| b C | = | b m sin b m Csin b C m =l cos ε′ cos ¹⁄₂ (α - α′)(cos (η + ¹⁄₂ (α - α′)) sin (α - α′) |
| = l cos ε′ cos ¹⁄₂ (α - α′)cos {η + ¹⁄₂ (α - α′)} 2 sin ¹⁄₂ (α - α′) cos ¹⁄₂ (α - α′) | ||
from which, putting b C = ρ
ρ = l cos ε′ 2 cos {η + ¹⁄₂ (α - α′)} sin ¹⁄₂ (α - α′)
Lastly, the position of C the centre of curvature for a ring is easily determined by two co-ordinates in reference to their origin, A, which is the vertex of the lens (see [fig. 60] below), by the equations:
| CG | = | ρ . sin α - a b | = | ρ . sin α - r₁ |
| CQ | = | ρ . cos α - q Q | = | ρ . cos α - t″ |
The elements of each successive zone are determined in the same manner. The annular lens of the first order of lights in Fresnel’s system consists, as already stated, of a central disc 11 inches in diameter, and 10 concentric rings, all of which have a common principal focus, where the rays of the sun meet after passing through the lens. With such accuracy are those rings and the disc ground and placed relatively to each other, that the position of the actual conjugate focus of the entire surface of the compound lens, differs in a very small degree from that obtained by calculation in the manner described below.[59]
Testing Lenses. 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.
Fig. 61.
