452 WAVE THEORY dividing edge. The planes of polarization of the two halves of the field are different, unless the original plane be parallel (or perpen dicular) to the principal section. In the Laurent analyser the half- wave plate is rigidly combined with a Nicol in such a position that the principal section of the latter makes a small but finite angle with that of the plate. The consequence is that the two halves of the field of view cannot be blackened simultaneously, but are rendered equally dark when the instrument is so turned that the principal section of the plate is parallel to the plane of original polarization, which is also that of the uncovered half of the field. A slight rotation in either direction darkens one half of the field and brightens the other half. In another form of "half-shade" polarimeter, invented by Poynting, the half-wave plate of the Laurent is dispensed with, a small rotation of one half of the field with respect to the other half being obtained by quartz (cut perpendicularly to the axis) or by syrup. In the simplest construction the syrup is contained in a small cell with parallel glass sides, and the division into two parts is effected by the insertion of a small piece of plate glass about y 3 ^ inch thick, a straight edge of which forms the dividing line. If the syrup be strong, the difference of thickness of r s ff inch gives a relative rotation of about 2. In this arrangement the sugar cell is a fixture, and only the Nicol rotates. The reading of the divided circle corresponds to the mean of the planes for the two halves of the field, and this of course differs from the original position of the plane before entering the sugar, This circumstance is usually of no importance, the object being to determine the rotation of the plane of polarization when some of the conditions are altered. A discussion of the accuracy obtainable in polarimetry will be found in a recent paper by Lippich. x Soleil s In Soleil s apparatus, designed for practical use in the estima- appa- tion of the strength of sugar solutions, the rotation due to the ratus. sugar is compensated by a wedge of quartz. Two wedges, one of right-handed and the other of left-handed quartz, may be fitted together, so that a movement of the combination in either direction increases the thickness of one variety traversed and diminishes that of the other. The linear movement required to compensate the introduction of a tube of syrup measures the quantity of sugar present. 24. Dynamical Theory of Diffraction. The explanation of diffraction phenomena given by Fresnel and his followers is independent of special views as to the nature of the ether, at least in its main features ; for in the absence of a more complete foundation it is impossible to treat rigorously the mode of action of a solid obstacle such as a screen. The full solu tion of problems of this kind is scarcely to be expected. Even in the much simpler case of sound, where we know what we have to deal with, the mathematical difficulties are formidable; and we are not able to solve even such an apparently elementary question as the transmission of sound past a rigid infinitely thin plane screen, bounded by a straight edge, or perforated with a circular aperture. But, without entering upon matters of this kind, we may inquire in what manner a primary wave may be resolved into elementary secondary waves, and in particular as to the law of intensity and polarization in a secondary wave as dependent upon its direction of propagation, and upon the character as regards polarization of the primary wave. This question is treated by Stokes in his " Dynamical Theory of Diffraction " 2 on the basis of the elastic solid theory. Equa- Let x, y, z be the coordinates of any particle of the medium in tions for its natural state, and , y, the displacements of the same particle elastic at the end of time t, measured in the directions of the three axes solid. respectively. Then the first of the equations of motion may be put under the form c(fOc ctsc dy ctz i where a 2 and & 2 denote the two arbitrary constants. Put for short- dy and represent by v 2 the quantity multiplied by I". this notation, the three equations of motion are . (1), According to -n -JTO- = &-VTJ + (a- - Zr) dt- d (2). It is to be observed that 5 denotes the dilatation of volume of the element situated at (x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible 8 vanishes ; but, in order that equations (2) may preserve their generality, we must 1 Wicn. Her., Ixxxv., 9th Feb. 1882. Sec also Phil. Trans., 1885, p. 360. 2 Camb. Phil. Trans., vol. ix. p. 1 ; reprint, vol. ii. p 243. suppose a at the same time to become infinite, and replace a"S by a new function of the coordinates. These equations simplify very much in their application to plane Plane waves. If the ray be parallel to OX, and the direction of vibration waves, parallel to OZ, we have | = 0, rj = 0, while f is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives of which the solution is (3), (4) where /is an arbitrary function. The question as to the law of the secondary waves is thus Stokes s answered by Stokes. "Let | = 0, rj = 0, =f(bt-x) be the dis- law of placements corresponding to the incident light ; let 1 be any point secondai in the plane P (of the wave-front), cZS an element of that plane wave, adjacent to : ; and consider the disturbance due to that portion only of the incident disturbance which passes continually across rfS. Let be any point in the medium situated at a distance from the point 1 which is large in comparison with the length of a wave ; let OjO = r, and let this line make an angle 6 with the direction of propagation of the incident light, or the axis of x, and <$> with the direction of vibration, or axis of z. Then the displacement at O will take place in a direction perpendicular to OjO, and lying in the plane ZOjO ; and, if be the displacement at 0, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive, C = -; (l + cos0)sin</>/ (M-r) . In particular, if we shall have f(U-x) = csin (U-x) (5), A cdS ,_. , . 2ir ,.. , , ._. ,, - - (1 + cos 6) sin < cos (ot - r) . . . (6). 2T A. It is then verified that, after integration with respect to dS, (6) gives the same disturbance as if the primary wave had been sup posed to pass on unbroken. The occurrence of sin <p as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direc tion 6 depend upon the condition of the former as regards polar ization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin (j> = 1 ; but, if the primary vibration be in the plane of diffraction, sin < = cos 0. This result was employed by Stokes as a criterion of the direction of vibration ; and his experiments, con ducted with gratings, led him to the conclusion that the vibrations of polarized light are executed in a direction perpendicular to the plane of polarization. The factor (1 + cos 6) shows in what manner the secondary dis turbance depends upon the direction in which it is propagated with respect to the front of the primary wave. If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which, the wave is supposed to be broken up, we may use simpler methods of resolu tion than that above considered. It appears indeed that the purely mathematical question has no definite answer. In illustra- Analog; tion of this the analogous problem for sound may be referred to. of soun< Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected. The introduction of the lamina (supposed to be devoid of inertia) will make no differ ence to the propagation of plane parallel sonorous waves through the position which it occupies. At every point the motion of the lamina will be the same as would have occurred in its absence, the pressure of the waves impinging from behind being just what is required to generate the waves in front. Now it is evident that the aerial motion in front of the lamina is determined by what happens at the lamina without regard to the cause of the motion there existing. Whether the necessary forces are due to aerial pressures acting on the rear, or to forces directly impressed from without, is a matter of indifference. The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element dS, the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest. The resulting aerial motion in front is readily calculated ; 3 it is symmetrical with respect to the origin, i.e., independent of Q. When the secondary disturbance thus ob- taiued is integrated with respect to dS over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to dS. The only assumption here involved is the evidently legitimate one that, when two systems of variously distri-
3 Theory of Sound, 278.