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linear motion. A ball, suspended by a string, describes in vibrating a curved line, or, in other words, it vibrates in the arc of a circle.
An assemblage of molecules, vibrating rectilinearly, in the same plane, and in all phases of their vibrations, constitutes a plane wave. An assemblage of molecules, vibrating curvilinearly or rotating, the rotation or vibration of every molecule being made in parallel planes, constitutes what may be termed a spiral or heticoidal wave. If the molecule revolve in a circle, the wave is circular; if in an ellipse, the wave is elliptical.
All motion being naturally rectilinear, it follows, that when we see a body moving in a curve of any kind, we conclude that it must be under the influence of at least two forces; one putting it in motion, and another drawing it off from the rectilinear course, which it would otherwise have continued to move in. The cause of these curvilinear movements of the ethereal molecules will be subsequently explained.
The Rev. Professor Baden Powell has contrived an ingenious machine, for showing in what manner rectilinear and curvilinear vibrations produce respectively plane and helicoidal (circular or elliptical) waves. It is founded upon this geometrical construction: a finite line, P Q, moves always through the point C, and with its end P always in the circumference of a given circle, whose centre is A; the end Q will describe a certain curve, which appears upon analysis to be one of a high order, but having in general some soit of oval form, which varies as the distance A C is altered. If A C be very great compared with the radius of the circle, Q will move up and down, almost in a straight line: if A C be somewhat less, its path will resemble an ellipse; if still less, it will be more rounded or resemble a circle.
Upon this the machine is constructed as follows (A and C correspond in both diagrams):
The lower part consists of a stout iron wire bent into a series of cranks, of which the two extremes are in the same position, e. g. downwards; the middle one vertical, and the intermediate ones at intermediate inclinations. Attached to each crank by a hinge or joint, is a long rod, R R', &c., which passes through an aperture in a cross-bar, C C, at the top. The top of this rod is made conspicuous by an ivory ball or a ball painted white, B, the rest of (he apparatus being painted black. The bar C C is attached to the supports A C A' C, by screws, and can be removed (without changing the rods) from D D to the positions D' D', or D' D". The proportions of the machine are not essential, but only that the length of the rods should be great compared with that of the cranks. When the bar is at D D, on turning the handle a plane polarized wave is produced by the balls; when at D' D' an elliptical one; and when at D" D" a circular one—that is, what, for illustration, and to the eye, may be considered so. If the distance A D" be eighteen inches, A D' should be about twenty-four, and A D about thirty-six inches; but these are not material as to exactness.
Mr. E. M. Clarke, philosophical instrument-maker, of the Strand, has constructed this instrument without cranks. The upright rods are attached inferiorly to metallic rings, through each of which runs an axis, A A'; and on this axis the rings are fixed in a spiral or helicoidal manner.
Professor Powell's Machine, ns constructed by Mr, Clarke,
Transversal Vibrations.—I come now to a most important part of the undulatory hypothesis—that by which the phenomena of polarized, as distinguished from those of common or unpolarized, light are accounted for. I refer now to the hypothesis of transversal vibrations, first suggested, I believe, by Dr. Young, but most admirably developed and applied by Fresnel.
"The existence of an alternating motion of some kind, at minute intervals along a ray, is," says Professor Powell,* "as real as the motion of translation by which light is propagated through space. Both must essentially be combined in any correct conception we form of light. That this alternating motion must have reference to certain directions transverse to that of
* A General and Elementary View of the Undulatory Theory, p. 4.
the ray is equally established as a consequence of phenomena
and these two principles must form the basis of any explanation which can be attempted."
In order to understand transversal vibrations, let us first consider how waves of water, and of other liquids, are formed. If a stone be thrown into a pond, there is formed a system or group of waves, which commences at the spot where the stone impinges, and gradually extends outwards in the form of concentric circles. The aqueous particles in the centre are forced down, and the surrounding ones thereby urged upwards above the normal level of the water. In this way the central depression, and the first or innermost circular heap, are formed. But gravity soon causes this heap to subside, and fill up the central depression, while by its downward progress it acquires momentum, and thereby descends below its normal level, thus not only giving rise to a circular depression, but causing the formation of another and outer circular heap by the elevation of the neighbouring particles. In this way the waves gradually extend outwards. It is obvious, then, that in waves of liquids, the directions of vibration of the molecules is vertical, or nearly so, while the propagation of the waves is horizontal.
In a vibrating cord, the vibrations are rectangular to the propagation of the undulations along the cord.
In luminous waves, the direction of vibration is supposed by Fresnel to be transverse to the direction of propagation; and the more recent researches of Cauchy seem to have established the doctrine of transversal vibrations; but he assumes a third vibration, namely, one parallel to the ray, so that, according to him, the motions of the molecules take place in three rectangular axes. The necessity for this third axis of vibration, parallel to the ray, seems to be derived from the phenomena of dispersion.
Now, polarized light, on the wave hypothesis, is light which has only one plane of vibration; whereas common or unpolarized light consists of light having two or more planes of vibrations, of which two must be rectangular—that is, after the molecules have vibrated in one plane, they change their vibration to another plane. So that common light consists in a rapid succession of waves in which the vibrations take place in different planes. It does not, however, appear that the planes of vibration are continually changing; but that in each system of waves, there are probably several hundred successive vibrations, which are all performed in the same plane; although the vibrations of one system bear no relation to those of another. Thus, then, we call that light polarized, in which all the vibrations take place in one plane; but when vibrations are succeeded rapidly by other vibrations in an opposite plane, the two waves though separately called polarized, are together, termed unpolarized or common light; so that, as Fresnel has observed, common light is merely polarized light, having two planes of polarization at right angles to each other.
Thus, then, I have now replied theoretically, as well as practically, to the question, " What is polarized light?"
Partially polarized light consists, according to Sir John Herschel, of two unequally intense portions; one completely polarized, the other not at all. Sir David Brewster, however, regards it as light whose planes of polarization are inclined at angles less than 90°. But to the latter view some objections have been raised by Mr. Lloyd.
In the following diagram, let the straight lines represent the directions in which the ethereal molecules are supposed to vibrate. Then A B and CD will represent the direction of vibration ot the ethereal molecules of two oppositely polarized rays; A B' C D' the two rectangular directions of vibration of a ray of common or unpolarized light; and A" B" C" D" a ray of partially polarized light, according to Sir D. Brewster's hypothesis.
"The difference between a polarized and an ordinary ray of light," says Sir John Herschel, "can hardly be more readily conceived than by assimilating the latter to a cylindrical, and the former to a four-sided prismatic rod, such as a lath or a ruler, or other long, flat, straight stick."
In order to illustrate Fresnel's notion of transversal vibrations, and of the hypothetical difference between common and polarized light, painted card models are very convenient. A piece of cardboard is cut out in a waved or undulated form, so that the curves of the upper and lower edges accord. Then, midway between these edges, a row of circular black spots is painted on the card: these are to represent the ethereal molecules, while the card-board represents the plane of vibration. A single card thus cut and painted serves to illustrate a ray of plane-polarized light (Fig A): two of them placed side by side, with their planes at right angles to each other, B, represent the two oppositely-polarized rays produced by a double refracting prism, while two so placed that they mutually cross, represent common light, C.
We are now prepared to understand how common light becomes polarized. In the case of the doubly refracting bodies the two planes separate, for reasons that will be explained in the next lecture; and as the two waves have the planes of their vibrations at right angles to each other, we see now how the rays are said to be oppositely polarized. As these two waves are propagated with different velocities, they in consequence follow different paths. The tourmaline likewise separates the two planes; but it gradually extinguishes the one, by offering such an impediment to its progress that its vibrations are destroyed. The agency of the reflecting plate in polarizing light may also be readily accounted for. When a ray of common light falls on a transparent surface, at a certain angle, its planes of vibration are resolved into two, one of which is transmitted, the other reflected; both are polarized, but oppositely.
The action of the analyzer or test may also be easily understood. Suppose the analyzer to be a reflecting plate: if this plate be at the same angle to the ray as the polarizing plate, the vibrations will be reflected when the planes of reflexion of the polarizing and analyzing plates coincide—but will be transmitted (that is not reflected) when the planes are at right angles to each other. Suppose the analyzer to be a tourmaline plate: in one position this plate permits the vibrations to be transmitted, but in a position perpendicular to this it destroys them. So that in these two rectangular directions, the crystal of tourmaline must possess unequal elasticities; for the motion or vibration is transmitted in the one, but stifled or destroyed in the other direction. Suppose the analyzer to be a rhombohedron of Iceland spar; in either of two rectangular directions the vibrations of the polarized incident ray are propagated unchanged, but at an angle of 45° to either of these positions, the plane of vibration of the incident ray is resolved into two rectangular planes, each of which forms an angle of 45° with the incident ray.