To find the versed sine of the curvature (which may be useful in the examination of the mirrors by a mould) we may proceed (see fig. 67) to put AG = _f_; BE = C; AC = R then BG² = AG . GD 4 BG² = BE² = 4 AG . GD C² = 4 _f_ . (2 R - _f_) C² = 8 _f_ R - 4 _f_² From which equation, C² C⁴ 2 _f_ - 2 R = ± √(4 R² - C²) = --- - ----- &c. 4 R 64 R³ C² C⁴ 2 _f_ = --- - ----- 4 R 64 R³ C² C⁴ _f_ = --- - -------. 8 R 128 R³ [Illustration: Fig. 68.] In order to test the accuracy of the workmanship of the mirrors, recourse must again be had, as in the case of the lenses and parabolic mirrors, to the formula of conjugate foci, in which we shall call R = the radius of curvature of the mirror M _m_ (fig. 68); _a_ = the distance of a light, _f_, which is arbitrarily placed in front of the mirror; and _b_ = the distance of a moveable screen S, on which the rays reflected from the mirror may converge in a focus. We must find the distance _b_, at which, with any given distance _a_, such convergence should take place. _f_ M′ = _a_ SM′ = _b_ OM′ = R Then (because _f_ MS is bisected by OM, and for points near the vertex of the mirror at M′) SM′ ∶ _f_ M′ ∷ SO ∶ O _f_ or _b_ ∶ _a_ ∷ R - _b_ ∶ _a_ - R _a_ _b_ - R _a_ = R _b_ - _a_ _b_. R _a_ From which _b_ = ---------, 2 _a_ - R the distance required, in which an error of ¹⁄₃₀ (of its whole length) may be safely admitted. [63] At such times when the horizon cannot be seen, the mirror may be placed, by means of a _clinometer_, with a spirit-level, set to the proper angle, which may be easily mechanically determined as follows: Draw a line from the focus F through the point O, where the centre of the mirror is to be, producing it beyond that point to a convenient distance at I; through O draw HOH, parallel to the horizon FH; bisect IOH by MOM, which coincides with a tangent to the mirror at its centre O; and MOH is the angle required to be laid off, or its complement. [Illustration: Fig. 69.] ~Cylindric Refractors for Fixed Lights.~