5 Some numerical values

For mercury $\sigma_{\rm F}\approx 13.5$ g/cm³ (at $0~^\circ$C); the most suitable material for the mirror would be crystalline glass: Zerodur or Cervit, for instance. Its $\sigma\approx 2.5$ g/cm³. A mirror made of this material would have very stable dimensions due to its small coefficient of thermal expansion (< 1 ppm). The height of the displaced liquid from Eq. (4) is $h_{\rm F}\cong 0.19H \approx H/5$. The following Table 2 contains, for illustration and comparison, the value of the metacentric height $\mathcal{H}$, and the value of the balancing moment M for some typical dimensions of square and circular shaped-mirrors.

  

Table 2: Metacentric height $\mathcal{H}$ and balancing moment M for square and circular glass plates of thickness H, length L, and radius R, respectively, floating on mercury (of displaced height $h_{\rm F}$) and inclined at an angle $\psi$ to horizontal direction

From Table 2 it is evident that the balancing moment of a plate of doubled size increases by more than an order of ten (more than twenty times), and for a cylindrical plate of 20 cm in diameter the metacentric height becomes about 0.4 m. With transoceanic ships the latter is given as 0.3 m, and with battle ships this is typically 1m. This indicates that a mirror of this diameter (but also that of 10 cm), will have good floating stability with concomitant sensitivity in re-establishing its equilibrium.

The discussed shape of the floating plate is also suitable for the shape of the mercury plate. This can then be comparatively flat and of minimum height; and the surfaces of the plate being parallel planes conform to the requirement that the mirror and a horizontal plane be parallel. ($\Delta z$ should be sufficiently small. Made of homogeneous material, and its surfaces being exactly plane-parallel, this would automatically be achieved.)

So far in our considerations, the floating body has only been a simple plate. Could it be of some purpose to change its form by attaching an additional mass below this? Doing so would lower the the centre of mass of the whole body, increasing thereby its metacentric height, and thus the balancing moment. But it would be quite ineffective. Calculating an increase to the balancing moment from Eqs. (9) and (14) - for instance for a square, or circular plate of the above table with thickness H=1.5, length L=10, or radius R=5 (cm), respectively: if we change L, or R, by $\Delta L= \Delta R=1$ mm, the increase in the balancing moment would be by more than 15 times greater than that due to a change of $\Delta H=1$ mm in the plate's thickness. This means that for a comparable effect, the additional mass attached below would have to be of considerable height, compared to that of the upper plate. It is evident that it would be impractical to enlarge the balancing moment this way, if this can be achieved much more effectively with a comparatively small change to the horizontal dimensions of the upper plate. Besides, the additional lower block would increase the vertical dimension - and amount - of the fluid (and thus of the mercury basin) required for the body to float, which would be untenable. A simple plate of cylindrical form is, indeed, the optimum shape for the floating mirror horizon.