Image-Equations.—If the co-ordinates of the pair of conjugate points Q, Q' (Fig. 4), re ferred to two systems of rectangular axes with their origins at the principal points H, H' art denoted by (u, y) and (u', y') ; that is, if u= HM, and u'= the following relations may be derived immediately from the figure: 1.= 0 (abscissa-equation), u' f f'"E y f' u ( magnification-formula ).
These are the so-called image-equations re ferred to the principal points. If we introduce the symbols n' ---• r =_- =- u' where n, n' denote the indices of refraction of the media of the object-space and image-space, these equations may also be put in the follow ing simple and convenient form: tr=U F, — U • U' The magnitude denoted by F is called the re fracting power of the optical system, and the symbols U, U denote the so-called reduced vergences of the object-rays and image-rays. If the linear magnitudes u, u, f and f are ex pressed in meters, the reciprocal reduced magni tudes U, U' and F' will be expressed in terms of a unit called a dioptry (sometimes written diopter or dioptre), which is defined to be the curvature of a sphere of radius one meter. This unit which was introduced into lens-optics by the opticians and spectacle-makers has been adopted into optical science.
Stops.— But the mere geometrical assump tion of a point-to-point correspondence be tween object-space and image-space by means of rectilinear rays (even were it realizable) is not by itself sufficient to explain the modus operandi of an actual optical instrument, be cause it leaves entirely out of account two fundamental conditions which are essential to all mechanical contrivances for the production of an optical image; namely, first, the fact (al ready alluded to) that the bundles of effective rays are necessarily limited by the transversal dimensions of the instrument (the diameters of the lens-fastenings, artificial diaphragms or and walls of the tube) ; and, second, the fact that the image, instead of being left floating in the air, is cast on a screen or sur face of some kind, for example, the ground glass plate of a photographic camera, or in the last analysis the surface of the retina of the observer's eye. Obviously, it will only be under very exceptional circumstances that this receiv ing surface will contain all the image-points that are conjugate to the points of the object relief ; so that even though the imagery were ideal in the sense of collinear correspondence between object-space and image-space, some parts of the image projected on the screen will be more or less blurred and indistinct due to this method of representation.
In addition to the circular fastenings of the lenses the system may also be provided with artificial diaphragms or disposed at various places along the axis of the instrument.
A ((stop' is usually formed by a plane opaque screen, perpendicular to the axis, pierced by a round hole concentric with the axis: To an eye looking into the instrument from the side of the object, only such stops as lie in front of the first lens will be directly visible. Any other stop or lens-rim will be seen only by means of the real or virtual image of it that is cast by that part of the optical system which is be tween it and the eye. Now these impalpable stop-images, whether visible or no; are just as effective in cutting out the rays as if they were actual material stops; because, obviously, any ray that goes through an actual stop must necessarily pass, either really or virtually, through the corresponding point of the stop image ; whereas a ray that is obstructed by a stop, will not go through the opening in the stop-image.
If the optical instrument is directed toward the object and focussed on some selected point M on the axis, this point of the object will be reproduced in the image-space at the point conjugate to M; on the assumption that the imagery is ideal, the transversal planes perpendicular to the axis at M and M' are a pair of conjugate planes which play a very es sential role in the theory of the imagery pro duced by an optical instrument. The plane in the object-space may be called the focus-plane (or the plane which is in focus on the (screens), and the conjugate plane in the image-space may be referred to as the screen plane.
Now if the eye is supposed to be placed at the axial object-point M and turned toward the instrument, the stop or stop-image whose aperture subtends at M the smallest angle is called the entrance-pupil of the system. All the effective rays in the object-space must be directed to- points which lie within the circum ference of the circular opening of the entrance pupil. On the other hand, when the eye is i placed at M' so as to look into the instrument through the other end, the stop or stop-image which subtends the smallest angle at M' de termines in the same way the so-called exit pupil, through which all the effective rays, when they emerge from the instrument, must pass, really or virtually. The exit-pupil is, in fact, the image of the entrance-pupil produced by the optical system as a whole, and the pupil-centres 0, 0' are, therefore, a pair of conjugate axial points. The apertures of the ray-bundles in the object-space and image-space are deter mined by the diameters of the entrance-pupil and exit-pupil, respectively. Each of the pupils is the conunon base or cross-section of the cones of effective rays in the, region to. which it belongs.