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The determination of stress states around cavities in the stressed elastic body, regardless of cavity shapes, that may be spherical, cylindrical elliptical etc. in its analytical approach has to be based on selection of a stress function that will satisfy biharmonic equation, under given boundary conditions. This paper is concerned with formulation and solution of the cited differential equation using elliptical coordinates in conformity with the cavity shape of oblong ellipsoid [1]. It is therefore considered that the formulation of the stress tensor will be done in conformity to the cited coordinates. The paper describes basic statements and definitions in connection to harmonic functions used for determination of stress states around cavities formed in the stressed homogeneous space. The particular attention has been paid to the use of Legendre`s functions, with definitions and derivation of recurrent formulas, that have been used for determination of stress states around an oblong ellipsoidal cavity, [1]. The paper also includes the description of procedures used in forming series based on Legendre`s functions of the first order.

Investigation of the stress field around the cavity that is loaded or partially loaded at the inner surface by the rotationally symmetric loading is still the contemporary problem in the theory of elasticity. As the contribution to the similar investigations, the paper introduces the new function of loading in the form of the infinite sine series. Besides the definition of the mentioned loading function as the boundary condition on inner surface of the cavity, the paper presents the comparative analysis of the new function with the previously used ones. The said loading function at the inner surface of a cavity is internal loading between rock mass and supporting structure, for the case of homogeneous isotropic elastic medium as the first approximation of the real situation.