Now, the angle _b_ C _m_ = α - α′ from which (since the triangle _b_ _m_ C is isosceles) _b_ _m_ C = 90° - ¹⁄₂ (α - α′); also, in the triangle _b_ _m_ _e_, the angle _b_ _m_ _e_ = _b_ _m_ C - _e_ _m_ C = 90° - ¹⁄₂ (α - α′) - η and _b_ _e_ _m_ = _k′_ _e_ E′ = 90° - ε′ We have therefore in the triangle _b_ _m_ _e_ _b_ _e_ sin _b_ _e_ _m_ _l_ cos ε′ _b_ _m_ = ----------------------- = --------------------- sin _b_ _m_ _e_ cos {η + ¹⁄₂ (α - α′)} and in _b_ _m_ C _b_ _m_ sin _b_ _m_ C _l_ cos ε′ cos ¹⁄₂ (α - α′) _b_ C = --------------------- = ---------------------------------- sin _b_ C _m_ 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 _l_ cos ε′ ρ = --------------------------------------- 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 - _r_₁ 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] [59] ~Testing Lenses.~