On superpotentials in the theory of Newtonian Gravitation. II. Tensors of higher rank

Abstract

In addition to the tensors considered in the earlier paper, the following tensors are defined and studied: D_ij(x)=G∫_vp(x')x'_ix'_j/|x-x'|dx' D_ijk(x)=G∫_vp(x')(x_i-x'_i) (x_j-x'_j)x'_k/|x-x'|³dx', and B_ijkl(x)=G∫_vp(x')(x_i-x'_i) (x_j-x'_j)(x_k-x'_k)(x_l-x'_l)/|x-x'|⁵dx'. These tensors are useful in problems (such as the stability of the Jacobi ellipsoids) in which it is necessary to examine the effects of perturbations belonging to the third harmonic.