The tests generally applied for examining the lenses used in Lighthouses, is to find the position of the conjugate focus _behind_ the lens, due to a given position of a lamp in _front_ of it. This test depends on the following considerations:--Draw a line from an object O in front of a lens, to any point Q in the lens; and from A, the centre of the lens, draw AR parallel to OQ, and cutting a line RF _r_ which passes through the principal focus F, at right angles to the axis of the lens; then join the points Q and R, and produce the line joining them: I, the image of O must be in that line. In the same way, draw a line from O to _q_, another point in the lens on the other side of its axis, and parallel to it draw A _r_ from the centre of the lens, cutting the plane of the principal focus in _r_. Join _q_ _r_, in which line the image will lie; and hence the intersection of OR and _q_ _r_, in I, will be the point in which the image of O is formed, or will be the conjugate focus of the lens due to the distance OA. This mode will serve to give the distance of the conjugate focus of a lens (_neglecting its thickness_) for rays falling on its surface at any angle. [Illustration: Fig. 61.] We shall suppose QA (fig. 61) to represent the half of a lens, and remembering the conditions described in reference to the last figure, we shall at once perceive the truth of the following analogy (fig. 62):-- [Illustration: Fig. 62.] OA ∶ AF ∷ AQ ∶ FR ∷ AI ∶ FI, and putting OA = δ, AI = φ′, and AF = φ, we have δ ∶ φ ∷ φ′ ∶ φ′ - φ, and, consequently, δ φ′ - δ φ = φ φ′; and hence the following equations, which express the relations subsisting between the principal focus of the lens and the distance of any object and its corresponding image: 1_st_, To find the principal focal distance of a lens from the measured position of its object and its image refracted through it, we have, δ φ′ φ = ------. δ + φ′ 2_d_, For the distance of the object, when that of the image is known, we have, φ φ′ δ = ------. φ′ - φ 3_d_, For the position of the image, when that of the object is known, we have, δ φ φ′ = -----. δ - φ [Illustration: Fig. 63.]