The paper substantiates the conditions of seismic stability in the rock mass with cavities, on which protected objects are located, under constant seismic effects of mass explosions in quarries. Based on the ray-tracing method, a mathematical model of the interaction between longitudinal and transverse waves with a spherical cavity has been developed. An algorithm has been elaborated for determining the total static and dynamic loads along the spherical cavity perimeter from seismic explosion waves falling on the interface. The regularities of joint action of static loads have been determined, when located above the objects cavity, with allowance for the effect of the mass of cavity ceiling, the mass of the protected object above it and dynamic loads in the rock mass, and under the plane wave front fall on the cavity from different directions. The dependences of the change of total stresses referred to the incident wave stress at the interface of the rock mass and the cavity depending on the coordinate on its surface have been set. It has been established that the most dangerous points of cavity surface, in which destruction is likely to occur, are the cases when the seismic wave front falls on the cavity from the bottom directions typical for blasting operations in deep quarries. Under the arbitrary angles of the wave incidence on the cavity the stresses rearrange along the perimeter, which under different circumstances might lead to the mass breach and the fall of the protected objects, located above it, under the ground. [ABSTRACT FROM AUTHOR]