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Autograph manuscript of an important article by Louis de Broglie at a time when he was at the height of his scientific powers and engaged on work that would earn him the Nobel Prize for physics in 1929. In his Ondes et mouvements of 1926, De Broglie had proposed that matter might behave as though it were a wave, an idea that was tested and confirmed by others in 1927. Thus the duality of both light (Einstein) and matter had been established. In the present essay he turns to the theory of five-dimensional relativity as developed by Kaluza, Kramers and Klein and what it can explain about wave mechanics. He particularly aims to improve on his predecessors' equations.