Having in this manner determined the angles of BCF, the obtuse angle BCA of the generating triangle of the zone is easily and directly deduced by the following expression, which results from the obvious relations existing among the known angles about C; and we have (see fig. 73), BCA = 90° + γ = 90° + 2 ξ - θ. We next proceed to consider the form of BA, the reflecting side of the zone, which is a point of the greatest consequence, as an error in the inclination of any part of its surface is doubled in the resulting direction of the reflected rays. The conditions of the question require, that every ray EW, after reflection at the surface AB, shall, like WI, be parallel to the first ray, which is reflected in the direction BC, and after a second refraction at C, emerges horizontally in CH. But, let us trace backwards the rays as they emerge in their horizontal directions IK, and it is obvious that if BA be made a straight line, then will every ray EW meet the first refracting side BC at the same angle, and there suffering the same refraction, they will go on parallel to each other, and never meet in the focus F. This convergence to F, which is a necessary condition of the problem, may, however, be produced by a curvature of AB, such that all the rays shall have a degree of convergence before falling on BC, sufficient to cause them to be finally refracted, so as to meet in F. On this account, they will occupy _less_ space in passing through BC, than they did in passing through AC; and thus BC will be _shorter_ than AC by some quantity which shall give to that part of AB which is at B the amount of _downward_ inclination required for causing the ray BF finally to converge to F; and the line joining B and A must be a curve, every point of which has its tangent inclined so as to serve the same purpose. [Illustration: Fig. 73.] To trace tangents to this curve, is therefore the next step in the process. The direction of the first tangent AZ depends upon very simple considerations; and all that is necessary to be done is to draw a line AU (fig. 73), parallel to BC (which is the parallel to the direction of the reflected rays), and forming an angle CAU, which is, of course, equal to the inclination of the extreme rays refracted by CB at C, with rays reflected from the arc which we have yet to trace. The line AX bisecting this angle, must therefore be a normal to the reflecting surface at A, and AB drawn perpendicular to AX, is consequently a tangent to the reflecting arc. We must next find the direction of the second tangent Z _b_, which must be so inclined that the ray F _b_ will, after refraction at _b_, be reflected into the direction, _b_ C; but as the rigorous determination of this is difficult, I shall describe two approximations suggested to me by M. LEONOR FRESNEL. The first method is based upon assuming the inclination of the ray refracted at _b_ to the ray refracted at C as equal to: _b_ FC ------ _m_