Characteristics of the ultrahyperbolic differential equation governing pole density functions

In this paper we define a hyperspherical x-ray transform, characterize the range of this transform in terms of an ultrahyperbolic differential equation which is derived here without referring to spherical harmonics, and provide its general solution in terms of both its characteristics or spherical harmonics. These results will aid in the solution of actual problems of texture goniometry, i.e., the analysis of preferred crystallographic orientation in polycrystalline materials, particularly in the solution of the inverse tomographic problem of texture goniometry.