The mode of grinding the zones I shall not notice here; but shall refer the more curious reader to the Appendix, in which I have given the details of the process followed by M. THEODORE LETOURNEAU, who now manufactures the apparatus for the Northern Lights Board, in the room of M. FRANÇOIS SOLEIL, who is engaged at St Petersburg in the same work. I accordingly proceed to consider what mode should be followed in testing the accuracy of the zones. For this purpose, various expedients suggested themselves, such as the application of gauges in the form of a radius, having at one end a plate with a triangular space cut through it, equal and similar to the cross section of the zone. The horizontal motion of this arm would, of course, detect the inaccuracies of the successive sections of the inclosed zone. The application of such a gauge, however, seemed difficult, and in order to test the _form_ of the zones, I satisfied myself with using callipers (similar to the sliding rules used by shoemakers) for measuring the _length_ of the sides of the zone, and a goniometer for the angles, which is represented in the figure (fig. 77), in which ABC represents the prism, with one angle inclosed between the arms AC and AB, moveable round a centre O, and RR the graduated limb. This instrument is inconvenient and defective, as the convexity of the sides AB and BC of the zone requires some skill in getting the arms to be tangents to them. [Illustration: Fig. 77.] [Illustration: Fig. 78.] A practical test, however, yet remained to be made of the zones when fixed in the brass frames (shewn at Plate XVIII.), and assembled around the common focus of the system, by measuring the final inaccuracy in the path of the rays emergent from them. I have successfully used the following mode. Having mounted the frames containing the zones on a carriage revolving round a small flame placed truly in the common focus, I carefully marked with a piece of soap the centre of the emergent _surface_ of each zone; and having attached to a vertical rod of metal a telescope, provided with a spirit-level and cross-hairs (for cutting the centre of the image of the flame reflected through the zone) in such a manner as to be capable of sliding on the rod, I observed the cutting of the centre of the flame by the cross-hairs. In the case of any aberration from a normal emergence of the central ray, I had thus the means of at once determining its amount and direction. The telescope was moved up or down, and its vertical inclination was varied until the axis of the instrument coincided with the direction of the ray emergent from the centre of each zone, which was made to circulate round the flame, the observer noting any change in the position of the reflected image of the flame, and causing an attendant to mark the zones in which the change occurred, that they might again be subjected to separate examination of the same kind, by adjusting the telescope to the error of each.