deletions | additions       diff --git a/These_details_allow_us_to__.tex b/These_details_allow_us_to__.tex  index 8d9dbfe..3dd255a 100644  --- a/These_details_allow_us_to__.tex  +++ b/These_details_allow_us_to__.tex 
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\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, &\int_{(\Gamma_0,  P_{\rm min})}^{(\Gamma_1, P_{\rm max})} \cdots db^\prime 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 for simplicity. We can then differentiate both sides with respect to $b$