A. A. Gurchenkov UDC 532.516 An investigation is made of the stability of steady rotation of a symmetrical body with a viscous fluid on the basis of integro- differential equations whose coefficients are determined by solving boundary-value problems of hydromechanics of an ideal fluid that are dependent on the geometry of a cavity. The perturbation method is employed to solve the problem on the stability of body rotation relative to the axis with the largest moment of inertia and on the instability relative to the axis with the smallest moment of inertia. A similar problem for a body with an ideal fluid is studied in [F. L. Chernous'ko, Prikl. Mat. Mekh., 31, Issue 3, 414-432 (1961); S. L. Sobolev, Prikl. Mekh. Tekh. Fiz., No. 3, 3-37 (1960); A. Yu. Ishlinskii and M. E. Temchenko, Prikl. Mekh. Tekh. Fiz., No. 3, 163-179 (1960)], while with a viscous fluid, it is studied in [N. N. Moiseev and V. V. Rumyantsev, Dynamics of a Body with Cavities Filled by Fluid [in Russian], Moscow (1965)], where consideration has been given to the problem on two-dimensional oscillations of a rectan- gular vessel under the action of the restoring force of an elastic spring. Computational Center of the Russian Academy of Sciences, Moscow. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 73, No. 3, pp. 28-32, May-June, 2002. Original article submitted January 31, 2001; revision submitted November 8, 2001. JEPTER74920200217 JEPTER7492017