NOTE.--The Six co-ordinates of the Generating Triangle of each Zone in these Tables have the Focus of the Principal Lenses of the System for their origin; it being considered more convenient, in executing the necessary protractions, preparatory to the construction of the grinding apparatus, at once to refer the whole of the grinding machinery to the axis of the apparatus. To prevent the appearance of any inconsistency, however, it is proper to mention, that the radiant points of the series of Zones, do not exactly coincide with the Focus of the Lenses nor with each other; and that to avoid the parallax which the distance of the radiant points from the origin of the co-ordinates would occasion, it is necessary to make some corrections upon the linear dimensions, so as to find the line corresponding to the angles θ, ξ, and the distance Δ. In the Hyperpyral series, which stands _above_ the flame, the Zones have the radiant point 10 millimetres above the Focus of the Lenses, and each _y_ of this series, therefore, requires a reduction of that quantity; while the _x_ remains unchanged. In the Hypopyral series, which stands _below_ the flame, the Focus or radiant points of each Zone, varies its place in the flame, moving upwards as the Zone is lower, so that the line joining the Zones and the Foci, revolves as a radius vector round a point between them. In this way, the _x_’s remain unaltered; but the _y_’s will be lengthened successively by the addition of 10, 14, 19, 25, 32, and 40 millimetres. As these Tables contain the dimensions of Zones which are intended as an addition to the apparatus of FRESNEL, I have adopted the metric scale, so as to render them at once applicable to the existing protractions of that system. It is only necessary to add, that the conversion of millimetres into imperial inches, is easily effected by adding the log. millimetres to the log. ̅2·59516, the sum being the log. of the equivalent of the first term in imperial inches.