deletions | additions       diff --git a/Ultimately_we_are_interested_in__.tex b/Ultimately_we_are_interested_in__.tex  index b31e5da..6cb95ae 100644  --- a/Ultimately_we_are_interested_in__.tex  +++ b/Ultimately_we_are_interested_in__.tex 
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\begin{eqnarray}  \label{eqn:convert_from_observed_to_actual_density}  \rho({\rm act}(b)) &=& \frac{1}{2} b_{\rm max}^2\ f(b)\ f^{-1}\ b^{-1}\  \dfrac{d\rho({\rm obs})}{db}\\ &=& \frac{1}{2} b_{\rm max}^2\ f(b)\ f^{-1}\  \left[ \left( \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right) \dfrac{d \Gamma_{\rm obs}}{db} + \left( \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} \right) \dfrac{d P_{\rm obs}}{db} \right] \\ &=& b_{\rm max}^2\ f(b)\ f^{-1}\  \sqrt{\dfrac{P_{\rm min}}{P_{\rm obs} - P_{\rm min}}} \left[ \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} + \left( \dfrac{2}{\Gamma_{\rm obs}} \right) P_{\rm min} \sqrt{\dfrac{P_{\rm obs} - P_{\rm min}}}{P_{\rm min}} \dfrac{\partial \rho({\rm obs})}{\partial P_{\rm obs}} \right] \end{eqnarray}  where $b = \left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}$.