It is still more obvious that a perfect paraboloïdal figure, and a luminous _point_ mathematically true, would render the illumination of the whole horizon by means of a fixed light _impossible_; and it is only from the divergence caused by the size of the flame which is substituted for the _point_, that we are enabled to render even revolving lights practically useful. But for this aberration, the slowest revolution in a revolving light would be inconsistent with a continued observable series, such as the practical seamen could follow, and would, as we have seen, render the flashes of a revolving light too transient for any useful purpose; whilst fixed lights, being visible in the azimuths only in which the mirrors are placed, would, over the greater part of the distant horizon, be altogether invisible. The size of the flame, therefore, which is placed in the focus of a paraboloïdal mirror, when taken in connexion with the form of the mirror itself, leads to those important modifications in the paths of the rays and the form of the resultant beam of light, which have rendered the catoptric system of lights so great a benefit to the benighted seamen. In order to obtain a mirror capable of producing a given divergence of the reflected beam, therefore, we must proportion its focal distance to the diameter of the flame in such a manner, that the sine of _one-half_ of the whole effective divergence of the mirror, may be equal to the _quotient of the radius of the flame, divided by the distance of a given point on the surface of the mirror from the focus_. The best proportions for paraboloïdal mirrors depend on the objects which they are meant to attain. Those which are intended to give great divergence to the resultant beams, as in fixed lights, capable of illuminating the whole horizon at one time, should have a short focal distance; while those mirrors which are designed to produce a nearer approach to parallelism (as in the case of revolving lights which illuminate but a few degrees of the horizon at any one instant of time), will have the opposite form. Those two objects may, no doubt, be attained with the same mirror, by increasing or diminishing the size of the burner; but that is by no means desirable, as any change on the size of a burner, which is found to be the best in other respects, must be considered as to some extent disadvantageous.