\documentclass[16pt,landscape,magscalefonts]{foils}
\usepackage{tabularx}
\usepackage[pdftex]{graphicx}
\usepackage[pdftex]{color}
\usepackage[pdftex]{geometry}
\geometry{headsep=-.01em,vscale=.96,hscale=0.9}
\usepackage{wrapfig,times,amsmath}
\setlength{\parindent}{0em}
\def\totalpages{\pageref{lastpage}}
\begin{document}%\footnotesize
\[R={D_M\over 2x_0+2x_A}\]
where $x_0$ is emittance contribution and $x_A$ is aberration
contribution.
\[x_0=\epsilon/\theta, x_A=D_M(\kappa\theta)^n\]
where $n$ is lowest uncorrected order. E.g.\ if corrected to octupole, $n=4$.
This has an optimum at some $\theta$ which we plug into $R$ to get:
\[R\propto\left({D_M\over\epsilon}\right)^{n\over n+1}\]
So it's easy to see that correcting to higher order yields diminishing
returns. On the other hand, because of ``feed-down'' effects, difficulty
of tuning rises very steeply with $n$.
$D_M$ is proportional to the size of the magnet and cost is proportional
to the cube of size. Therefore
\[\makebox{Cost}\propto R^{3+3/n}\]
\end{document}