New methods in potential theory

Abstract

A theory of equigravitating bodies by which external force fields of volumetric axially symmetric figures can be represented by unitary integrals is developed. This theory is being developed in three directions. The first is connected with the proof of the existence of equigravitating line segments. Such line segments can have both real and imaginary distributions of density; however, the mass and external potential remain real values. The ends of line segments coincide with special points (these are cusp points on the surfaces or special points of the analytical continuation of the external potential inside the body). At two special points, the body has only one line equigravitating segment, otherwise the line segments are compound or form equigravitating “skeletons.” At the isolated special points, external gravitational fields can be presented by a set of line segments and mass points. The second direction is based on a representation of the external gravitational field of volumetric axially symmetric figures with an equator plane by means of potentials of flat round disks. Such disks are obtained on the line segments with symmetric density distributions. The return is always true: for homogeneous or any nonuniform round disk, it is possible to find an equigravitating line segment. It manages to construct chains of “spheroid-disk-line segment” equigravitating bodies. The third direction of this theory is connected with the development and expansion on the scope of the method of confocal transformations. This method is modified and applied not only to continuous homogeneous ellipsoids, but also to non-uniform stratified ellipsoids with a stratification of the general type, as well as to homogeneous and nonuniform shells. Any elementary or thick ellipsoidal shells (and continuous nonuniform stratified ellipsoids) connected by special confocal transformations equigravitate each other.