The equations of rotation in a medium with resistance of a free system of two rigid bodies connected by an elastic spherical joint or Hooke's joint are obtained. Under the assumption that the center of mass of solid bodies is located on the third axis of inertia, equations of perturbed motion are obtained. These equations are represented as eight ordinary differential equations with periodic coefficients. For two Lagrange gyroscopes, a fourth-order characteristic equation is obtained. Based on the Liénard-Schipart criterion, written in the form of innors, conditions for the asymptotic stability of uniform rotations of Lagrange gyroscopes are obtained. These conditions are presented in the form of three inequalities. Analytical studies of stability conditions were carried out. It has been proven that the first inequality is always satisfied. From the third inequality it follows that if the gyroscopes have equal axial moments of inertia and rotate with the same angular velocities in different directions or there is no elasticity in the hinge, then the characteristic equation has a multiple root. In this case, the issue of sustainability requires additional research. The conditions for stability in terms of kinetic moments are written out and it is shown that the leading inequality coefficients are positive, from which it follows that stability will always be possible for sufficiently large values of one or two kinetic moments. Similar conclusions were obtained with the Hooke's hinge, and it was also shown that with identical gyroscopes the characteristic equation splits into two equations. In the absence of elasticity, a multiple zero root appears in the hinge and stability requires additional study.