

\subsection{\bf (ii) Analysis of Scattering}
\subsection{\bf Experiments---II}

\begin{slide}
\begin{itemize}
\item Even though $P_{n_1} (\cos\theta)$ and $P_{n_2} (\cos\theta)$ are
orthogonal polynomials, the means $\left\langle P_{n_1}
(\cos\theta)\right\rangle$, $\left\langle P_{n_2}
(\cos\theta)\right\rangle$ are not uncorrelated.

\item Both are calculated from same set
$\left\lbrace\theta_k \right\rbrace$ and fluctuate in a correlated
fashion as observations $\theta_k$ fluctuate.

\item We can calculate correlations
\begin{eqnarray*}
f & = & P_{n_1} (\cos\theta) \\
g & = & P_{n_2} (\cos\theta) \\
F & = & \frac{1}{N} \sum_k P_{n_1} \left(\cos\theta_k \right) \\
G & = & \frac{1}{N} \sum_k P_{n_2} \left(\cos\theta_k \right) 
\end{eqnarray*}

\item $F$ and $G$ will be independent if ${\displaystyle \int\limits}
p(\theta) P_{n_1} (\cos\theta) P_{n_2} (\cos\theta) = 0$.

\item This is not in general true.  

\item Thus, our formalism gives correlation between $a_{n_1}$ and
$a_{n_2}$.
\end{itemize}
\end{slide}

