transmitted light. Various modes of removing similar inconveniences in telescopic lenses have been devised; and the suggestions of DESCARTES, as to combinations of hyperbolic and elliptic surfaces with plane and spherical ones, more especially fulfil the whole conditions of the case; but the excessive difficulty which must attend grinding and polishing those surfaces, has hitherto deprived us of the advantages which would result from the use of telescopic lenses entirely free from spherical aberration. In Lighthouse lenses, where so near an approach to accurate convergence to a single focus is unnecessary, every purpose is answered by the partial correction of aberration which may be obtained, by determining an average radius of curvature for the central disc, and for each successive belt or ring, as you recede from the vertex of the lens. In the lenses originally constructed for FRESNEL by SOLEIL, the zones were united by means of small _dowels_ or _joggles_ of copper, passing from the one zone into the other; but the greater exactness of the workmanship now attained, has rendered it safe to dispense with those fixtures; and the compound lens is now held together solely by a metallic frame and the close union between the concentric faces of the rings, which, however, are in contact with each other at surfaces of only ¹⁄₄ inch in depth, as shewn in Plate XII. It is remarkable, that an instrument, having about 1300 square inches of surface, and weighing 109 lb., and which is composed of so many parts, should be held together by so slender a bond as two narrow strips of polished glass, united by a thin film of cement. I now proceed to the formulæ employed by FRESNEL, to determine the elements of the compound lens,[56] in the calculation of which two cases occur, viz., the central disc and a concentric ring. The focal distance of the lens and the refractive index of the glass are the principal data from which we start. [56] It may be proper to mention that, while the formulæ given in the text are those of M. FRESNEL, I am responsible for the investigations in the Notes; I have, at the same time, much pleasure in acknowledging my obligations, at various times (about ten years ago), to Mr EDWARD SANG, and (more recently) to Mr WILLIAM SWAN, for their kind advice on this part of the subject. [Illustration: Fig. 54.] I begin with the case of the central disc or lens round which the annular rings are arranged. Its principal section is a mixtilinear figure (fig. 54) composed of a segment _b_ _a_ _c_, resting on a parallelogram _b_ _c_ _d_ _e_, whose depth _b_ _d_ or _c_ _e_ is determined by the strength which is required for the joints which unite the various portions of the lens. Those particulars have, as I already stated, been determined with so much judgment by FRESNEL and the dimensions of the lenses so varied to suit the case of various lights, that nothing in this respect remains to be done by others. [Illustration: Fig. 55.]