I must now endeavour to trace the various steps by which the elements of the zones given in the appended Table have been determined; and this, I fear, I cannot do without considerable prolixity of detail. Referring to Plates XV., XVI., XVII., and XVIII., in which F shews the flame, RR, the refractors, and MRM and MRM, the spaces through which the light would escape uselessly _above_ and _below_ the lens, but for the corrective action of the mirrors MM, which project the rays falling on them to the horizon, I have to observe that a similar effect is obtained, but in a more perfect manner, by means of the zones ABC and A₂B₂C₂ (fig. 71, on page 271), whose action on the divergent rays of the lamp causes the rays FC, FB and FC₂, FB₂ to emerge horizontally, by refracting them at the inner surfaces BC, B₂C₂, reflecting them at AB, A₂B₂, and a second time refracting them at AC, A₂C₂. [Illustration: Fig. 71.] The problem proposed is, therefore, the determination of the elements and position of a triangle ABC, which, by its revolution about a vertical axis, passing through the focus of a system of annular lenses or refractors in F, would generate a ring or zone capable of transmitting in an horizontal direction by means of _total reflection_, the light incident upon its inner side BC from a lamp placed in the point F. The conditions of the question are based upon the well-known laws of _total reflection_, and require that all the rays coming from the focus F shall be so refracted at entering the surface BC, as to meet the side BA at such an angle, that instead of passing out they shall be _totally reflected_ from it, and passing onwards to the side CA shall, after a second refraction at that surface, finally emerge from the zone in an horizontal direction. For the solution of this problem, we have given the positions of F the focus, of the apex C of the generating triangle of the zone, the length of the side BC or CA, and the refractive index of the glass. The form of the zone must then be such as to fulfil the following conditions:-- 1. The extreme ray FB must suffer refraction and reflection at B, and pass to C, where being a second time refracted, it must follow the horizontal direction CH. 2. The other extreme ray FC must be refracted in C and passing to A, must at the point be reflected, and a second time reflected, so as to follow the horizontal course AG (see fig. 72, on opposite page). These two propositions involve other two in the form of corollaries. 1. That every intermediate ray proceeding from F, and falling upon BC in any point E, between B and C, must, after refraction at the surface BC in E into the direction EW, be so reflected at W from AB into the direction WI, that being parallel to BC, it shall, after a second refraction in I, at the surface AC, emerge horizontally in the line IK. And, 2. That the paths of the two extreme rays must therefore trace the position of the generating triangle of the zone.