For a reason which will become apparent later it is necessary for the reflecting and refracting surfaces used in optical instru ments to approach very closely to ideal geometrical forms. The manufacturing processes by which the necessary degree of perfection can be reached im pose severe limitations on the types of surface which may be employed, and in practice any surface but a portion of a sphere —with the plane as a special case—is rarely employed. We will therefore consider the refrac tion of light at a spherical surface.
In fig. 2 let a ray passing through the point P be refracted at Q, a point on a spherical surface whose centre is at C. The refracted ray lies in the plane PQC containing the incident ray PQ and the normal QC, and it will therefore in general meet PC at some point P'. Let PC meet the surface in R and make an angle a with QC, and let 4 and 41 be the angles of incidence and refraction. Then where r is the radius of the surface. It follows from this expres sion that all rays which, before refraction in the neighbourhood of R, pass through P, will afterwards pass through P'. Physically this means that light energy diverging from a particle of matter placed at P will converge to P' or alternatively will diverge in the new medium as though it were liberated at P'. The reunion of the rays at P' is thus of the greatest significance, and P' is called the image of the object P. If P' is so situated that the rays can actually pass through it the image is called real, but if it is so placed that they may merely be regarded as having originated there the image is called virtual. It should be observed that there is no need for the rays to have actually passed through the point P, that is to say we may deal with virtual objects as well as vir tual images.
Consider now a succession of spherical surfaces which are all met by rays under the conditions just described. Corresponding to an object point P, real or virtual, the first surface forms an image at a definite point The point may be regarded as a source of rays falling upon the second surface, which forms an image of Each surface in turn forms a point image of that due to the preceding surfaces, and we conclude that the whole series of surfaces will form at a definite point P' in the final medium, an image, either real or virtual, of an arbitrary point P in the object space. The relation connecting P and P' may be shown to be unique and reversible, so that it is a matter of con vention which of the spaces external to the system is regarded as the object space and which as the image space. It will be ob
served that we have not assumed axial symmetry in the system, so that this conclusion holds whether the centres of curvature of the various refracting surfaces are collinear or not.
The theory of the symmetrical instrument has been treated very comprehensively by Maxwell and later by Abbe on the as sumption that this two dimensional point to point correspondence holds. From symmetry it is clear that the image of each point on the axis is itself a point on the axis. Thus the axis is a self conjugate ray for the system, that is to say the axis, regarded as a whole is its own image. Corresponding to the point at infinity on the axis in the object space there corresponds a point F' (see fig. 3), usually at a finite distance, in the image space. This is named the second principal focus of the system. Then all rays which in the object space are parallel to the axis will be refracted so as to pass through F' in the image space, and conversely all rays in the image space which pass through F' correspond to rays which are parallel to axis in the object space. Similarly there is a point F on the axis in the object space such that all rays passing through F emerge in the image space as rays parallel to the axis. This point is called the first principal focus of the system. Since the incident portion of any ray refracted parallel to the axis lies in the same axial plane as the emergent portion, the two will meet if produced in some point H. The point thus determined on the incident ray is at the same distance from the axis as the whole of the emergent portion of the ray, and the height of the image of an object extending from H to the axis is equal to the height of the object itself, a fact usually expressed by saying that the transverse magnification is 1. The locus of points H determined in this manner is therefore called the first unit surface. It is to be considered as situated entirely in the object space.