deletions | additions       diff --git a/Ultimately_though_we_re_interested__.tex b/Ultimately_though_we_re_interested__.tex  index a2f1f76..971e400 100644  --- a/Ultimately_though_we_re_interested__.tex  +++ b/Ultimately_though_we_re_interested__.tex 
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It's clear that plugging Equation \ref{eqn:n-Pobs_from_uniform_n-Pact} into Equation \ref{eqn:n-Pact_from_n-Pobs} recovers the original uniform distribution. On the surface, though, Equation \ref{eqn:n-Pact_from_n-Pobs} suggests the strange result that for the power-law distribution $n(P_{\rm obs}) \sim P_{\rm obs}^{-2}$, similar to that reported in \citet{Jackson_2015}, the underlying distribution $n(P_{\rm obs}) = 0$. However, the approach here assumes that $P_{\rm obs}$-values only span a finite range, which is violated by the simple $P_{\rm obs}^{-2}$ power-law. Instead, a form similar to Equation \ref{eqn:n-Pobs_from_uniform_n-Pact} can be used to describe a power-law, with the 2 replaced by the desired index.   Interestingly, applying Equation \ref{eqn:n-Pact_from_n-Pobs} to the form $n(P_{\rm obs}) \propto \left[ \left( P_{\rm max}/P_{\rm obs} \right)^\gamma - 1 \right]$ will return an equation of the same form, i.e. $n(P_{\rm act}) \propto \left[ \left( P_{\rm max}/P_{\rm act} \right)^\gamma - 1 \right]$. Also, Equation \ref{eqn:n-Pact_from_n-Pobs} assumes $n(P_{\rm obs})$ declines for increasing $P_{\rm obs}$. If, instead, it INCREASES, increases,  the minus sign in Equation \ref{eqn:n-Pact_from_n-Pobs} should be dropped. discarded.