The operator $g^{\alpha\beta}(\dots)_{\beta\alpha}$ has been previously denoted by~$\Wave$. Also, by~\Eq{(74.1)} $\kappa_{\alpha}^{\alpha} = 0$. Hence \[ \Wave\kappa_{\mu} = J^{\mu} - G_{\mu}^{\epsilon}\kappa_{\epsilon}. \Tag{(74.31)} \] \PageSep{176} In empty space this becomes \[ \Wave \kappa_{\mu} = 0, \Tag{(74.32)} \] showing that $\kappa_{\mu}$~is propagated with the fundamental velocity. If the law of gravitation $G_{\mu\nu} = \lambda g_{\mu\nu}$ for curved space-time is adopted, the equation in empty space becomes \[ (\Wave + \lambda) \kappa_{\mu} = 0. \Tag{(74.33)} \] \Subsection[(b)]{Propagation of electromagnetic force.} To determine a corresponding law of propagation of~$F_{\mu\nu}$ we naturally try to take the curl of~\Eq{(74.31)}; but care is necessary since the order of the operations curl and~$\Wave$ is not interchangeable. By~\Eq{(74.2)} \begin{align*} J_{\mu\nu} &= g^{\alpha\beta}(\kappa_{\mu\beta\alpha\nu} - \kappa_{\beta\mu\alpha\nu}) \\ % %[** TN: Next line not broken in the original] &= g^{\alpha\beta}(\kappa_{\mu\beta\nu\alpha} - \kappa_{\beta\mu\nu\alpha}) \\ &\qquad- g^{\alpha\beta} (B_{\mu\nu\alpha}^{\epsilon} \kappa_{\epsilon\beta} + B_{\beta\nu\alpha}^{\epsilon} \kappa_{\mu\epsilon} - B_{\beta\nu\alpha}^{\epsilon} \kappa_{\epsilon\mu} - B_{\mu\nu\alpha}^{\epsilon} \kappa_{\beta\epsilon}) \quad\text{by~\Eq{(34.8)}} \\ % &= g^{\alpha\beta}(\kappa_{\mu\beta\nu} - \kappa_{\beta\mu\nu})_{\alpha} - g^{\alpha\beta} (B_{\mu\nu\alpha}^{\epsilon} F_{\epsilon\beta} - B_{\beta\nu\alpha}^{\epsilon} F_{\epsilon\mu}) \\ % &= g^{\alpha\beta}(\kappa_{\mu\nu\beta} - \kappa_{\beta\mu\nu} + B_{\mu\beta\nu}^{\epsilon} \kappa_{\epsilon})_{\alpha} - B_{\mu\nu\alpha\epsilon} F^{\epsilon\alpha} - G_{\nu}^{\epsilon} F_{\epsilon\mu}. \end{align*} Hence \begin{multline*} J_{\mu\nu} - J_{\nu\mu} = g^{\alpha\beta}(\kappa_{\mu\nu\beta} - \kappa_{\nu\mu\beta} - B_{\beta\mu\nu}^{\epsilon} \kappa_{\epsilon} + B_{\mu\beta\nu}^{\epsilon} \kappa_{\epsilon} + B_{\nu\beta\mu}^{\epsilon} \kappa_{\epsilon})_{\alpha} \\ - (B_{\mu\nu\alpha\epsilon} - B_{\nu\mu\alpha\epsilon}) F^{\epsilon\alpha} - G_{\nu}^{\epsilon} F_{\epsilon\mu} + G_{\mu}^{\epsilon} F_{\epsilon\nu}. \end{multline*} But by the cyclic relation~\Eq{(34.6)} \[ B_{\beta\mu\nu}^{\epsilon} + B_{\mu\nu\beta}^{\epsilon} + B_{\nu\beta\mu}^{\epsilon} = 0. \] Also by the antisymmetric properties \[ (B_{\mu\nu\alpha\epsilon} - B_{\nu\mu\alpha\epsilon}) F^{\epsilon\alpha} = 2B_{\mu\nu\alpha\epsilon} F^{\epsilon\alpha}. \] Hence the result reduces to \[ J_{\mu\nu} - J_{\nu\mu} = g^{\alpha\beta} (\kappa_{\mu\nu} - \kappa_{\nu\mu})_{\beta\alpha} - G_{\nu}^{\epsilon} F_{\epsilon\mu} + G_{\mu}^{\epsilon} F_{\epsilon\nu} - 2B_{\mu\nu\alpha\epsilon} F^{\epsilon\alpha}, \] so that \[ \Wave F_{\mu\nu} = J_{\mu\nu} - J_{\nu\mu} - G_{\mu}^{\epsilon} F_{\epsilon\nu} + G_{\nu}^{\epsilon} F_{\epsilon\mu} + 2B_{\mu\nu\alpha\epsilon} F^{\epsilon\alpha}. \Tag{(74.41)} \]