[69] The angle OAX is easily found, as will be seen by referring to fig. 73, p. 277; for, AH being horizontal by construction and AO vertical, HAO = 90°; and HAC and CAU being both known, we have OAX = 90° - (HAU + UAX) = 90° - (HAU + ¹⁄₂ CAU). [Illustration: Fig. 76.] The positions of the apices A and B of the angles of the zones are also easily found in reference to the focus, and are given in the Table in the Appendix. In fig. 76 we may, in reference to the known position of C, find that of A or B, by simply adding the quantities AH, HC, and BK, to C _y_ or C _x_, and by deducting CK from C _y_; while it is obvious that those quantities are respectively proportional to the length of the known sides AC and BC, modified by the inclination of those sides with the horizon. Hence we have AH = AC . sin ACH; HC = AC . cos ACH; BK = BC . sin BCK; and CK = BC . cos BCK. In the process of grinding the zones, it is found convenient for the workman to give a curved form to the refracting sides BC and AC, the one being made convex and the other concave, so that both being ground to the same radius, the convergence of the rays produced by the first shall be neutralized by the divergence caused by the second. By this arrangement we have three points given in space from which, with given radii, to describe a curvilinear triangle whose revolution round the vertical axis of the system generates the zone required. Co-ordinates to those two centres of curvature for the surfaces AC and BC were determined in reference to arris A of each zone, and will be found in the Appendix. The mode of finding those co-ordinates is, of course, similar to that already given; and, the radii being assumed at 4000 millimètres, the co-ordinates are respectively proportional to the sine and cosine of the inclination of the radius at A to the vertical line, which inclination depends upon the relations of known angles around A and C. The section ABC (fig. 71, p. 271) of the first zone being thus determined, we proceed by fixing the point C₂ of the second zone, which is at the intersection of the horizon GAG₂ with the ray FBC₂ passing through B. This arrangement prevents any loss of light between the adjacent zones. The calculation of the elements of the second and of every following zone, is precisely similar to that of the first. ~Testing of Zones.~