deletions | additions       diff --git a/These_details_allow_us_to__.tex b/These_details_allow_us_to__.tex  index c028bdb..02a2fb3 100644  --- a/These_details_allow_us_to__.tex  +++ b/These_details_allow_us_to__.tex 
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These details allow us to write 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}) &=& &\int_{(\Gamma_{\rm obs}, P_{\rm obs})}^{(\Gamma_1, P_{\rm max})} \cdots db^\prime& - &\int_{(\Gamma_0, P_{\rm min})}^{(\Gamma_1, P_{\rm max})} \cdots db^\prime& \\ &=& &\int_{(\Gamma_0, P_{\rm min})}^{(\Gamma_{\rm obs}, P_{\rm obs})} \cdots db^\prime& = &\int_{b^\prime = 0}^{b} \cdots db^\prime &,  \end{eqnarray}  where we have suppressed the integrand integrands  for clarity. We can then differentiate both sides with respect to $b = \left( \Gamma_{\rm obs}/2\right) \left[ \left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min} \right]^{1/2}$, but, for the left-hand side, we will convert the $b$-derivative into a $P_{\rm obs}$-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}} \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).