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Assis, A.K.T. 1989, Found. Phys. Lett. 2, 301 Assis, A.K.T. 1993, J. Phys. Soc. Japan, 62, 1418
Costa de Beauregard, O. 1995, in Advanced Electrodynamics, Eds. Barret, T.W. and Grimes, D.M. (World Scientific, Singapore) pp. 77 – 104
Costa de Beauregard, O. 1999, comments following Mikhailov’s paper, Ann. Fond.
Louis de Broglie, 24, 161
Mikhailov, V.F. 1999, Ann. Fond. Louis de Broglie, 24, 161
Schrödinger, E. 1925, Ann. der Phys. 77, 325. For his ad hoc [factor equal to unity.
Tisserand, M.F. 1872, Comptes Rendus, 74, 750
Weber, W.E. 1848, Ann. der Phys. 73, 229; 1893 (Julius Springer, Berlin) Vols. 1-6 Wesley, J.P. 1998a, Apeiron, 5, 89
Wesley, J.P. 1998b, Apeiron, 5, 219 Wesley, J.P. 1999, Apeiron, 6, 237
