deletions | additions       diff --git a/Convective Velocities.tex b/Convective Velocities.tex  index 5311a50..fb411be 100644  --- a/Convective Velocities.tex  +++ b/Convective Velocities.tex 
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\begin{equation}  \bar{v}^2 = - \frac{1}{8} g (\Delta \rho (\lambda)/\rho) \lambda.  \end{equation}  Relating $\Delta \rho$ and $\Delta T$ requires the equation of state for the gas $\rho = \rho (\mu, T, \P)$. In general we \P)$, which in differential form  can be written as  \begin{equation}  \D\rho = \bigg(\frac{\partial \rho}{\partial \mu} \bigg)_{\P,T} \D\mu + \bigg(\frac{\partial \rho}{\partial T} \bigg)_{\P,\mu} \D T + \bigg(\frac{\partial \rho}{\partial \P} \bigg)_{\mu,T} \D\P,   \end{equation}  or  \begin{equation}  \D\rho = \frac{\rho}{\mu}\bigg(\frac{\partial \ln \rho}{\partial \ln \mu} \bigg)_{\P,T} \D\mu + \frac{\rho}{T} \bigg(\frac{\partial \ln \rho}{\partial \ln T} \bigg)_{\P,\mu} \D T + \frac{\rho}{\P} \bigg(\frac{\partial \ln \rho}{\partial \ln \P} \bigg)_{\mu,T}  \D\P \end{equation}  write, assuming again pressure equilibr