deletions | additions       diff --git a/For_each_of_the_equal__.tex b/For_each_of_the_equal__.tex  index 356bdc2..5b334ac 100644  --- a/For_each_of_the_equal__.tex  +++ b/For_each_of_the_equal__.tex 
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\begin{equation}   \frac{dN}{dM} = M^{\beta} \exp\left(-\frac{M}{M_c}\right),  \end{equation}  which is a power-law mass distribution with an exponential truncation. The cutoff (truncation) mass in the distribution is $M_c$ and the index of the distribution is $\beta$. We consider a pure power-law distribution, letting $M_c\to\infty$ and we also consider a Schechter function, constraining $\beta=-2$ and optimizing for $M_{c}$. The pure power-law function has been traditionally considered in previous on molecular cloud mass distributions \citep{Solomon_1987,Rosolowsky_2005a}. The \citet{Schechter_1976} form of the mass distribution is expected for the gravitational fragmentation of a gas distribution below a characteristic mass (e.g., the Jeans mass). In all cases, we limit the fits to clouds with masses $M>3\times 10^{5}~M_{\odot}$ as determined by the minimum mass of clouds admitted by our cataloging procedure. The {\sc powerlaw} package optimizes the fits to the data using the procedure described in \citet{Clauset_2009}, which uses a maximum likelihood framework and a Kolmogorov-Smirnoff test to assess goodness-of-fit. Of note, the method provides likelihoods for favoring one functional representation over another. In Figure \ref{fig:massfits}, we show the fits of the pure power law and the truncated power law distributions. There are no regions where a pure power-law is clearly a better representation of the data, though in the outer galaxy individual bins don not contain enough points to clearly distinguish between the two functional forms. We also fit the data using exponential, stretched exponential and log-normal functional forms, but none of these are clearly superior to a truncated power-law. Both the index of the pure power law functional form and the characteristic mass of the clouds $M_c$ decreases with galactcocentric radius. We compare this behavior to that seen in the massive clusters, which also show a decrease in the characteristic mass, albeit this value was derived for a Schechter mass distribution.   Since it is not clear that the truncated power law is a good representation of the distributions and no other functions are identified that are clearly superior, we also report a non-parameteric characteristic mass as the geometric mean of the five most massive clouds in each bin $\langle M\rangle_5$.