L I G H T 599 In any isotropic body, homogeneous or not, it is clear that rfV = v(adx + &dy + ydz) ; and then we have, to determine Y, the partial differential equation The treatment of this equation is precisely the same as that of the corresponding one which will presently be derived from the undulatory view of the question. We will now illustrate the application of Hamilton s method to the undulatory theory, in which the time of passage from one point of the path to another is the characteristic which fulfils the station ary condition. For the sake of limitation, we will confine ourselves to its application to single refraction in a non-homogeneous medium. In such a medium the velocity of light, at any point, is the same whatever be the direction of the ray. Hence it depends only upon the coordinates of the point, and upon some characteristic (say the wave-length) of the light considered. If T be this time of passage, ds an element of the path, and v the velocity of light in that element, we must have /* a quantity fulfilling the stationary condition. This gives fdSs rdsSv Now, by what has just been said, if A be the wave-length, we have an equation expressing the data of the problem, v=f(, x, y, z), where the form of/ depends on the arrangement of the parts of the medium. Hence dxdSx + dyd.Sij + dzdSz -,- ds dx The unwritten part consists of an integral which, by the station ary condition, vanishes if the ray be of a definite wave-length and the terminal points through which it passes be given, i.e., if 5f, 8y, S~, be each equal to 0. The rest of the expression depends on the terminal points of the ray, and on the wave-length, only. It gives the equations or 1 dx ST_ 1 dij 5r 1 dz Sx v ds Sy v ds 8z v ds and ST f I dv 7 __. / ds S J r- d Squaring and adding the first three, we have It is easily shown, by a process similar to that used for varying action (see MECHANICS), that, if we can find a complete integral of this equation, containing therefore two arbitrary constants, in the form r-(x,y,z,,a,0), then are the equations of two series of surfaces whose intersections give the paths of the rays. and 13 here are also arbitrary constants. (These four constants are necessary, and sufficient, for the purpose of making the two intersecting surfaces pass each through any two given points.)
- As an illustration, let us suppose the light to be homogeneous,
and the medium to be arranged in concentric spherical shells such that the velocity at a distance r from their centre is expressed by where b and c are absolute constants. It is easy to see that, on account of the symmetry, the path of every ray is in a plane through the centre of the spheres. We may therefore restrict our work to the plane of x, y passing through that centre. The equation is then or, by change to polar coordinates, ^Y+I/^ T dr) + r*de What we require is a sufficiently general solution. Assume, therefore, dr and we have From these -*!= A / _ ^ a " dr V (u-+r 2 )- T* The equation of the path is therefore ar , . c 2 *. V (b* + r* r- r a 2 This is the equation of a series of circles, whose one common characteristic is that the rectangle under the segments of any chord which passes through the origin is b 2 . Hence every ray in any diametral plane describes a circle ; and pairs of conjugate foci are situated on a line through the centre, the rectangle under their distances from the centre being i 2 . The property holds therefore for all rays in the medium. This very singular ideal arrangement was suggested to Clerk Maxwell by the eye of a fish. He has given an investigation of it, by a totally different analysis, in the Cambridge and Dublin Mathematical Journal, vol. ix. As an illustration of those effects of want of homogeneity to which (as already stated) all the complex pheno mena of mirage, &c. , are due, it may be well to consider this simple case more closely. We will therefore consider how images are seen in such a medium. To get rid of the difficulty which would arise from finite change of density if an eye were supposed to be plunged in the medium, we will suppose it to be cut across by a crevasse whose surface is everywhere nearly at right angles to the rays by which the image is to be seen, the eye being then placed (in air) close to such a cutting surface. Let AB (fig. 25) be a small object, the centre of the spherical layers of equal refractive index. Then every ray from A describes a circle which passes through A , where j AOA is a straight] line, and AO.OA = 6 2 . A similar construction | gives B from B. To an eye placed] at Ej (in a little cre vasse as before ex-! plained), and looking towards the object, it] will be seen erect, A being seen in the direction of a tangent] to the circle through AEjA , and similarly] for B. Here the rays j have not passed through their con- 1 jugate focus. But if| the eye be now. turned away from _. the object, it (or rather its image) will be seen, A in the direction opposite to that in which A was seen, B in the opposite direction to B. The image will now be an inverted one, but it will easily be seen to possess a strange peculiarity. For what is now seen will be the back of the object, the side farthest from the eye. The reader may easily trace for himself the course of the rays which would fall on the eye in any other assigned position. Vision in such cases would usually be of a peculiar character from another point of view, viz. , the amount of divergence in the plane of the figure will in general differ from that perpendicular to its plane, and therefore the rays would have different divergence for the height and for the breadth of the image. These would therefore appear at different distances from the spectator. This, however, could be cured by a proper cylindrical lens. It is clear from this example (which has been chosen for its special simplicity) that want of
