The behavior of a fluid-filled gyroscope having a natural coning frequency near that of the inertial oscillation of a contained rotating liquid is calculated. The limitations of the calculations are that the coning angle be small enough to make the linearization of the Navier-Stokes equations appropriate and that the liquid viscosity be low enough to make boundary-layer techniques appropriate. The theory is compared to observations from the literature to assess the limitations of the theory. Some comparison with earlier work is given. It is concluded that the present theory reflects the data reasonably well and that it is an improvement over previous work. PINNING fluid-filled containers (examples include fluidfilled artillery shells and spacecraft with liquid-fueled apogee engines) can be unstable to growing pitch and yaw. For successful flight, the angle between the spin axis and the flight path (herein called the coning angle) must remain small. The motion of the container axis forces differential motions within the fluid, which in turn produce torques on the container. If the torques are such as to increase the coning angle, the container can be unstable. Coning at a frequency at or near an inertial mode frequency of the rotating liquid leads to a resonant liquid response and much larger torques. A flying container is acted on by many forces and torques other than the fluid dynamical ones that form the main topic of this paper. Attention will be restricted to fluid-filled gyroscopes, which cone freely in the laboratory. The coning angle grows with time; and this growth rate is determined almost entirely by the response of the contained fluid to the coning motion. Since the initial coning angle is essentially zero, the Navier-Stokes equations can be linearized about a solid rotation using the coning angle as the small parameter, previous work addressing the fluid dynamics includes that of Gans,1 which was specifically designed to deal with the resonance for forced coning, Murphy,2 and Gerber et al. 3'4 The present work is an extension of Ref. 1. The Murphy work contains a comprehensive bibliography of the literature, especially that of the projectile community. The differences between Refs. 3 and 4 and the present work are discussed in the final section. The motion of a gyroscope can be described in terms of the motion of its symmetry axis. The axis can be described instantaneously in an inertial system in which it is initially vertical by a unit vector K- sinacosfltf/ + sinfl/// + cosa/c/ (1)