The influence of concave lenses will evidently be precisely the converse of that of convex. Rays which fall upon them in a parallel direc tion will be made to diverge as if from the principal focus, which is here called the nega tive focus. This will be, for a plano-concave if uninterrupted, they would have met in the principal focus, will be rendered parallel ; if converging more, they will still meet, but at a greater distance ; and if converging less, they will diverge as from a negative focus at a greater distance than that for parallel rays. If already diverging, they will diverge still more, as from a negative focus nearer than the principal focus; but this will approach the principal focus, in proportion as the distance of the point of di vergence is such, that the direction of the rays approaches the parallel.
If a lens be convex on one side and concave on the other, forming what is called a meniscus, its effect will depend upon the proportion be tween the two curvatures. If they are equal, as in a watch-glass, no perceptible effect will be produced ; if the convex curvature be the greater, the effect will be that of a less powerful convex lens ; and if the concave curvature be the more considerable, it will he that of a less powerful concave lens. The focus of conver gence for parallel rays in the first case, and of divergence in the second, may be found by dividing the product of the two radii by half their difference.
Hitherto we have considered only the effects of lenses upon a pencil of rays issuing from a single luminous point, and that point situated in the line of its axis. If the point be situated above the line of its axis, the focus will be below it, and vice versa. The surface of every luminous body may be regarded as comprehend ing an infinite number of such points, from every one of which a pencil of rays proceeds, and is refracted according to the laws already specified ; so that a perfect but inverted image or picture of the object is formed upon any surface placed in the focus, and adapted to re ceive the rays.
lens, at the distance of the diameter of the sphere of curvature ; and for a double concave, in the centre of that sphere. In the same manner, rays which are converging to such a degree that, In optical diagrams it is usual, in order to avoid confusion, to mark out the course of the rays proceeding from two or three only of such points. By an inspection of the subjoined figures, it will be evident that, if the object be placed at twice the distance of the principal focus, the image being formed at an equal dis tance on the other side of the lens, will be of the same dimensions with the object : whilst, on the other hand, if the object be nearer the lens, the image will be farther from it, and of larger dimensions ; and if the object be farther from the lens, the image will be nearer to it, and smaller than itself. Further, it is to be re marked, that the larger the image in proportion to the object, the less bright it will be, because the same amount of light has to be spread over a greater surface ; whilst a smaller image will be much more brilliant, in the same proportion.
The knowledge of these general facts will enable us readily to understand the ordinary operation of the microscope; but the instru ment is subject to imperfections of various kinds, the mode of remedying which cannot be comprehended without an acquaintance with their nature. One of these imperfections re sults from the spherical aberration of the rays which have passed through lenses, whose curva tures are equal over their whole surfaces. If the course of the rays he carefully laid down, it will be found that they do not all meet exactly in the foci already stated, but that the focus of the rays which have passed through the peri pheral portion of the lens is much closer to it than that of the rays which are nearer the line of its axis ; so that, if a screen be held in the former, the rays which have passed through the central portion of the lens will be stopped by it before they have come to a focus ; and if the screen be carried back into the focus of these, the rays which were most distant from the axis will have previously met and crossed, so that they will come to it in a state of divergence.
In either case, therefore, the image will have a certain degree of indistinctness ; and there is no one point to which all the rays can be brought by a lens of spherical curvature. The difference between the focal points of the cen tral and of the peripheral rays is termed the spherical aberration. It is obvious that, to produce the desired effect, the curvature is re quired to be increased around the centre of the lens, so as to bring the rays which pass through it more speedily to a focus, and to he diminished towards the circumference, so as to throw the focus of the rays influenced by it to a greater distance. The requisite conditions may be exactly fulfilled by a lens one of whose surfaces, instead of being spherical, is a portion of an ellipsoid or hyperboloid of certain pro portions ; but the difficulties in the way of the mechanical execution of lenses of this descrip tion are such, that, for all practical purposes, they have been entirely abandoned in favour of lenses with spherical surfaces. Various means have been devised for diminishing the aber ration of these. In microscopes of ordinary construction, the method employed is to dimi nish the aperture or working surface of the lens, so as to employ only the rays that pass through the central part, which, if sufficiently small in proportion to the whole sphere, will bring them all to nearly the same focus. The use of this may be particularly noticed in the object-glasses of common microscopes ; where, although the lens itself be large, the greater portion of its surface is rendered inoperative by a stop, which is a plate with a circular aperture interposed between the lens and the rest of the instrument. If this aperture be gradually enlarged, it will be seen that, although the image becomes more and more illuminated, it is at the same time becoming more and more indistinct; and that, in order to gain defining power, the aperture must be !educed again. Now this reduction is attended with two great inconveniences ; in the first place, the loss of intensity of light, the de gree of which will depend upon the quantity transmitted by the lens, and will vary therefore with its aperture ; and, secondly, the diminu tion of the number or quantity of rays, which will prevent the surfaces of objects from being properly seen. Thus, for example, we shall suppose the observer to be looking at the scales of a butterfly's wing with a microscope fur nished with two object-glasses of the same focal length,—one corrected, the other not so. If, with the same illumination of the object, he apply to it the uncorrected objective, the aper ture of which is necessarily small, after having looked at it with the corrected lens, he will, in the first place, perceive that the whole field is much darker; but if, by increasing his illumi nation, he give the image an equal brightness, and see its outline with equal distinctness, he will be completely unable to see with the un corrected lens a series of delicate lines upon the surface of the scale, which the other makes evident. The power of exhibiting these and similar objects is termed penetration ; it de pends upon the size of the conical pencils of light admitted by the lens, and therefore upon its aperture.