We shall take as an example, which is well calculated to illustrate the difficulties of the grinding process, one of the prismatic rings of a Catadioptric Light of the first order. The first operation will be to take off the rough arris at the angles of the pieces as they come from the moulds, and to reduce to equality the length of each of the four quadrantal prisms or segments by removing from each the quantity that may be necessary to make those four pieces, when placed on a circle, exactly equal to that of the finished zones. Each of them must have an excess of material at the various surfaces just sufficient to insure the rubber having scope enough to remove all the flaws or defects of the two surfaces to be first ground, which are the (concave and convex) refracting faces of the zone (the sides AC and BC in the Table in the Appendix, No. IV.) [Illustration: Fig. 110.] The pieces must be placed end to end on the horizontal plane or table of the lathe at AA (fig. 110), and must rest on the exterior arris A of the reflecting side, on which arris there is ground a narrow plane whose width is proportionate to the projection of the outer edge beyond the inner edge of the zone, _foreshortened_ by the _bevel_ or inclination of the reflecting side, when resting (as in fig. 110) on the circular iron belt, which is screwed to the table of the lathe provided for its reception. This narrow plane at the arris A should be sufficient to give the prism a solid and regular bearing on the circular iron belt. In this figure (fig. 110) _a_ _b_ is the vertical axis of the lathe, _n_ the point from which the co-ordinates for O, the grinding centre for the exterior _concave_ refracting surface AC, are measured, and _e_ AC the arc swept by the grinding surface. Conversely, _n′_ is the origin of the co-ordinates for the grinding centre O of the interior _convex_ refracting surface BC, and _e_ CB the arc swept by its grinding surface. Some skill is required in fixing the prism on the belt, for, on the one hand, there is an obstacle to correct workmanship from the dragging motion of the platform, and, on the other, by the unequal subdivision of the weight of the glass, which should be nearly balanced. This narrow plane being perfectly adjusted for all the segments so as to bed them quite level, the circular iron belt on which the ring should be ground, is placed on its platform, in the manner represented in the figure. It should be as truly levelled as possible, otherwise all the subsequent operations will be deranged by it. This iron belt is heated by means of heating _pans_; and the degree of heat may be practically judged of by the ebullition of drops of water let fall on it. The segments of glass are also at the same time placed in a stove heated with steam, and are generally raised to about 120° centigrade. The difference between the time required for the two operations of heating the iron belt and the glass segments is