another ray from an opposite point reflected at _b′_ would be seen in the parallel direction _b′_ G′, thus indicating the boundary of a circular portion of the mirror _b a b′_, the whole of which would reflect light to the distant observer’s eye. Again, let us suppose a ray to come from another part of the flame, and be reflected at the mirror’s edge _e_ into the direction _e_ H, and another from the opposite side of the flame to be reflected at its opposite edge _e′_, into the direction _e′_ G″, and we obtain the full effect of the whole reflecting surface, which will continue unabated until the mirror in the course of its revolution shall reflect at _e′_ to the observer’s eye, a ray from a point in the margin of the flame (through which a tangent drawn from _e′_ to the flame would pass) in such a direction, that the angle which it makes with the axis of the mirror is equal to that subtended by the radius of the flame at the distance F _e_ or F _e′_. After this the light would recede from the edges of the mirror in the same gradual manner, until it should vanish in the direction _a_ D′, which is the opposite limit of the extreme divergence of the instrument. In the above explanation, I have confined myself simply to the effects of the outer ring of the flame, which is the source of divergence; but I need not remind the reader that every portion of the flame radiates light, which, being reflected, conduces to the effect. Some rays also are passing from the opposite sides of the flame through the true focus, so as to be normally reflected in lines parallel to its axis. The solid lines in the diagram shew the theoretical reflection of rays proceeding from F to _b_, _b′_, _e_, _e′_, where they are diverted into the directions _b_ B, _b′_ B′, _e_ E, and _e′_ E′; and by contrast with the dotted lines, serve to render more perceptible the path of the divergent rays which come from the edge of the flame. The Greek letters indicate the angles of divergence, and point out their relations to each other on either side of the mirror. The arcs of greatest and least divergence are marked in the diagram. This subject will be found treated less directly, but, certainly, more concisely and neatly, by Mr W. H. BARLOW, in a paper on the Illumination of Lighthouses in the London Transactions for 1837, p. 218.