In the hydrostatic theory of liquid crystals, we characterize those orientation patterns which may occur, in the absence of externally applied body forces, no matter what the form of the stored energy function may be. It turns out that there are three types of such patterns. In one, the orientation vector is perpendicular to a family of concentric spheres. Another results from replacing these spheres by concentric circular cylinders. In the third, the orientation vector is constant in each of a family of parallel planes, the direction changing continuously with distance measured normal to these planes.