Mit when light diverging from a point fails on a surface, after reflec tion it generally does not again converge to a point, or diverge from one accurately, nor does even an infinitesimal pencil iu general after reflection pass through a focal point, but only through two infinitesimal focal lines, lying in two rectangular planes passing through the pencil. These lines lie respectively on two sheets of a eurface, which may be called the caustic surface, which is touched in each sheet by every ray of the system. The consideration of caustic surfaces ia, however, ordinarily restricted to the case of a system of rays symmetrical about an axis, in which case one sheet merges in a line along the axis, and the other, to which the term caustic is ordinarily restricted become. a surface of revolution, generated by the revolution of the curve of ultimate intersection of the rays which lie in any plane passing through the axis. The equations and properties of caustics are, however, rather objects of analytical exercise than of any practical use. [Orrice.] In the case of reflection, the light is returned to the medium In which it moved previous to incidence ; hut when a ray of light is incident on a transparent medium of greater density than that of the medium in which it originally moved, a portion of it is reflected, but another portion enters the medium, and then proceeds generally in one straight course in the Platte of incidence, but not in the original direction, hawing a deviation in course, though not in plane, and some times as in certain crystallised media, it splits into two rays, one in the plane of incidence as before, the other in a plane determined by the nature of time crystal, while in other crystallised media it splits into two rays, of which neither lies in the plane of incidence. The same phenomena take place when light passes from a dense to a rarer medium, except that in this case the whole of the light may under a certain incidence be totally reflected.
This alteration of the path of light passing from one medium to another, which is familiarly observed in the apparently bent form of a straight stick partially immersed in water in an oblique direction, is called refraction. As double refraction is of comparatively rare occur rence among the instances of refraction which ordinarily fall under our notice, we shall first attend to the laws of single refraction.
Let e r C represent a solar beam in vacuo and incident at e on a transparent medium (as water), to the surface of which n o a is normal. When the medium is fluid, place a graduated circle D s a in the plane of incidence with its centre at C; a portion of the light will be reflected in the direction C L, and another entering the medium will be refracted in the direction c n. If uninfluenced by the medium, its direction would have been c s. The angle nor is the angle of refraction, or a C a of incidence, and s C rt of deviation. The arcs D r, D L, are equal by the law of reflection, and If we compare the arcs D a D, their sines will be found in a constant ratio, depending on the nature of the medium, but independent of the angle of incidence. Thus if be the angle of incidence, and it that of refraction, the two are con nected by the simple relation sin 1-=a sin it. The constant a peculiar to the medium is called its index of refraction. When the medium is solid, we can easily compare the tangents of the angles, and thence their sines. The above law will be found rigorously exact.
This law may be accounted for on the theory of emission. Let v be the velocity of the ray before incidence, which is decomposable into a horizontal velocity, v sin 1, and a normal one, v cos a. The
former will not be affected by the medium ; the square of the latter • will be increased at the confines of the medium by a quantity which is the sum of the products of twice the force into the element of the normal throughout that inappreciable space in which the forces of the medium do not destroy each other, in consequence of proximity to the urface. Therefore the normal velocity of the refracted ray is V 14- and its actual velocity J 0 + ; so that the horizontal velocity in the medium is V sin which being equated with v sin a, its value before incidence gives sin t=i.t sin n, V where /A= v How are we to account for the reflected ray o 1.1 Why is not the whole incident light refracted ? Even when the incident light is per pendicular to the refracting surface, a portion of the light is reflected ; and when the ray has but a very small inclination to the surface, a portion will yet be intromittecl. Hence we may consider generally that the incident light consists of portions which are differently dis posed to be subject to the repulsive and attractive forces of the medium, or, in language, are in fits of easy rejtexion or transmission. When the angle of incidence increases, the normal velocity of the ray diminishes, the effect of the repulsive forces ie therefore augmented, or the reflexion is more copious.
If r, r' be any portions of the incident and refracted rays measured to fixed points in their directions, and v, the corresponding dr velocities, and we make A c a the axis of x, we have sin a = del sin it — 71:, and since v sin sin it, therefore (v r + dr). 0 ; and v r+ minimum, which result is agreeable to the dynamical principle of least action.
On the undulatory theory each portion of an incident wave, as it crrives at the reflecting and refracting surface, is conceived to be the centre of a disturbance which spreads each medium with the velocity of propagation appropriate to that medium. Thus, setting aside the incident waves, the disturbance in the two media is conceived of as the aggregate of the disturbances due to these various elementary or secondary waves, a anode of conception the legitimacy of which rests directly on the general dynamical principle of the superposition of small motions. In this way the laws of reflection pnd refraction are simply deduced [UNDULATORY THEORY OF LimiT], and with respect to refraction, it is shown that the ratio of the sine of incidence to the sine of refraction is that of the velocity of light in the first medium to its velocity in the second. , Hence, according to this theory, light must travel more swiftly in vacuo than in a refracting medium in the ratio of a to 1, while, according to the theory „of emission, it must travel more slowly in the inverse ratio. This leads to a crucial experiment for deciding between the two theories, which is decisively against the theory of emission. [UNDULATORY Tlizonv.] The fact that the differently coloured rays have different refractive indices has been thought by some to offer a great difficulty to the undulatory theory, inasmuch as the velocity of propagation within refracting media must depend on the periodic time of the disturbance, which is contrary to the laws of elastic fluids. The circumstances are, however, different, as the fluid in this case envelops the material particles of the medium, instead of constituting one uninterrupted mass. Besides, the fact of dispersion has been accounted for on this theory, and even its law (in the case of substances of low dispersive power) deduced, by adopting certain dynamical hypotheses not im probable in themselves.