Figure \[fig:VarSelect\] shows the selected variables on $g-r$ and $g-i$ CMDs. After rejecting variables that were classified in Paper II as eclipsing binaries or short period pulsators and rejecting variables without period determinations we are left with 575 variables that are candidate cluster members. A catalog of these 575 variables is given in tables \[tab:M37\_rotation1\]-\[tab:M37\_rotation3\], we show only the first ten rows of each table for illustration, the full tables are available in the online edition of the journal. Note that only stars with spectroscopy are included in table \[tab:M37\_rotation3\].

Revised Periods
---------------

In Paper II we calculated the periods of the variable stars using the L-S, phase-binning AoV, and BLS algorithms. We then selected, by eye, the most likely period for each star from the values returned by the three algorithms. Here we recalculate the periods using the multi-harmonic AoV algorithm due to @Schwarzenberg-Czerny.96. This method is equivalent to fitting to each light curve a harmonic series of the form:

$$\tilde{r}(t) = a_{0} + \sum_{i=1}^{N}a_{i}\cos(2\pi i t / P + \phi_{i})
\label{eqn:fourier}$$

where $P$ is the period, $a_{i}$ are the amplitudes, and $\phi_{i}$ are the phases. This method is more general than the L-S algorithm which is equivalent to 