deletions | additions       diff --git a/These_considerations_suggest_the_following__.tex b/These_considerations_suggest_the_following__.tex  index 1f3f71b..9c2a1c4 100644  --- a/These_considerations_suggest_the_following__.tex  +++ b/These_considerations_suggest_the_following__.tex 
...
These considerations suggest the following:  \begin{eqnarray}  \label{eqn:difference_between_observed_density_points}  \rho(\Gamma_{\rm obs}, P_{\rm obs}) - \rho(\Gamma_0, P_{\rm min}) th})  &=& &\int_{(\Gamma_{\rm obs}, P_{\rm obs})}^{(\Gamma_1, P_{\rm max})} \cdots db^\prime& - &\int_{(\Gamma_0, P_{\rm min})}^{(\Gamma_1, th})}^{(\Gamma_1,  P_{\rm max})} \cdots db^\prime& \\ &=& &\int_{(\Gamma_0, P_{\rm min})}^{(\Gamma_{\rm th})}^{(\Gamma_{\rm  obs}, P_{\rm obs})} \cdots db^\prime& = &\int_{b^\prime = 0}^{b} \cdots db^\prime &, \end{eqnarray}  where we have suppressed the integrands for clarity. We can then differentiate both sides with respect to $b = \left( \Gamma_{\rm obs}/2\right) \left[ \left( P_{\rm obs} - P_{\rm min} th}  \right)/P_{\rm min} th}  \right]^{1/2}$, but, for the left-hand side, we will convert the $b$-derivative: \begin{equation}  \label{eqn:b_derivative_into_P_obs_derivative}  \dfrac{d}{db} = 2 \left( \dfrac{2}{\Gamma_{\rm obs}} \right) \left( \dfrac{P_{\rm obs} - P_{\rm min}}{P_{\rm min}} th}}{P_{\rm th}}  \right)^{1/2} \left( P_{\rm obs}\ \dfrac{\partial}{\partial P_{\rm obs}} - \left( \dfrac{\Gamma_{\rm obs}}{2} \right) \dfrac{\partial}{\partial \Gamma_{\rm obs}} \right). \end{equation}