three reflections, and so on for the other images since in the formation of each of these images, respectively, the angles of incidence and reflection are equal, it follows that the focus, or point of place of the image _b_, formed by one reflection of light, is at one-third of the distance from the eye to the further end of the bore; that formed by two reflections _d_, is one-fifth; that by three reflections is one-seventh, and the succeeding ones, one-ninth, one-eleventh, etc., of the same distance. Therefore, it will be observed that all these images are located within the third part of the length of the bore nearest to the eye. Consequently there are two-thirds of the entire length of the bore in which none of these images appear. It is to this part of the bore, only that the workman directs his attention, for it is here only that he can cause the “shade” to appear which discloses the crooks in the bore if any exist. When this part is straightened, he inverts the barrel and works from the other end. [Illustration: FIGURE 32.] The practical application of the process is made in this manner: the workman has a rest, generally consisting of an upright strip of board of convenient height with a V cut in its upper end for convenience in resting the barrel in case of rotating it. Across a window opposite, at almost any distance, say about ten or twelve feet, is nailed horizontally, a strip of board like a common lath, as the horizontal bar of the window sashes where they come together at the middle of the window (provided there be upper and lower sashes) will answer nearly as good a purpose. Now place one end of the barrel in the V of the rest, look into the bore, directing the eye to the lower side of it and to the point just beyond the image _b_; gradually depress the end held in the hand, bringing the direction nearer and nearer to the horizontal strip, or the sash bar as the case may be, and a dark shade is soon seen as shown at _m_, Fig. 32. This is the reflected image of the horizontal strip, or sash bar, the curved part of the outline being the image of the straight-edge. Depress the end more and more and the shade lengthens to _n_, _o_, _p_, etc. If the bore be perfectly straight, the shade will always maintain a true and symmetrical parabolic form, growing more and more pointed at its apex, until it reaches the further end of the bore. But if there be even the slightest flexure or crook in the bore the parabolic figure of the shade will be distorted. If a distortion be discovered, the barrel is slowly revolved about its axis as it is retained in the rest; at the same time slightly elevating and depressing the end held in the hands, until the shade assumes a form in which the two sides near the apex are equally drawn in toward each other as shown at _q_. If the crook be considerable the two sides may be drawn quite together, cutting off a portion of the shade of the apex as at _r_.