OPTICS 799 thetical course A BCDE is the same as along A B C D E . Hence 2fis along A BCDE is the same as along ABCDE, or (on subtraction of the common part) the same along A B, DE as along AB, DE. But since AB is perpendicular to AA , the value along A B is the same as along AB, and there fore the value along DE is the same as along DE ; or, since the index is the same, DE = DE , that is, EE is perpendi cular to DE. The same may be proved for every point E which lies infinitely near E, and thus the surface EE is perpendicular to the ray DE, and by similar reasoning to every other ray of the system. It follows that reflexions and refractions cannot deprive a system of rays of the pro perty of being normal to a surface, and it is evident that a system issuing from a point enjoys the property initially. Consecutive rays do not in general intersect one another ; but if we select rays which cut the orthogonal surface along a line of curvature, we meet with ultimate intersec tion, the locus of points thus determined being a caustic curve to which the rays are tangents. Other lines of cur vature of the same set give rise to similar caustic curves, and the locus of these curves is a caustic surface to which every ray of the system is a tangent. By considering the other set of lines of curvature we obtain a second caustic surface. Thus every ray of the system touches two caustic surfaces. In the important case in which the system of rays is symmetrical about an axis, the orthogonal surface is one of revolution. The first set of lines of curvature coincide with meridians. The rays corresponding to any one meridian meet in a caustic curve, and the surface which would be traced out by causing this to revolve about the axis is the first caustic surface. The second set of lines of curvature are the circles of latitude perpendicular to the meridians. The rays which are normal along one of these circles form a cone of revolution, and meet in a point situ ated on the axis of symmetry. The second caustic surface of the general theorem is therefore here represented by a portion of the axis. The character of a limited symmetrical pencil of rays is illustrated in fig. 6, in which BAC is the ortho gonal surface, and HFI the caustic curve having a cusp at F, the so-called geometrical focus. The distance FD between F and the point where the extreme ray BHDG cuts the axis is called the longitudinal aberration. On account of the sym metry FD is an even function of AB. If the pencil be small, we may in general consider FD to be proportional to AB 2 , although in particular cases the aberration may vanish to this order of approximation. Let us examine the nature of the sections at various points as they may be exhibited by holding a piece of paper in the solar rays converging from a common burning-glass of large aperture. In moving the paper towards the focus nothing special is observed up to the position HI, where the caustic surface is first reached. A bright ring is there formed at the margin of the illuminated area, and this gradually contracts. At D the second caustic surface DF is reached, and a bright spot develops itself at the centre. A little farther back, at EG, the area of the illuminated patch i a minimum, and its boundary is called the least circle of aberration. Farther back still the outer boundary corre sponding to the extreme rays begins to enlarge, although Fi the circle of intersection with the caustic surface continues to contract. Beyond F the caustic surfaces are passed, and no part of the area is specially illuminated. As a simple example of a symmetrical system let us take the case of parallel rays QR, OA (fig. 7), incident upon a spherical mirror AR. By the law of reflexion the angle OR<? = angle ORQ = angle ?OR. Hence the triangle R<?0 is isosceles, and if we denote the radius of the surface OAvl by r, and the angle AOR by we have Q If F be the geometrical focus, ^ = AF=$r. If a be a - small angle, the longitudinal aberration q = Oq - OF r (sec a-I) = ^a-r, in which AR = ra. Focal Lines. In the general case of a small pencil of rays there is no one point which can be called the geo metrical focus. Consider the corresponding small area of the orthogonal surface and its two sets of lines of curvature. Of all the rays which are contiguous to the central ray there are only two which intersect it, and these will in general intersect it at different points. These points may be regarded as foci, but it is in a less perfect sense than in the case of symmetrical pencils. Even if we limit ourselves to rays in one of the principal planes, the aberration is in general a quantity of the first order in the angle of the pencil, and not, as before, a quantity of the second order. If, however, we neglect this aberration and group the rays in succession according to the two sets of lines of curvature, we see that the pencil of rays passes through two focal lines perpendicular to one another and to the central ray, and situated at the centres of curvature of the orthogonal surface. At some intermediate place the section of the pencil is circular. It happens not unfrequently that the pencil under con sideration forms part of a symmetrical system, but is limited in such a manner that the central ray of the pencil does not coincide with the axis of the system. The plane of the meridian of the orthogonal surface is called the primary plane, and the corresponding focus, situated on the caustic surface, the primary focus. The secondary focus is on the axis of symmetry through which every ray passes. The distinction of primary and secondary is also employed when the system, though not of revolution, is symmetrical with respect to a plane passing through the central ray, this plane being considered primary. The formation of focal lines is well shown experimentally by a plano-convex lens of plate-glass held at an obliquity of 20 or 30 in the path of the nearly parallel rays, which diverge from a small image of the sun formed by a lens of short focus. The convex face of the lens is to be turned towards the parallel rays, and a piece of red glass may be interposed to mitigate the effects of chromatic dispersion. To find the position of the focal lines of a small pencil incident obliquely upon a plane refracting surface of index /*. The complete system of rays issuing from Q (tig. 8 and refracted at the plane surface CA is symmetrical about the line QC drawn through Q perpendicularly to the surface. Hence, if QA be t central ray of the pencil, the secondary focus q., lies at the inter section of the refracted ray with the axis. If < be the angle of incidence, <p of refraction, A.Q = u, Aq = Vy then .(1). To find the position of the primary focus q v let QA be a neigh- bou in- ray in the primary plane (that of the paper) with angles of incidence and refraction <f> + 6<t> and </> + 5tf> , Aq l = v v We have AA cos <f> = u5<j>, AA cos $ = v^Q ; moreover, by the law of refraction, cos (f> 8<f> fj. cos 5(/> ;
