Radius of flame. = ------------------------------------------- Distance from focus to point of reflection. [Illustration: Fig. 26.] It is obvious that this quantity which varies _inversely_ with the distance of the reflecting surface from the focus, is greatest at the vertex of the curve, and least at the sides or edges of the paraboloïd. The most useful part of the light, or that which conduces to the strongest part of the flash in a revolving light, is that which is derived from the cone of rays which is bounded by the limits of this _minimum_ divergence; for the faint light which first reaches the eye of a distant observer, in the revolution of a reflector, is not that which is reflected by the sides or edges, as might at first be supposed, but proceeds from the centre. The light, in fact, gradually increases in power in proportion as additional rays of reflected light are brought to bear on the observer’s eye, until, last of all, the extreme edge of the mirror adds its effect. The light continues in its best state until the opposite limit of minimum divergence has been reached, when it begins gradually to decline, receding from the margin of the mirror towards the centre, and, having at length reached the limit of its maximum divergence, it finally disappears at the centre. The increase and decline of the power of a mirror in the course of its movement round the circle of the lantern, as seen by a distant observer, will, therefore, in all its different states, be measured by the areas of a series of circles described from its focus, with radii equal to the distance of the focus from the point of the mirror which reflects to the observer’s eye the extreme ray which can reach him in any given position of the mirror. This will be more easily understood by referring to the accompanying diagram, Fig. 26, in which _e a e′_ is the principal section of a paraboloïdal mirror, F its focus, αFA its axis, and FK the radius of the flame. If the reflector revolve round a vertical axis at O, an observer placed in front of it (at a distance so great that the subtense of the mirror’s width would be small enough to allow us safely to consider the lines drawn from _e_ and _e′_ to his eye as parallel), would receive the first ray of light in the direction _a_ D, as reflected at _a_, from a single point on the edge of the flame (where a tangent to the flame would pass through _a_); and conversely he would lose the last ray at D′, as reflected at _a_, from a single point on the opposite margin of the flame; and hence, as above, the greatest divergence is measured by the angle which the flame subtends at the vertex a of the mirror, being the sum of the angles _α_ and _α′_. We shall next suppose the mirror to move a little, so that the observer may receive at G a ray of light from some other point in the flame which is reflected at _b_; while