to be diluted by commentary
and by the jokes, anecdotes and other devices which
provide pauses for the audience to catch up.
It is instructive
to look at a TV science programme or to read a
popular science essay and then list the number
of ideas that the author tries to get across.
Usually there are only one or two. Such a style
is unsuitable for mathematics but it should,
nonetheless, be
observed that mathematicians are prone to confuse
high information content with high information transmission.

\section{An essay with commentary}\label{hostage}
\textsf{[We write
the commentary in sanserif} and
the essay in ordinary type. \textsf{Recall the
question} 
\begin{question*}{\bf\ref{Moebius}.}
Write an essay on M\"{o}bius maps.
\end{question*}
\sffamily
We start by jotting
down all the topics we can remember concerned
with M\"{o}bius maps.

(1) They take circles and straight line to
circles and straight lines.

(2) They form a group.

(3) They can be broken up into simpler maps.

(4) They can be used to map any three points
into any three points.

(5) They are connected with $SL({\mathbb C},2)$.

(6) We have to deal with $\infty$.

We can not remember anything about (5)
so we decide not to include it. Point (1)
seems the most important so we decide to make it
the centre of our essay. We also take the key
decision to make as much use of (3) as we can.
We shall treat the points in order (3),
(2), (1), (4). Since we can not think of any
interesting historical titbit we go straight
to the definitions.]

\normalfont
We work on the extended complex plane 
${\mathbb C}^{*}={\mathbb C}\cup{\infty}$.
Suppose that $a$, $b$, $c$, $d\in{\mathbb C}$
with $ad-bc\neq 0$. If $c=0$, we define
\begin{align*}
f(z)&=(az+b)/d \ \text{if $z\in{\mathbb C}$}\\
f(\infty)&=\infty.
\end{align*}
If $c\neq 0$, we define
\begin{align*}
f(z)&=(az+b)/(cz+d)\ \text{if $z\in{\mathbb C}$ and $z\neq -d/c$}\\
f(-d/c)&=\infty\\
f(\infty)&=a/c.
\end{align*}
We write
\[f(z)=\frac{az+b}{cz+d}\]
and call $f$ a M\"{o}bius map. In what
follows we shall often leave the treatment of special cases
like $c=0$ or $z=\infty$ to the reader.
\textsf{[This is the kind of fiddly but easy stuff
which can be omitted from an essay.]} 

We write
$\mathcal M$ for the set of M\"{o}bius maps.
We are particularly interested in the following
elements of $\mathcal M$ which we call 
`elementary maps' (here $\lambda\neq 0$)
\begin{align*}
T_{a}(z)&=z+a\\
D_{\lambda}&=\lambda z\\
J(z)&=1/z.
\end{align*}
We observe that $T_{a}$ is a translation and that
if $\lambda=r\exp i\theta$ with $r>0$, $\theta$ real
then $D_{\lambda}$ is a rotation through $\theta$ about $0$
followed by a dilation. We observe also that 
$T_{a}T_{-a}=I$, $D_{\lambda}D_{\lambda^{-1}}=I$
and $J^{2}=I$ where $I$ is the identity on ${\mathbb C}^{*}$.

\begin{lemma}\label{Elementary}
(i) If $S$ is an elementary M\"{o}bius map
and $T\in \mathcal M$ then $TS\in \mathcal M$.

(ii) Every M\"{o}bius map is the composition of
elementary maps.

(iii) $\mathcal M$ is precisely the set of compositions
of elementary maps.
\end{lemma} 
\textsf{[I would be inclined to leave this completely
unproved. If we decide to prove anything it should be
the least obvious thing which in my opinion is (ii).]} 

Combining our observations on the nature of the
elementary maps with part~(iii) of Lemma~\ref{Elementary}
we obtain the following result.
\begin{theorem}\label{Group}
$\mathcal M$ is a set of bijections
${\mathbb C}^{*}\rightarrow {\mathbb C}^{*}$
forming a group under composition.
\end{theorem}

\textsf{[The proof of Lemma~\ref{Group} given here
could hardly be presented to beginning students without
a great deal of discussion. However the presumed
audience for an essay consists of good mathematicians
who merely know nothing about the particular subject of
the essay\footnote{\textsf{As Szilard,
is claimed to have said
`You may assume zero knowledge and infinite intelligence'.}}.]} 

We now study the effect of M\"{o}bius maps on circles
and straight lines. We begin by finding a suitable 
characterisation of these objects.

The equation of a circle is
\[|z-a|^{2}=r^{2}\]
with $a\in{\mathbb C}$ and $r>0$.
Expanding, using the relation $|w|^{2}=ww^{*}$ gives
\[zz^{*}-a^{*}z-az^{*}-(r^{2}-|a|^{2})=0.\]
Finally, multiplying by a real non-zero constant $A$
gives
\[Azz^{*}+Bz^{*}+B^{*}z+C=0\]
with $A$, $C$ real, $A\neq 0$ and $|B|^{2}>AC$.
We note that the working is reversible
so we have a complete characterisation of a circle.

The equation of a real line is
\[\alpha x+\beta y=\gamma\]
with $\alpha$, $\beta$, $\gamma$ real and $\alpha$, $\beta$
not both zero. This may be rewritten
\[\alpha(z+z^{*})-i\beta(z-z^{*})=2\gamma\]
so that, collecting terms we have
\[(\alpha-i\beta)z+(\alpha+i\beta)z^{*}-2\gamma=0\]
that is
\[B^{*}z+Bz^{*}+C=0\]
with $C$ real and $B\neq 0$ or equivalently $|B|^{2}>AC$
where $A=0$.
Once again the working is reversible.

Combining these facts with some further simple
observations we obtain our key theorem.
\begin{theorem}\label{general} 
The general equation of a circle or line
is
\[Azz^{*}+Bz^{*}+B^{*}z+C=0\]
with $A$, $C$ real and $|B|^{2}>AC$.

We have a line if and only if $A=0$. The circle or line
passes through $0$ if and only if $C=0$.
\end{theorem}
\textsf{Since Theorems~\ref{general} and~\ref{circles to}
form the centre of our essay we give more detail
in the proof of Theorem~\ref{general} than elsewhere
in the essay.}
\begin{theorem}\label{circles to}
(i) The elementary M\"obius maps
translation $z\mapsto z+a$ and dilatation and rotation
$z\mapsto \lambda z$ $[\lambda\neq 0]$ takes circles 
to circles and straight lines to straight lines.

(ii) The elementary M\"{o}bius map $z\mapsto 1/z$
takes circles and straight lines passing through $0$
to straight lines and all other circles and straight lines
to circles. It takes straight lines to circles and straight lines
passing through $0$.

(iii) M\"{o}bius maps take circles and straight lines
to circles and straight lines.
\end{theorem}
\begin{proof} (i) Obvious.

(ii) By Theorem~\ref{general}
the general equation of a circle or line
is
\begin{equation*}
Azz^{*}+Bz^{*}+B^{*}z+C=0 \tag{$*$}
\end{equation*}
with $A$, $C$ real and $|B|^{2}>AC$.

The mapping $z\mapsto 1/z$ takes the curve described 
by equation $(*)$ to the curve described by
\begin{equation*}
Czz^{*}+B^{*}z^{*}+Bz+A=0 \tag{$**$}
\end{equation*}
with $C$, $A$ real and $|B^{*}|^{2}>CA$,
that is another circle or straight line.
The remaining statements follow from the second
paragraph of Theorem~\ref{general}.

(iii) Every M\"{o}bius map is the composition of
elementary M\"{o}bius maps.
\end{proof}
It can be shown that M\"{o}bius maps preserve
the angle between curves. \textsf{[An essay can
contain a certain number of unproved results
though a long sequence of such assertions makes
rather dull reading.]}

Here is an example of the use of Theorem~\ref{circles to}.
\textsf{[Since this example is not
in the syllabus, it should be possible to get
full marks without including it. However, it makes the
essay more interesting and we would certainly get credit
for it.]}
\begin{theorem}[Steiner's porism] Suppose $C$ and
$C'$ are non intersecting circles with $C'$ inside
$C$. If $C_{1}$, $C_{2}$, \dots form a sequence of circles
arranged in clockwise order
with $C_{1}$ tangent to $C$ and $C'$ and $C_{j+1}$
tangent to $C$ and $C'$ and $C_{j}$ $[j\geq 1]$
we say that they form a chain of circles. If $C_{n}=C_{1}$
we say that the chain is closed. Steiner's porism
states that if one choice of $C_{1}$ gives a closed chain
then all choices do.
\end{theorem}
\textsf{[This should be illustrated by a diagram.
Use diagrams freely throughout your essay. Remember,
a picture is worth a thousand words.]}
\begin{proof}[Sketch proof] We can find a M\"{o}bius
transform which takes $C$ and $C'$ to concentric
circles. The transform takes chains of circles to
chains. Since Steiner's porism is trivial for
concentric circle we are done.
\end{proof}
\textsf{[This really is a sketch. It is not obvious without
some further argument that there is a M\"{o}bius
transform which takes $C$ and $C'$ to concentric
circles. We have not shown explicitly that tangent
circles are taken to tangent circles.]}

Just as a simple translation allows us to move
any point in ${\mathbb R}^{n}$  to the origin
so M\"{o}bius maps allow us to move
any three points in ${\mathbb C}^{*}$ to
any other.
\textsf{[Note that we are trying to show that
M\"{o}bius maps are useful. A skeptical reader
would not be fully convinced without the inclusion
of material from later courses.]} Here
is a particular example.

\begin{lemma}\label{mix}
(i) If $w\neq\infty$, the M\"{o}bius map
$z\mapsto 1/(z-w)$ takes $w$ to $\infty$.

(ii) If $w\neq \infty$ the M\"{o}bius map
$z\mapsto z-w$ takes $w$ to $0$ and fixes $\infty$.

(iii) If $w\neq 0,\ \infty$ the M\"{o}bius map
$z\mapsto z/w$ takes $w$ to $1$ and fixes $0$ and $\infty$.

(iv) If $w_{1}$, $w_{2}$, $w_{3}$ are distinct then
there exists a M\"{o}bius map $T$ with $Tw_{1}=0$,
$Tw_{2}=1$ and $Tw_{3}=\infty$
\end{lemma}

Using Lemma~\ref{mix} it can be shown that the following 
general result holds.
\textsf{[We are not obliged to prove everything,
but it helps the reader if we make it clear when
we are not proving something.]}
\begin{theorem}\label{general mix}
If $w_{1}$, $w_{2}$, $w_{3}$ are distinct
and $z_{1}$, $z_{2}$, $z_{3}$ are distinct then
there exists a M\"{o}bius map $T$ with $Tw_{j}=z_{j}$
\end{theorem}

The map $T$ of Theorem~\ref{general mix} is unique.
To show this we introduce the idea of the cross ratio
$[z_{1},z_{2},z_{3},z_{4}]$.
\textsf{[Generally speaking it is bad exposition
to introduce a definition that is used only once,
but this is an exam and we wish to display our
knowledge.]}
\begin{definition} We set
\[[z_{1},z_{2},z_{3},z_{4}]=
\frac{(z_{1}-z_{2})(z_{3}-z_{4})}{(z_{1}-z_{4})(z_{3}-z_{2})}\]
treating $\infty$ as as in the definition of the M\"{o}bius
map.
\end{definition}
\textsf{[This is more of sketch of a definition than a
definition]}

\begin{lemma} If $T$ is a M\"{o}bius map
\[[Tz_{1},Tz_{2},Tz_{3},Tz_{4}]=[z_{1},z_{2},z_{3},z_{4}].\]
\end{lemma}
\begin{proof}[Sketch proof] The result is true for elementary
M\"{o}bius maps by direct calculation and so true
for all M\"{o}bius maps by Lemma~\ref{Elementary}~(iii).
\end{proof}
\begin{lemma} The M\"{o}bius map
$T$ of Theorem~\ref{general mix} is unique.
\end{lemma}
\begin{proof}[Sketch proof] We shall show that
the result is true for the
case $w_{1}=z_{1}=0$, $w_{2}=z_{2}=1$, $w_{3}=z_{3}=\infty$
since then
\[z=[z,0,1,\infty]=[Tz,T0,T1,T\infty]=[Tz,0,1,\infty]=Tz.\]
The general result may be obtained from this.
\end{proof}

This shows that, in general
there does not exist a
M\"{o}bius map allowing us to move
any four points in ${\mathbb C}^{*}$ to
any other.
 
\textsf{
[The syllabus also includes \emph{discussion of relations
between eigenvalues of a matrix and fixed points
of a M\"{o}bius map}. This idea did not occur
in the list of jottings with which we started.
My feeling is that the essay is complete without
it but of course the examiner could disagree.
One could write a very nice essay ending with
a discussion of the behaviour of the
sequence $T^{n}z$ for different M\"{o}bius
maps $T$.
Many students will have seen a discussion of
the homomorphism
\[\theta:SL({\mathbb C},2)\rightarrow{\mathcal M}\]
given by 
\[\theta(A)(z)=\frac{az+b}{cz+d}
\ \text{where}\ A=
\left(
\begin{matrix}
a&b\\c&d
\end{matrix}
\right).\]
This could also have been included in the essay
but I have chosen not to.}

\textsf{Not only could we have chosen different
topics, but we could have chosen much the same
topics but dealt with them very differently.]}
\begin{exercise} Write an essay on the M\"{o}bius
transform based on the following outline.

(i) Define the cross ratio.

(ii) Show that the set of maps 
$T:{\mathbb C}^{*}\rightarrow {\mathbb C}^{*}$ which leave 
the cross product of every four points unchanged
is a group of bijections of ${\mathbb C}^{*}$.
We call this group ${\mathcal M}$.

(iii) Show that ${\mathcal M}$ consists precisely
of the maps
\[z\mapsto\frac{az+b}{cz+d}\]
with $ad-bc\neq 0$.

(iv) Show by elementary geometry (recall that angles on the
same chord of a circle are equal or complementary)
that four points lie on a circle or straight
line if and only if their cross ratio is real.

(v) Prove that the M\"{o}bius maps take circles and straight lines
to circles and straight lines.
\end{exercise}
\section{What can you expect from your supervisor?}
When your supervisor looks at your essay he
or she will, like the examiner, have a series
of points in mind. Have you included results $A$
and $B$? If you used method $C$ did you take
care to define $D$? If, on the other hand, you
used method $E$ how did you deal with the rather
tricky argument to establish $F$? 

If your arguments are fallacious or you leave out
an important point your supervisor will tell you
but often he or she will be less prescriptive
merely saying that you could have included this
or excluded that but that it was very reasonable
to proceed as you did. Some students find
this very upsetting but it reflects the fact
that there are many different ways of writing
a good essay. Your supervisor is more concerned
with getting you to think about the general
idea of essay writing than polishing a particular
essay.

An ordinary supervision runs more effectively
if students bring along a list of the problems
they can not resolve and the points they
wish to discuss. In the same way the discussion
of an essay question is likely to be more
productive if you make a list of the decisions
you made in writing your essay and, in particular,
any decisions you are worried about. Your
supervisor may not be able to resolve all
your worries (there is, I repeat, no unique
recipe for writing an essay) but the discussion
will usually be both helpful and reassuring.

If you have no specific problems it may
be instructive to discuss the following 
kinds of question.

(1) If one extra result was to be added to the essay
to make it longer
which should it be and why?

(2) If one result was to be removed from the essay
to make it shorter which should it be and why?

(3) If one proof was to be expanded
which should it be and why?

(4) If one proof was to be shortened
which should it be and why? 

It may be noted that you do not need the presence
of a supervisor to ask yourself these questions.
\begin{exercise} What would your replies to
questions (1) to (4) if they were asked about 
the essay in Section~\ref{hostage}.
\end{exercise}

Good luck with your supervisions and your exams.
\end{document} 
 
