In certain investigations which I have presented to the Society, relative to the modifications which light undergoes when it meets with a medium more dense than that in which it is travelling, the law of force has been supposed to be that of the inverse square of the distance. On re-examination of this subject, I find that there is no necessity for restricting the computations by the hypothesis of any particular law. The conclusions are perfectly independent of the law, provided one of the equations of reduction, the value of which it is not possible to compute by any known methods of analysis, be admitted as an experimental result. The object of the present Memoir is twofold : 1st, To present the analysis of the general theorem of vibrations at the surface of an interrupted medium in its most simple form; and, 2d, To apply the results to the case of reflection unaccompanied by refraction.