

\subsection{\bf $\chi^2$ Formulation of Linear Fit}
\subsection{\bf for Cross Section Data---II}

\begin{slide}
\begin{itemize}
\item The ``$\chi^2$'' matrix $M_{ml}$ is given by:
\[
M_{ml} = \sum_i \frac{C_{im} C_{il}}{\sigma^2_i}
\]

\item where $M_{ml}$ is a positive definite symmetric matrix
\[
= \sum_i \frac{(\partial\xi_i / \partial a_m ) (\partial\xi_i /
\partial a_l )}{\sigma^2_i}
\]

\item The right hand side
\[
b_m = \sum_i \frac{x_i C_{im}}{\sigma^2_i} = \sum_i
\frac{x_i}{\sigma^2_i}\ \frac{\partial\xi_i}{\partial a_m}
\]

\item Remember:

\begin{description}
\item{$\bullet$} $\xi_i = \sigma_{\rm theoretical} \left( \theta_i
\right) = a_0 + a_1 \cos\theta_i + a_2 \cos^2 \theta_i + a_3 \cos^3
\theta_i$

\item{$\bullet$} $C_{im} = \cos^m \theta_i \hspace{1em} \mbox{for}\ \ 
m=1,2,3$
\end{description}
\end{itemize}
\end{slide}

