incident rays is turned or made to rotate either to the right or left, according as the quartz plate is either right-handed or lefthanded. This rotation of the plane of polarization of the incident ray is proportional to the thickness of the plate. The rings produced by thin plates are broader and less numerous than those produced by thick plates. If two plates be superposed the effect is, very nearly, the same as that produced by a single plate whose thickness is either the sum or the difference of the thicknesses of the two plates; according as they are of the same kind (that is, both either right-handed or left-handed), or of opposite kinds (that is, one right-handed, the other left-handed). Thus, if the rotation of the red rays, effected by a plate of quartz of th of an inch thick be equal to 1730, that produced by two superposed plates of equal thickness, taken from the same crystal, will be 2x 17° 35°. On the other hand, if we combine a plate of right-handed quartz of th of an inch thick with a plate of left-handed quartz ofths of an inch thick, the same effects are produced as if we had employed a left-handed plate ofths of an inch thick. When the thicknesses of the two dissimilar plates are equal " the plates of course destroy each other's effects, and the system of rings with the black cross will be distinctly seen." (Brewster).

The rotation of the plane of polarization increases with the refrangibility of the rays. Thus it is greater with violet than with blue, with blue than yellow, and with yellow than red.

I come now to the explanation which the wave hypothesis offers of these phenomena.

When the light, rectilinearly polarized by the first tourmaline plate, is incident on the quartz plate, it suffers double refraction. To prove this, Fresnel contrived a combination of a right-handed prism, and two halves of a left-handed one, by which he doubled the separation of the two rays, and in this way managed to demonstrate the actual existence of double refraction in the principal or prismatic axis of quartz. This is a most remarkable fact. In the principal or prismatic axis of every other known crystal of the rhombohedric system double refraction does not

exist.

Fio. 50.

following order-red, violet, indigo, blue, green, yellow, orange, red again, and so on. So that to obtain the same order of succession, the analyser must be turned in the one case right-handed, or as we screw up, in the other left-handed, or as we unscrew. This will be rendered more obvious by the following diagrams:

Right-handed.

FIG. 52.

Red.

Left-handed.
Red.

In each of these diagrams the arrow shows the direction in which the analyzer is to be rotated, in order to obtain the spectral tints in the descending order. In one complete revolution of the analyzer each of the colours of the spectrum occurs twice. In other words, all the colours are seen in one semi-revolution of the analyzer.

Hence those specimens of quartz which present the colours in the descending order by a right-handed rotation of the analyzer, are denominated dextrogyrate, or right-handed quartz; while those which present them by a left-handed rotation are called lævogyrate, or left-handed quartz.

FIG. 53.

Between these two varieties there has been discovered by Sir John Herschel another difference. In that form of quartz, termed by Haüy plagiedral (from Tλáyios oblique, and edpa a base), it has been found that when the unsymmetrical or plagiedral faces (fig. 53 x) lean to the right, the polarization is righthanded, and vice versa, when they lean to the left the polarization is left-handed. So that the Plagiedral Quartz. cause, whatever it may be, which determines the optical phenomena is also connected with the production of the plagiedral faces.

If, instead of using white light in our experiments, we employ homogeneous light, we find that the plane of polarization of the

incident rays is turned or mao 1. TOISE left, according as the quart: piat is eve handed. This rotation of tax plan: C dent ray is proportiona, u the CKREST ( produced by thi plates art brUAGE those produced by thick plates ! zm the effect is, very neary. the saint a 1212 plate whose thickness is ete u thicknesses of the two plates

kind (that is, voti ele D--LKIMAT opposite kinas (tha: O TL-EUE_ Thus, if the rotation of the re: TE

of th of an art tuck equi two superposed piates of equa

crystal, will be 2> 17g=25+ G. We owe a plate of right-nauer qua

(

plate of left-hander quart elke &
effects are producer a
ofths of ar ier turer.
similar plates art equa

effects, and the system o ng WIL

tinctly seen."

Brewster,

But the two rays thus obtained differ in their properties from those produced by Iceland spar and other doubly refracting crystals, for while the latter are rectilinearly polarized, those of quartz are circularly polarized. Now every circularly polarized ray is equal to two rectilinearly polarized waves, differing in their progress an odd number of undulations. It follows therefore, that the two circularly polarized waves are equal to four rectilinearly polarized waves. Hence then to explain the phenomena, we must assume that the rectilinearly polarized ray (which I shall call R) incident on the quartz, is resolved into two others (A and B) of equal intensity, the one (A) polarized in a plane 45° inclined to the right, the other (B) 45° inclined to the left of the plane of polarization of the primitive ray (R). Let us further conceive that each of the two rays (A and B) is resolved into two other rays, namely A into Aa and Ab, and B into Ba and Bb. Aa and Ab are polarized in one plane, viz., + 45o, while Ba and Bb are polarized in another plane, viz., 45°. Aa and Ba have each their phases advanced, or + undulation, while Ab and Bb have each their phases retarded, or undulation *. Now if we suppose these four rays to be combined two and two in a cross order, we shall have resulting two circularly polarized rays, one right-handed, the other left-handed. Thus Aa and Bb combine to form a left-handed ray, while Ab and Ba form a right-handed one. For when the advanced system of waves has its plane of polarization to the right of that of the retarded system, the ethereal molecules rotate from right to left; whereas they rotate from left to right when the first plane is to the left of the second.

These two circularly polarized rays are propagated along the axis of quartz with unequal velocities. In right-handed quartz, the right-handed ray is transmitted with greater velocity, in lefthanded quartz with lesser velocity than the left-handed ray; and thus at their emergence one is in advance of the other. If the surface of egress or ingress be oblique to the axis, the two circularly polarized rays will emerge in different directions; but if it be perpendicular (as in the experiment under examination) they will emerge superposed, and will compound a single ray polarized

"It results from the laws of interference," says Fresnel, "that a system of waves, polarized rectilinearly, may be replaced by two others, polarized at right angles to each other, and coinciding in their route; and that for each of these we may substitute two other systems of waves having the same plane of polarization, but the one advanced, the other retarded 4th of an undulation; and thus separated 4th of an undulation. In this way are obtained four systems of waves of equal intensity, of which two, polarized at right angles to each other, are th of an undulation behind the two others polarized in the same planes."

in a single plane. Now this plane is removed from the plane of primitive polarization by an angle proportional to the interval of retardation (therefore, proportional to the thickness of the crystal) and to the refrangibility of the ray.

Thus then the differently coloured rays emerge from the quartz plate polarized in different planes; hence, by rotating the analyzer, they are successively transmitted and brought into view.

The following diagram may, perhaps, serve to render more intelligible the explanations of the action of a plate of quartz, of one millimetre (0.03937 of an English inch) in thickness, on the incident rectilinearly polarized red light.

A succession of quartz prisms does not give a further multiplication of images-a circumstance which distinguishes the double refraction of quartz from that of Iceland spar and other crystals.

The above explanation is applicable only when the direction of the rays coincides with the principal axis of the crystal. When it is inclined to this axis, Mr. Airy has shown that the two resulting rays are elliptically polarized, the elliptical vibrations in the two rays being in opposite directions (that is, one righthanded, the other left-handed), and the greater axis of the ellipse is for the extraordinary ray, in the principal plane of the crystal, and, for the ordinary ray, in a plane perpendicular to the principal one. The ratio of the axes in these ellipses, varies with the inclination of the ray to the principal axis of the crystal. When the direction of the ray coincides with this axis, the ratio is one of equality, and the ellipses become circles. But when the ray is inclined to the axis, the ratio increases indefinitely with the inclination. "It is also necessary to suppose that the axis of revolution of the spheroid (prolate for quartz) in which the extraordinary ray is supposed to diverge, is less than the radius of the sphere into which the ordinary wave diverges."

Hitherto we have had no satisfactory theory of the cause of the unequal velocities with which the two rays are transmitted along the principal axis of quartz. We conceive that it must depend either on some peculiarity in the molecules themselves,