deletions | additions       diff --git a/Ultimately_we_are_interested_in__.tex b/Ultimately_we_are_interested_in__.tex  index a80d77e..474b79b 100644  --- a/Ultimately_we_are_interested_in__.tex  +++ b/Ultimately_we_are_interested_in__.tex 
...
\label{eqn:convert_from_observed_to_actual_density}  \rho({\rm act}(b)) &=& \frac{1}{2} b_{\rm max}^2\ f(b)\ \dfrac{d\rho({\rm obs})}{db}\\  &=& \frac{1}{2} b_{\rm max}^2\ f(b)\ \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] \\  &=& \frac{1}{2} b_{\rm max}^2\ f(b)\ \left[ 2\left( \dfrac{P_{\rm min}}{P_{\rm obs} - P_{\rm min}} \right)^{1/2} \dfrac{\partial \rho({\rm obs})}{\partial \Gamma_{\rm obs}} \right]  \end{eqnarray}  where $b = \left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}$.