deletions | additions       diff --git a/These_details_allow_us_to__.tex b/These_details_allow_us_to__.tex  index 9930494..c028bdb 100644  --- a/These_details_allow_us_to__.tex  +++ b/These_details_allow_us_to__.tex 
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where we have suppressed the integrand 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).  \end{equation}