vertical. Therefore there are two independent local rotations. In addition to these rotations we consider the rotation of the Earth about its north-south (NS) axis, and we model the planet as a perfect sphere. We neglect friction and also the rotation of the Earth about the Sun. In this case a non-rotating observer located at the center of the Earth can be approximated as being an inertial frame. We establish cartesian coordinates for such an observer (whom we name as 1-O), and the barycenter of the gyroscope is located at a distance from the center of the Earth. First time-dependent rotation Consider another (non-inertial) observer (the 2-O) located at the center of the Earth but rotating about the NS-axis by We establish coordinates attached to this observer as so that the unit versor is mapped to the point . For the 2-O neither the Earth nor the barycenter of the gyroscope is moving. The rotation of 2-O relative to 1-O is performed with angular velocity . We suppose that the axis denotes points with zero longitude (the prime, or Greenwich, meridian). Second and third fixed rotations We now rotate about the axis, so that the -axis has the longitude of the barycenter. In this case we have With the next rotation (about the axis of an angle , the co-latitude) we bring the axis along the local zenith (-axis) of the barycenter. This can be achieved by the following orthogonal matrix (with unit determinant) so that the versor is mapped to the point Constant translation We now choose another coordinate basis whose origin is located at the barycenter of the gyroscope. This can be performed by the following translation along the zenith axis so that the origin of the new system, is located at the point and is the radius of the Earth. Now the -axis points towards the south direction. Fourth time-dependent rotation Now we rotate about the zenith -axis so that the new coordinate system is attached to the structure of the gyroscope, so that for an observer at rest in this coordinate system, the gyrocompass is only rotating about its own axis of symmetry. In this case we find The axis of symmetry of the gyrocompass is now along the -axis. Last time-dependent rotation The last rotation is a rotation on the axis of symmetry of the gyroscope as in Dynamics of the system Since the height