deletions | additions       diff --git a/kinematics.tex b/kinematics.tex  index 9bba83b..c9bbc99 100644  --- a/kinematics.tex  +++ b/kinematics.tex 
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\subsection{Kinematics of an Off-Axis Neutrino Beam \footnote {Theses derivation {These derivations  are adapted from \cite{McDonald_2001}}} \subsubsection{Energy of the Neutrino} 
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\begin{equation}  \begin{split}  \mathbf{\pi^{+}} &= (m_{\pi} , 0, 0, 0) \\  \mathbf{\nu_{\mu}} &= (E_{\nu}, (E_{\nu}',  E_{\nu}' \sin \theta, 0, E_{\nu} E_{\nu}'  \cos \theta) \end{split}  \end{equation}  Therefore, the dot-product is given by  \begin{equation}  ( \pi \cdot \nu ) = m_{\pi} E_{\nu} E_{\nu}'  \end{equation}  Plugging this into equation \ref{eq:energy-momentum-expanded-more},  \begin{equation}  \begin{split}  m_{\nu}^2 = m_{\pi}^2 - 2 m_{\pi}E_{\nu} m_{\pi}E_{\nu}'  \end{split}  \end{equation}