These laws, combined with Malus's law already mentioned, and with the formula which gives the intensity of the light resulting from the interference of two streams of light of known intensities and difference of path, enable ua to calculate completely the colours of crystalline plates in polarized light without making any assumption as to what constitutes polarization.
Let Cr be the plane of primitive polarization, ea that of analyza• Lion, c o, c E the rectangular planes of polarization of the rays ante which a ray of any kind is divided by the double refraction of tin crystalline plate. Let i be the azimuth of c o, and a that of c A, botl measured from c e. We may without loss of generality suppose i to la between the limits 0° and 90°, and s between the limits i j 90' Further, let o, e, be the lengths of path in air equivalent in time o being described to the paths of the two rays respectively within tilt crystal ; let A be the wave-length in air belonging to any particular kind of light, and take the original intensity of that light as unity.
By 3Ialna' s law the intensities of the two streams, polarized aloni c o, c E, into which the original stream is divided by double refraction will be coal i sing i, respectively. If each of these be again dividec into two, polarized along and perpendicularly to c A, the intensities o: the former portions will by the same law be cos° i cos' (i—a), sing sin' (i—s), respectively. The difference of phase of these portion, w will be — (o—e). Now if 1, be the intensities of two streams of common light from the same source, p their difference of phase, tie intensity of the light resulting from their interference will be + + 'J (I f) cos p. But by the second law of interference of polarized ight this formula may be applied to the interference of two streams of solarized light which are capable of interfering, as by law 4 the two are which are polarized along o A. In the application of ;he formula we must take V r = cos i cos (i—s), J 1' = sin i sin (i—s) sr = sin i sin (s—i), according as i > s or i < s. But by law 5 when < s we must change the actual difference of path by half an undula km, that is, change p by sr, or, which comes to the same, change the igu of one of the radicals V i or V 1'. Hence it will suffice to take V = sin i sin (i—a) in all cases, and omit the addition or subtraction )f the half undulation. Hence the expression for the intensity will
Become cos i sing i sing (i—s) 2sr + 2 cos i sin i cos sin (i — a) cos 7 (o—e) (A) which may be readily transformed into the more simple expression cos"- s — sin 2i sin 2(1—s) sing (o—e) . . (B) The discussion of either of these expressions would give account of the observed phenomena of the coloration of crystalline plates in polarized light, but would exceed our limits. Suffice it to remark that in the expression (a) the first term denotes the illumination, dike for all colours, which would exist if the plate were removed, while the second changes materially from colour to colour, in con sequence of the variation of A, in comparison of which the variation of may usually be neglected.
Now the study of the phenomena of light which are Independent of polarization leads us, and that in different ways, to the conclusion that with light of given wave length the square of the amplitude of vibra tion must be taken as the measure of intensity, and consequently the amplitude of vibration will vary as the square root of the intensity. If now, bearing this in mind, we go over the whole investigation of the colours of crystalline plates, beginning with the first application of Malus's law, and deducing at every step from the intensities which are the objects of direct observation the corresponding amplitudes of vibration, we can hardly fail to be impressed with the idea that in polarized light the vibrations of the ether take place in a rectilinear manner in a direction transverse to that of propagation, and related in some constant manner to the plane of polarization. Unquestionably there must be a transverse something about polarized light which admits of composition and resolution in that way. Now polarized light in all its relations is symmetrical with respect to the plane of polarization and the perpendicular plane, and to no other. For ex ample, when light is polarized by reflection at the surface of glass, everything must evidently be symmetrical with respect to the plane of reflection. Hence our rectilinear vibrations must also he symmetrical with respect to the plane of polarization, and therefore must either be in or perpendicular to that plane.