It is well known that a beam of light falling on a reflecting mirror forms standing waves. This effect has been very beautifully made use of in Lippmann's colour photography process. The standing light waves, in this case, produce a periodic effect in the emulsion of the photographic plate which, when developed, scatters light and produces a similar colour effect. Instead of using a beam of light, it would seem possible to scatter electrons from the emulsion and obtain a reflection of electrons similar to that of a space grating. But it seemed to us that it would be of much greater interest to consider an experiment in which electrons are reflected from the standing waves of light. The direct scattering of free electronic waves by light has strictly never been observed, and it was thought possible that by this method, owing to the interference of the electrons and to the fact that the scattered electrons are focussed to one spot, the magnification of the phenomenon would be sufficient to make it observable. From the theory developed below, it will be seen that the experiment is just on the verge of possibility, and would be very difficult to carry out. The main interest of the experiment would come from the possibility of observing stimulated scattered radiation which up to the present has never been verified experimentally.