the Lighthouses of France, has, with the greatest liberality, put me in possession of the various formulæ used by his lamented predecessor, in determining the elements of those instruments which have so greatly improved the lighthouses of modern days. [55] _r_ F = ------- _m_ - 1 in which _r_ is the radius of curvature, and _m_ is the refractive index.--_Coddington’s Optics_, Chap. VIII. If the radiant be brought near the lens, so as to cast divergent rays on its surface, then the conjugate focus will recede behind the _principal focus_; and when the luminous body reaches the _principal_ focus _in front_ of the lens, the rays will emerge from its posterior surface in a direction parallel to its axis. If it be brought still nearer the lens, the rays would emerge as a divergent cone. Hence converging lenses can only collect rays into a focus, when they proceed from some point _more_ distant than the principal focus. Spherical lenses, like spherical mirrors, collect truly into the focus those rays only which are incident near the axis; and it is, therefore, of the greatest importance to employ only a small segment of any sphere as a lens. The experience of this fact, among other considerations, led CONDORCET, as already noticed, to suggest the building of lenses in separate pieces. FRESNEL, however, was the first who actually constructed a lens on that principle; and he has subdivided, with such judgment, the surface of the lens into a centre lens and concentric annular bands and has so carefully determined the elements of curvature for each, that no farther improvement is likely to be made in their construction. For the drawings of the great lens, I have to refer to Plate XII., which also contains a tabular view of the elements of its various parts. The central disc of the lens, which is employed in lights of the first order, and whose focal distance is 920 millimètres, or 36·22 inches, is about 11 inches in diameter; and the annular rings which surround it vary slightly in breadth from 2³⁄₄ to 1¹⁄₄ inches. The breadth of any zone or ring is, within certain limits, a matter of choice, it being desirable, however, that no part of the lens should be much thicker than the rest, as well for the purpose of avoiding inconvenient projections on its surface, as to permit the rays to pass through the whole of the lens with nearly equal loss by absorption. The objects to be attained in the polyzonal or compound lens, are chiefly, as above noticed, to correct the excessive aberration produced by refraction through a hemisphere or great segment, whose edge would make the parallel rays falling on its curve surface converge to a point much nearer the lens than the principal focus, as determined for rays near the optical axis, and to avoid the increase of material, which would not only add to the weight of the instrument and the expense of its construction, but would greatly diminish by absorption the amount of