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\end{equation}  while if the total event rate is not included in the likelihood, namely by removing the first term of the product in Eq.~\ref{eq:likelihood}, these elements take the form  \begin{equation}  V_{ij} = {\cal L} \left[ \int { {\rm d}\Omega f_i {f_i  \times f_j \over S^0} - {\int { \int  {\rm d}\Omega f_i \int {\rm d}\Omega f_i \over \int {\rm d}\Omega S_0 }  \right], \end{equation}