445 where . (12), (13). These series are convenient when y is less than z. The second set of expressions are 2 . z- sin l>y cos |y ,, / . (H), (15), where z $ These series are suitable when z/y is small. When the primary wave is complete, r = <x, and we have at once from the second set of expressions x K 2/c ^ K IK. so that T2_ oo " ro co _ * (19) as we know it should be. In the application to the problem of the shadow of a circular disk the limits of integration are from r to co . If these integrals be denoted by C , S , we have and (23) - When the point where the illumination is required is situated upon the axis, I, z are zero. Hence V = 1,V 1 = 0, and P _ the same as if the primary wave had come on unbroken. This is Poisson s theorem, already found ( 10) by a much simpler method, in which attention is limited from the first to points upon the axis. The distribution of light at other points upon the screen is to be found from (23) by means of the series (16), (17) for V and Vj. Lommel gives curves for the intensity when y = Tr, 2ir, 3w, . . . 6ir. The bright central spot is accompanied by rings of vary ing intensity. The limit of the geometrical shadow [/(a + &) = r/] corresponds to y = z. In this case . . (24), Vi-J^-Js^+W- ...=4sin ..... (25). ondition The numbers computed for special values of y and z apply to " mathe- a whole class of problems. Since <atical ^^L+k > _ 2?r C milarity. V = ~ ^ b ~ " r ~> < - ^ j r , both y and z remain unchanged, even when A is constant, if we suppose 6 oca, rx^x^/a ..... (26). We may fall back upon Fraunhofer s phenomena by supposing a = & = o 0) or more generally b= -a, so that y=0- Under these circumstances But it is unnecessary to add anything further under this head. 19. Polarization. A ray of ordinary light is symmetrical with respect to the direc tion of propagation. If, for example, this direction be vertical, there is nothing that can be said concerning the north and south sides of the ray that is not equally true concerning the cast and west sides. In polarized light this symmetry is lost. Huygens showed that when a ray of such light falls upon a crystal of Iceland spar, which is made to revolve about the ray as an axis, the pheno mena vary in a manner not to be represented as a mere revolution with the spar. In Newton s language, the ray itself has sides, or is polarized. Malus discovered that ordinary light may be polarized by reflexion Brew- as well as by double refraction ; and Brewster proved that the ster s effect is nearly complete when the tangent of the angle of incidence law. is equal to the refractive index, or (which comes to the same) when the reflected and refracted rays are perpendicular to one another. The light thus obtained is said to be polarized in the plane of re flexion. Reciprocally, the character of a polarized ray may be revealed by submitting it to the test of reflexion at the appropriate angle. As the normal to the reflecting surface revolves (in a cone) about the ray, there are two azimuths of the plane of incidence, distant 180, at which the reflexion is a maximum, and two others, distant 90 from the former, at which the reflexion (nearly) vanishes. In the latter case the plane of incidence is perpendicular to that in which the light must be supposed to have been reflected in order to acquire its polarization. The full statement of the law of double refraction is somewhat Double complicated, and scarcely to be made intelligible except in terms refrac- of the wave theory ; but, in order merely to show the relation of tioii. double refraction in a uniaxal crystal, such as Iceland spar, to polarized light, we may take the case of a prism so cut that the refracting edge is parallel to the optic axis. By traversing such a prism, in a plane perpendicular to the edge, a ray of ordinary light is divided into two, of equal intensity, each of which is re fracted according to the ordinary law of Snell. Whatever may be the angle and setting of the prism, the phenomenon may be repre sented by supposing half the light to be refracted with one index (1 - 65), and the other half with the different index (1 48). The rays thus arising are polarized, the one more refracted in the plane of refraction, and the other in the perpendicular plane. If these rays are now allowed to fall upon a second similar prism, held so that its edge is parallel to that of the first prism, there is no further duplication. The ray first refracted with index 1 65 is refracted again in like manner, and similarly the ray first refracted with index 1 48 is again so refracted. But the case is altered if the second prism be caused to rotate about the incident ray. If the rotation be through an angle of 90, each ray is indeed refracted singly ; but the indices are exchanged. The ray that suffered most refraction at the first prism now suffers least at the second, and vice versa. At intermediate rotations the double refraction reasserts itself, each ray being divided into two, refracted with the above- mentioned indices, and of intensity dependent upon the amount of rotation, but always such that no light is lost (or gained) on the whole by the separation. The law governing the intensity was formulated by Malus, and Law of has been verified by the measures of Arago and other workers. If Mains. 6 be the angle of rotation from the position in which one of the rays is at a maximum, while the other vanishes, the intensities are proportional to eos 2 and sin 2 0. On the same scale, if we neglect the loss by reflexion and absorption, the intensity of the incident light is represented by unity. A similar law applies to the intensity with which a polarized ray is reflected from a glass surface at the Brewsterian angle. If 6 be reckoned from the azimuth of maximum reflexion, the intensity at other angles may be represented by cos 2 0, vanishing when 0-90. The phenomena here briefly sketched force upon us the view Traus- that the vibrations of light are transverse to the direction of pro- verse pagation. In ordinary light the vibrations are as much in one vibra- transverse direction as in another ; and when such light falls upon tioiis. a doubly refracting, or reflecting, medium, the vibrations are resolved into two definite directions, constituting two rays polarized in perpendicular planes, and differently influenced by the medium. In this case the two rays are necessarily of equal intensity. Consider, for example, the application of this idea to the reflexion of a ray of ordinary light at the Brewsterian, or polarizing, angle. The incident light may be resolved into two, of equal intensity, and polarized respectively in and perpendicular to the plane of incidence. Now we know that a ray polarized in the plane per pendicular to that of incidence will not be reflected, will in fact be entirely transmitted; and the necessary consequence is that all the light reflected at this angle will be polarized in the plane of inci dence. The operation of the plate is thus purely selective, the polarized component, which is missing in the reflected light, being represented in undue proportion in the transmitted Ijght. If the incident light be polarized, suppose at an angle with the plane of incidence, the incident vibration may be resolved into cosfl in the one plane and sin(? in the other. The latter polarized component is not reflected. The reflected light is thus in all cases polarized in the plane of reflexion ; and its intensity, proportional to the square of the vibration, is represented by hcos"6, if h be the intensity in which light is reflected when polarized in the plane of reflexion. The law of Malus is thus a necessary consequence of the principle of resolution. The idea of transverse vibrations was admitted with reluctance, Elastic even by Young and Fresnel themselves. A perfect fluid, such as solid
the ethereal medium was then supposed to be, is essentially incap- theory.