Thus, to gravitationally force on the liquid mirror a converging (parabolic) "free" liquid surface, one with minimum focal-length: The splines are rocked or pivoted such that outer ends are brought in contact with the mirror's limb or circumference, while concomitantly, teeter-totter-wise, the opposite ends (at mirror center) are rotated out a maximum distance away from the center. In this (converging) case, the attached, circular, flattened bladder carrying the (gravitationally) "figuring" liquid, is maximally thickened at its center while it is maximally thinned at its circumference. By local gravitational attraction of the "figuring" mass, the primary mirror's entire center is pulled infinitesimally backward (or maximum "fine-tuning" paraboloidal curvature is gravitationally achieved, as will be shown by application of Ta Li's analysis to synthesis, discussed below). Hyperbolic (coarser, convex) surface control, when required, is effected by the technique of first adding or withdrawing liquid from the shallow liquid mirror "pool". One thereby axially translates the contact ring to a given, desired "altitude" on the toroidal mirror boundary-ring (Figure 18), which will give a greater or lesser mirror radius of curvature, depending on whether the translation rotates the contact angle out of (away from) the mirror surface, or into (toward) the minor surface, respectively (Figure 18). "Altitude" of mirror surface intersection on circumferential, toroidal boundaryring, is adjusted by adding or withdrawing liquid, until desired (paraboloidal or hyperboloidal or even plane) mirror curvature is achieved. "Fine tuning" of this surface is then effected by squeezing the "gravitational figuring" liquid (probably mercury), more from the limb to the center (decreases focal length) or vice-versa (increases focal length), then locking splines at corresponding angle. Figuring telescope/laser liquid mirrors for orbit; perturbations Astronomical and energy-handling reflective primary and secondary liquid optics generally will be nearly spherical (e.g., needing only a slight retouching from the "epihydrostatic" spherical figure to generate required parabolic or hyperbolic surfaces [2, 3]), or purely spherical [14], Using an expression (the "Fundamental Expression of Liquid Space Optics," derived below), developed from Dr. Ta Li's paper on what Dr. Elliott Benedikt [7] first called "epihydrostatics", one can calculate shapes, sizes and positions of masses designed to lend the required retouching (to render spherical liquid optical surfaces, Cassegrainian or Newtonian-that is, paraboloidal). [6] Dr. Li showed in his General Dynamics ('60, San Diego) formulation for deviation of (spherical-Boundary) liquid surfaces from a spherical liquid-vapor interface under given small net axial acceleration (in zero-gravity for example, orbit or unpowered trajectory in space), that we are facing a standard isoperimetric problem (with a mobile upper limit) in the calculus of variations. Couching the rapidly-converging series solution (given by Li, op. cit.) to the resulting differential equation in both physico-chemical and optical parameters, one can show arbitrary (integrated) "figuring" gravitational acceleration, may be practically applied to such (reflective), almost arbitrarily large diameter, liquid surfaces.
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