deletions | additions       diff --git a/frac_d_2_r_d__.tex b/frac_d_2_r_d__.tex  index 9a7f3f1..1921c79 100644  --- a/frac_d_2_r_d__.tex  +++ b/frac_d_2_r_d__.tex 
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$$  \frac{d^2 r}{d\tau^2} \ddot{r}  = - \Gamma^r_{\alpha\beta} \frac{d x^\alpha}{d\tau}\frac{d x^\beta}{d\tau} \dot{x^\alpha}\dot{x^\beta}  = - \Gamma^r_{tt} \frac{d t}{d\tau}\frac{d t}{d\tau} - \dot{t}^2  - \Gamma^r_{rr} \frac{d r}{d\tau}\frac{d r}{d\tau} \dot{r}^2  - \Gamma^r_{\theta\theta} \frac{d \theta}{d\tau}\frac{d \theta}{d\tau} \dot{\theta}^2  - \Gamma^r_{\phi\phi} \frac{d \phi}{d\tau}\frac{d \phi}{d\tau} \dot{\phi}^2  $$