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Theorems relating the plane wave modes (scalar or vector) transmitted and reflected by periodic structures of lossless elements bounded by two parallel planes are derived. One special case is continuity of energy flux normal to the boundary planes, and others relate the phases, or both phases and magnitudes, of the propagating modes. These relations are essentially discrete analogs (i.e., with integrals over all angles of observation replaced by sums over propagating mode directions) of the relations for a bounded scatterer given by Saxon. The essential difference in the two situations is that by restricting the parameters so that only a few modes propagate we obtain special cases which can be treated relatively explicitly. Thus the case of one propagating mode, and the cases corresponding to Bragg reflections are discussed in some detail; here there is both conservation of magnitude and phase.