deletions | additions       diff --git a/Convective Velocities.tex b/Convective Velocities.tex  index a645a27..28b377a 100644  --- a/Convective Velocities.tex  +++ b/Convective Velocities.tex 
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\begin{equation}\label{eq:velocity2}  \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 $\rho = \rho (\mu, T, \P)$, \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}  ,  \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}  and finally   \begin{equation} 
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\begin{equation}  Q = \frac{4-3 \beta}{\beta} - \bigg(\frac{\partial \ln \mu}{\partial \ln T} \bigg)_{\p},  \end{equation}  with $\beta = \P_{Gas}/\P$. \p_{Gas}/\p$.  Plugging Eq.~\ref{eq:deltarho} in \ref{eq:velocity2} one obtains \begin{equation}  \bar{v}^2 = \frac{1}{8} g Q (\Delta T (\lambda) / T) \lambda .  \end{equation}