deletions | additions       diff --git a/If_a_devil_with_a__.tex b/If_a_devil_with_a__.tex  index 29e86eb..fa7204b 100644  --- a/If_a_devil_with_a__.tex  +++ b/If_a_devil_with_a__.tex 
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If a devil with a given $P_{\rm act}$ and $\Gamma_{\rm act}$ is detected, $b$ is probably not zero, in which case $P_{\rm obs} < P_{\rm act}$. We call this effect the signal distortion, and we can quantify the probability for a devil with $P_{\rm act}$ and $\Gamma_{\rm act}$  to be observed with $P_{\rm obs}$ and $\Gamma_{\rm  obs}$. From the encounter geometry, we can see that the infinitesimal probability $dp$ for the center of a devil to pass within a certain range of radial distances, between $b$ and $b + db$, is proportional to the infinitesimal area of the corresponding disk, giving $dp = 2 b\ db/b_{\rm max}^2$. Outside of $b_{\rm max}$, the probability to detect the devil is zero. For the Lorentz profile, we can relate the differential range of distances $db$ to $dP_{\rm obs}$: \begin{equation}  db = \frac{1}{2}\Gamma_{\rm act}\left[ \dfrac{P_{\rm act}}{P_{\rm obs}} - 1 \right]^{-1/2} \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) dP_{\rm obs} = \frac{1}{2}\left( \dfrac{\Gamma_{\rm act}}{2} \right)^2\dfrac{P_{\rm act}}{P_{\rm obs}^2} \dfrac{ dP_{\rm obs} }{b}.  \end{equation}