\subsection{Sorted masses and the reduction}\label{ss:reduction}

For $m\in(\R_{>0})^5$ let $m_{(1)}\le\dots\le m_{(5)}$ be its coordinates in increasing order. The \emph{sorted scaled image} of $m$ is
\[
\hat m=\big(m_{(1)},m_{(2)},m_{(3)},m_{(4)},m_{(5)}\big)/m_{(5)}.
\]
The sorted scaled images are the mass vectors with $0<m_1\le m_2\le m_3\le m_4\le m_5=1$. Among them are the points $\bone=(\tfrac14,1,1,1,1)$ and $\btwo=(\tfrac14,\tfrac14,1,1,1)$.

A \emph{face} $F$ is a set of consecutive pairs $(i,i+1)$, $1\le i\le4$. We write it as the partition of $12345$ into the runs of bodies joined by it, its \emph{blocks}. For instance $1|23|45$ is the face $\{(2,3),(4,5)\}$ and $1|2|3|4|5$ the empty face. The \emph{stratum} of $F$ is
\[
\Sigma_F=\{m:\ 0<m_1\le\dots\le m_5=1,\ m_i=m_{i+1}\text{ for }(i,i+1)\in F,\ m_i<m_{i+1}\text{ for }(i,i+1)\notin F\}.
\]
At a point of $\Sigma_F$ two bodies have equal masses if and only if they lie in one block of $F$. The sorted scaled images form the disjoint union of the sixteen strata. The point $\bone$ lies in $\Sigma_{1|2345}$ and $\btwo$ in $\Sigma_{12|345}$.

\begin{lemma}\label{lem:R}
Let $m\in(\R_{>0})^5$.
\begin{enumerate}[label=\textup{(\alph*)}]
\item The planar five-body problem with masses $m$ has finitely many central configurations up to similarity if and only if it has finitely many with masses $\hat m$.
\item $\hat m=\bone$ if and only if $m$ is a positive multiple of a permutation of $(1,4,4,4,4)$, and $\hat m=\btwo$ if and only if $m$ is a positive multiple of a permutation of $(1,1,4,4,4)$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) This is Lemma~\ref{lem:slice}(c), since $\hat m$ is obtained from $m$ by a permutation and a positive scaling.

(b) The sorted scaled image of a positive multiple of a permutation of $(1,4,4,4,4)$ is $(\tfrac14,1,1,1,1)=\bone$. Conversely, if $\hat m=\bone$, then $m$ is a permutation of $m_{(5)}\bone=\tfrac{m_{(5)}}4(1,4,4,4,4)$. The same argument applies to $(1,1,4,4,4)$ and $\btwo$.
\end{proof}

Theorem~\ref{thm:S} below concerns sorted masses. On the rays of Lemma~\ref{lem:R}(b) we will count the central configurations directly in \S\ref{sec:rays}.