If the problem is restricted (for example, by a suitable arrangement of ustopsri with very narrow -openings) to such rays as are never at any stage in their passage thrOugh the sym metrical optical instrument very far from the optical axis (so-called paraxial rays), it may be shown that under these circumstances (assum ing also that we have to deal only with monochromatic light) there will be ultimately strict collinear correspondence between object space and image-space; so that the bundle of image-rays corresponding to a homocentric bundle of object-rays will also be homocentric, and accordingly every point P of the object will be reproduced by an ideal image P' which is said to be aconjugatel to P. According as the image-rays, on issuing from the instrument, converge to a (real) focus at P' or diverge as though they had come from a (virtual) focus at P the image is described as a real image or a virtual image, respectively.
Gauss Theory, Cardinal Points, Focal Lengths, etc.—Although it is difficult to realize even approximately the conditions that are necessary for a collinear correspondence of object-space and image-space, nevertheless the assumption of this geometrical relation does afford a fairly accurate conception of the fundamental connections of the position and size of the image with those of the object and enables us to form an idea of the general and salient characteristics of the imagery. A gen eral method of treating the problem of the imagery produced by the refraction of paraxial rays through a centered system of lenses was published by C. F. Gauss in his famous Dioptrische in 1841. Accord ing to this theory the action of such a system is completely determined by the positions on the axis of four cardinal points, namely, the two focal points F and F' (Fig. 4) and the two principal points H and H'. These points must now be defined.
In every centered system of spherical re fracting surfaces there are two (and only two) transversal planes at right angles to the optical axis which are characterized by the following properties: A bundle of paraxial object-rays which all meet in a point in one of these planes (the primary focal plane) will emerge from the sys tem as a cylindrical bundle of parallel image rays; and, similarly, a cylindrical bundle of parallel object-rays will emerge from the sys tem as a bundle of image-rays which all meet in a point in the other one of these planes (the secondary focal plane). The points where the focal planes are pierced by the optical axis are the primary and secondary focal points F and F' above mentioned.
Moreover, in every symmetrical system of lenses there is one (and only one) pair of conjugate planes perpendicular to the axis which are distinguished by the property that in these planes object and image are congruent; and, therefore, any straight line drawn parallel to the axis of symmetry will intersect these planes in a pair of conjugate points. These
are the so-called principal planes, one belong ing to the object-space (the primary principal plane) and the other belonging to the image space (the secondary principal plane). The primary and secondary principal points H, H' are the axial points of the principal planes.
Construction of the itnage.—Provided the positions of the cardinal points F, F' and H, H' are given, the image-point Q' conjugate to an extra-axial object-point Q may be found by the following geometrical construction (Fig.4).
Through Q draw the straight line QV' parallel to the axis meeting the secondary prin cipal plane in V', and also the straight line QF meeting the primary principal plane in W. The required point Q' will be at the intersection of the straight line V'F' with the straight line WQ' drawn parallel to the axis. The feet of the perpendiculars let fall from Q, Q' on to the axis will locate also a pair of conjugate axial points M, M'Q will be the image of MO.
Thus, point by point, the image-relief cor responding to a given 3-dimensional object may be constructed; and in this image-relief object planes perpendicular to the axis will be repro duced by similar plane figuresplaced at right angles to the axis, but in general the magnifica tion-ratio (M'Q':MQ) will be different for different pairs of conjugate transversal planes, assuming all possible values positive and nega tive depending on the place of the object.
However, in the special case when the object point is infinitely far away, the construction as given above fails. Suppose, for example, that the object is a distant star seen by the naked eye in the direction of the straight line FI (Fig. 5) which makes an angle u = I HFW with the optical axis of the instrument Its image will be formed at the point I' (conjft'ate to the infinitely distant object-point I) ',Ave the straight line WI' drawn parallel to the axis meets the secondary focal plane.
Focal Lengths.— The focal lengths, denoted by f and r, are defined to be the abstisme IA the principal points with respect to the cope sponding focal points as origins; thus, f'=',FIL f ----- F'H'; and if the indices of the object-space and image-space are denoted by n and n', respectively, the theory showpr that f n I' =— n' -- •This fundamental relation may be expressed by saying that the focal lengths of a centered system of spherical refracting surface* are proportional to the indices of refraction di the first and last media, and are opposite in except in the single case when the optical tern includes an odd number of reflecting f faces, under which circumstances the focal lengths have the same sign (— = + n'In particular, when the media of object space and image-space are identical (rs'==a), the focal lengths are equal in magnitude (f'–• —f, or, in case of an odd number of reflecting surfaces, f g t).