Owing to the necessary arrangements of a lantern, only a very small portion of those rays, which escape from below the lenses, can be rendered available for the purposes of a Lighthouse; and any attempt to subject it to lenticular action, so as to add it to the periodic flashes, would have led to a most inconvenient complication of the apparatus. FRESNEL adopted the more natural and simple course of transmitting it to the horizon in the form of flat rings of light, or rather of divergent pencils, directed to various points of the horizon. This he effected by means of small curved mirrors, disposed in tiers, one above another, like the leaves of a Venetian blind--an arrangement which he also adopted (shewn in Plates XV. and XVI.) for intercepting the light which escapes above as well as below the dioptric belt in fixed lights. Those curved mirrors are, strictly speaking, generated (see fig. 65) by portions, such as a b, of parabolas, having their foci coincident with F, the common flame of the system. In practice, however, they are formed as portions of a curved surface, ground by the radius of the circle, which osculates the given parabolic segment.[62] The mirrors are plates of glass, silvered on the back and set in flat cases of sheet-brass. They are suspended on a circular frame by screws, which are attached to the backs of the brass cases, and which afford the means of adjusting them to their true inclination, so that they may reflect objects on the horizon of the Lighthouse to an observer’s eye, placed in the common focus of the system.[63] [62] To find the radius and centre of a circle, which shall osculate a given parabola, whose focus is in F, draw the normals to the curve from _p_ and P, meeting in O, and draw N _e_ parallel to a tangent of the curve, or to _p_ P, then P O or _p_ O is the radius required. Now, we have similar triangles P _p_ _d_ and N _e_ _n_, and P H and _p_ _h_ are (proximate) ordinates; hence we have the following analogies:-- P _d_ ∶ P _p_ ∷ PH ∶ PN N _e_ ∶ N _n_ ∷ PH ∶ PN [Illustration: Fig. 66.] Hence compounding those ratios (in which P _d_ = N _n_ nearly) N _e_ ∶ P _p_ ∷ PH² ∶ PN² also N _e_ ∶ P _p_ ∷ NO ∶ PO, (for O P _p_ and N _o_ _e_ are similar triangles) PH² ∶ PN² ∷ NO ∶ OP, then PN²- PH² = HN² and PO - NO = NP, therefore HN² ∶ PN² ∷ NP ∶ PO, and finally, PN³ PO = ---. HN² Then put FP = HC = FN = ρ; HN = ρ - _z_; then as FP² - FH² = PH² = ρ² - _z_² PN² = PH² + HN² = (ρ² - _z_²) + (ρ² - 2 ρ _z_ + _z_²) = 2 ρ² - 2 ρ _z_ PN = √(2 ρ (ρ - _z_)) Therefore √{2 ρ (ρ - _z_)}³ PO = ---------------- (ρ - _z_)² √({2 ρ (ρ - _z_)}³) = ------------------ (ρ - _z_)⁴ ( ρ³ ) and finally, PO = 2 √2 √(-------) (ρ - _z_) [Illustration: Fig. 67.]