THE study of the motion of a liquid in a partially filled vessel in weak fields of force requires the surface forces to be taken into account [1]. If the intensity of the potential field of the external forces (such as gravitational forces) is sufficiently small, then the contribution of the surface energy to the potential energy of the system turns out to be of an order not less than that of the contribution of the potential energy from the external force field. The problem of small oscillations of an ideal liquid in a vessel allowing for surface energy is studied in [1, 2]. In this case the Cauchy problem reduces to the solution of a hyperbolic equation on the equilibrium surface of the liquid Ut + Su = /tf, u(t) ¦ t = 0 = ϑ, u t¦ t = 0 = ψ , where S is a positive definite unbounded operator. The problem of small movements of a viscous liquid in a partially filled vessel without considering the surface energy is studied in [3], and for a completely filled vessel in [4, 5]. A linearized boundary condition on the equilibrium surface of the liquid for the case when surface forces are taken into account is obtained in [6], where, too, the corresponding problem of “normal” oscillations [3], i.e. the eigenvalue problem, is also studied. Taking a model problem it is found that the surface forces cause an essential change in the structure of the eigenvalue spectrum. In this paper we consider the Cauchy problem for non-stationary linearized Navier — Stokes equations in a fixed, partially filled vessel. We also study the motion of the liquid when the deviation of the free surface from the equilibrium position is small, surface forces being taken into account. The mathematical feature of the problem is the presence in the boundary condition on the equilibrium surface of a second-order differential operator (of elliptic type). In Section 1 we obtain the decomposition of the Hilbert vector function space L 2(/GW) into orthogonal subspaces which arise naturally in the study of small oscillations of a liquid in a partially filled vessel. In Section 3 we prove existence and uniqueness theorems for the general solution of the Cauchy problem with finite total energy [7]. In Section 4 we use the functional method [8] and the Galerkin method [5, 8] to prove an existence theorem for a weak solution of the Cauchy problem (in Hopf' s sense).