The importance of experimental investigation of the intensity of scattering of X-rays by crystals has long been realized, since it affords valuable information regarding the charge distribution in the atoms of the crystal. Recently Debye has pointed out the importance of similar investigations of the scattering by gases. Here diffraction effects may be observed due to regularities in the structure of the gas molecules as distinct from crystal diffraction, where the regularities are due to the orderly arrangement of crystal atoms. This provides a means of obtaining important information as to the structure of molecules, and Debye and his collaborators have obtained interesting experimental results in this field. Debye has also introduced an approximate theory of the results obtained, but owing to the great complexity of the molecules considered it is at present impossible to proceed any further in the theoretical discussion. However, the case of molecular hydrogen may be worked out for short X-rays, since the charge distribution in the ground state of the molecule is fairly well known. It is the purpose of this paper to calculate the intensities of short X-rays scattered from hydrogen gas. The case of other diatomic molecules will not be very dissimilar.