A minimal height for the mercury column below the bottom of the mirror is desirable. (An optimal height is the layer at which mercury covers the bottom surface of the basin completely. The depth is about 0.5 mm, if there is a sublayer of amalgam. Mercury forms amalgam in contact with certain metals - electrolytical copper, silver, gold and platinum. Silver, gold and platinum can be coated as a thin layer on another material - such as steel or glass - that will not form amalgam with mercury. If the basin were made of the latter material alone, the height of the mercury column would increase significantly.) A higher mercury column would increase vibrations of mercury when the mirror was rotated or tilted.

The centring device will centre the mirror within the mercury basin, damping its vibrations at the same time, before getting into its state of equilibrium; however, it must allow for a free floating of the mirror. A centring accuracy of the mirror of about 0.01- 0.02 mm is required, to limit the influence of the mentioned capillary effects to an anticipated 0.01-0.02 ''. (The above mentioned authors, applying a differential equation establishing conditions of equilibrium between the capillary and gravity forces, derived that at a distance of , e.g., 5 and 7 cm from the internal wall of the basin, the deviation of a perpendicular to the mercury surface with respect to a vertical line is < 0.6'' and < 0.01 '', respectively.) A setup of the whole appliance, conforming to requirements on prismatic astrolabes, is shown in Fig. 5 (representing a plan and Sect. A-A).

$\begin{figure} \centering\includegraphics[]{1485f5.eps}\end{figure}$

Figure 5: Possible design of a "floating mirror horizon" (plan and Sect. A-A); (a) is the mercury basin, (b) is the floating mirror, (c) is the centring device - consisting of centring arms, fastening ring and centring balls in three fixed conical bearings

The centring device (c) is part of a fixed support extending outside the basin with mercury, and the whole appliance would rotate with the instrument. Other kinds of instruments would require modifications. With a PZT, or a meridian circle, the centring device would have to rest on a rotatable support - separate from the basin (a) and fitted with a bearing - to enable its rotation (with the mercury basin being stationary) through $180^\circ$ . With a circumzenithal astrolabe, the centring device would have to be fixed symmetrically to the circular opening for the mirror and to the baseplate of its upper rotatable part (carrying the optical system), either on the upper surface of the baseplate, or beneath this. These modifications would ensure, with all instruments, proper orientation of the floating mirror (b) in the required direction.

The centring device consists of three identical blocks, with their centres situated at equal radii, and at angles of $120^\circ$ , to the axis of rotation. At the centre of each block there is a bearing with a steel ball. The centring of the mirror is then achieved by limiting its movement relative to the three steel balls at the centres of the bearings by points of contact with the mirror's centring arms. The points of contact are tangent points of vertically oriented sides of the arms with the balls. The clearance between two sides of a centring arm and a ball is the (above) centring tolerance of 0.01-0.02 mm, which allows for free floating of the mirror. The arms are connected to the mirror by a ring. In this the mirror rests by its extended rim, and is fastened. (No fastening is indicated in Fig. 5. Three partially recessed and loose segments inside the ring, with fastening screws attached to its outside face, could be used; and care would have to be taken to avoid imposing an excessive strain when the segments are fastened to the mirror.) If the fasten is released, the mirror can rotate freely, and its sloping line can be set to the required direction. The arms - and the connecting ring - should be made of light, homogeneous material inert to mercury and its vapours.

The centring balls (Fig. 5) rest freely in fixed bearings, having the form of a reversed truncated cone, with its sides at an angle of $45^\circ$ to the vertical. Upon impact by a centring arm, if the instrument is rotated, the centring balls would elevate, spiralling up from their lowest position, thus absorbing part of the impact energy; then they would oscillate until coming to rest.

In order to estimate their most effective size, we can make use of the following relations, neglecting friction in the bearings, and assuming that the first impact is only with one ball: $E_{\rm P} = E_{\rm R} + E_{\rm T}$ , where $E_{\rm P}$ is potential energy of a ball, and $E_{\rm R }$ , $E_{\rm T}$ , are rotational and translational energies of the mirror. $E_{\rm P} = m.g.\Delta h$ ,where m is the mass of the ball, g is acceleration due to gravity and $\Delta h$ denotes the ball's rise above its lowest (equilibrium) point; $E_{\rm R} = 1/2 \Theta. \omega^2$ ; $\Theta = 1/2G.R^2$ is the moment of inertia of the mirror; $\omega$ is its angular rotational velocity; G, and R, are the mirror's mass and radius; and $E_{\rm T} = 1/2G.v^2$ , where v is the mirror's translational velocity. The mass of the ball is open to choice with regard to its allowable free path in the conical bearing (say, <2 mm). Neglecting $E_{\rm T}$ (the motion is mainly rotational), for a typical rotational velocity of the instrument about its vertical axis of 0.5 rad s^-1 (during its setting to a star, for example), with earlier constant $\sigma=2.5$ g cm^-3, for mirrors with diameters of l0 and 20 cm (of the previous thicknesses of 1.5 and 3.5 cm), and $\Delta h$ of 2 mm and 1 mm respectively, we obtain the following orientative values of diameters of steel balls ( $\sigma =8.0$ g cm^-3): 8.2 and 10.4 mm for a mirror of 2R=10 cm, and 27.6 and 34.7 mm for a mirror of 2R=20 cm. This shows that a diameter of 10 mm would be suitable in the first case, and one of 30 mm in the second.

The described gravitational damping of the mirror's oscillations could be combined with oil-damping, if the ball bearings were enclosed in small containers which could be filled with oil (of a suitable kind) to a sufficient level. Within a symmetrical design, the oil-induced adhesive forces should not (theoretically) disturb significantly the position of the mirror's equilibrium. But the functionality of such a system (and whether it would be of any advantage) would have to be tested experimentally. The symmetrical design also minimizes - indeed practically eliminates - the effects of thermal expansion of the system upon the equilibrium position of the mirror.

Another alternative is silicon oil of a suitable viscosity, which, in a thin layer, could be used (especially at fixed stations) to coat the free surface of mercury in the basin. It is inert to mercury and would diminish the mercury's evaporation. If the speed of damping of the mirror's vibrations is unimportant, the centring balls can be fixed to the bearings. They can then be of a comparatively small diameter. But it would not be suitable to fix them to the centring arms, because this would unnecessarily increase the load on the mercury.

6 Practical design of a floating mirror horizon