The diffraction of electrons by anthracene crystals gives results which are in agreement with the structure as previously determined with x rays. In addition to the normal type of multispot pattern characteristic of electron diffraction by thin single crystals, however, there appears superimposed another pattern consisting of areas or islands of diffuse scattering. It is shown that this diffuse-area pattern is due to molecules which, though oriented with respect to the beam in accordance with the crystal-orientation, have nevertheless no definite phase relationship with each other. The diffuse-area pattern is thus virtually equivalent to that which would be given by a gaseous stream of oriented molecules flowing past the electron beam. Since the crystal setting cannot as a rule be directly observed as it can in x-ray analysis, it has been necessary to develop methods for determining the crystal orientation from a consideration of the normal electron-diffraction pattern features. In this way it was established that the diffuse-area pattern was furnished by molecules which, though in, or very near, their normal positions and orientations within the lattice, were acting as independent scattering groups. It is shown, by an extension of the Debye theory to molecular lattices, that the occurrence of the diffuse-area pattern in electron-scattering by molecular crystals can be satisfactorily accounted for on the supposition that the molecules vibrate thermally as practically rigid units about their mean positions within the lattice. It is pointed out that the diffuse-area pattern can be regarded as an important adjunct to the x-ray Fourier synthesis method of determining molecular and crystal structure, in cases where the atomic arrangement within the molecule exhibits a sufficient degree of symmetry. Furthermore, it offers an approach to the study of molecular structure in cases in which the method of electron diffraction by gases fails owing to the complex nature of the pattern resulting from the lack of orientation of the molecules in the gaseous stream.