The case of two quartz plates cut at an angle of 67° 30' to the axis, whereof one is free to revolve in front of the other, gives rise to some interesting transformations of the secondary figures. When the principal planes of the crystals are coincident, the field generally shows rings double in number to those due to one plate. But towards the side away from which the axes are directed the rays are more nearly parallel, while towards the opposite side the rays are more nearly perpendicular to the axis. Hence, towards the first side, the rings will approximate more nearly to circles, and towards the other they will show indications of becoming branches of hyperbolas. As the principal planes are turned round in opposite directions from their initial position, the secondary figures begin to appear. At 45° discontinuous bands with hyperbolic curvature towards the part of the field most distant from the ring-centres are seen. At 90° these bands become continuous and sharply defined, while towards the ring-centres a portion of the ellipses may be observed entering the field. Beyond 90° the hyperbolic branches leave the field, and the rectilinear part of the bands is replaced by the diaper pattern described in a former part of this lecture; while the ellipses are gradually elongated to parallel straight lines, and then are converted into hyperbolas. At 180° the hyperbolas occupy the

centre of the field.

Many similar experiments may be made with biaxal crystals; but it would exceed our present limits to describe them here.

CHAPTER IX.

COMPOSITION OF COLOURS BY POLARISED LIGHT

THE results of combining two or more colours of the spectrum have been studied by Helmholtz, ClerkMaxwell, Lord Rayleigh, and others. And the combinations have been effected sometimes by causing two spectra at right angles to one another to overlap, and sometimes by bringing images of various parts of a spectrum simultaneously upon the retina. Latterly also W. von Bezold has successfully applied the method of binocular combination to the same problem (Poggendorff, Jubelband, p. 585). Some effects, approximating more or less to these, may be produced by chromatic polarisation.

Complementary Colours.-First, as regards complementary colours. If we use a Nicol's prism N as polariser, a plate of quartz Q cut perpendicularly to the axis, and a double-image prism P as analyser, we shall, as is well known, obtain two images whose colours are complementary. If we analyse these images with a

Extracted from the Proceedings of the Royal Society, No. 153, 1874. See also Proceedings of the Royal Institution, 1874.

prism, we shall find, when the quartz is of suitable thickness, that each spectrum contains a dark band, indicating the extinction if a certain narrow portion of its length. These bands will simultaneously shift their position when the Nicol N is turned round. Now, since the colours remaining in each spectrum are complementary to those in the other, and the portion of the spectrum extinguished in each is complementary to that which remains, it follows that the portion extinguished in one spectrum is complementary to that extinguished in the other. And in order to determine what portion of the spectrum is complementary to the portion suppressed by a band in any position we please, we have only to turn the Nicol N until the band in one spectrum occupies the position in question, and then to observe the position of the band in the other spectrum. The combinations considered in former experiments are those of simple colours; the present combinations are those of mixed tints, viz. of the parts of the spectrum suppressed in the bands. the mixture consists of a prevailing colour corresponding to the centre of the band, together with a slight admixture of the spectral colours immediately adjacent to it on each side.

But

The following results given by Helmholtz may be approximately verified :—

When in one spectrum the band enters the green, in the other a band will be seen on the outer margin of the red, and a second at the opposite end of the violet; showing that to the green there does not correspond one complementary colour, but a mixture of violet and red, i.e. a reddish purple.

Combination of two Colours.-Next, as to the combination of two parts of the spectrum, or of the tints which represent those parts. If, in addition to the apparatus described above, we use a second quartz plate Q, and a second double-image prism P1, we shall form four images-say O O, O E, EO, E E. And if A, A' be the complementary tints extinguished by the first combination Q P alone, and B, B' those extinguished by the second Q, P, alone, then it will be found that the following pairs of tints are extinguished in the various images.

It is to be noticed that in the image O E the.combination Q, P, has extinguished the tint B' instead of B, because the vibrations in the image E were perpendicular to those in the image o formed by the combination Q P. A similar remark applies to the image

E E.

The total number of tints which can be produced

by this double combination Q P, Q, P1 is as follows:

4 single images
6 overlaps of two

4 overlaps of three

I overlap of four

Total 15

Collateral Combinations.-- The tints extinguished in the overlap 00 + EO will be B, A, B', A; but since B and B' are complementary, their suppression will not affect the resulting tint except as to intensity, and the overlap will be effectively deprived of A alone; in other words, it will be of the same tint as the image O would be if the combination Q, P, were removed. Similarly, the overlap OE + EE will be deprived effectually of A' alone; in other words, it will be of the same tint as E, if Q, P, were removed. If, therefore, the Nicol N be turned round, these two overlaps will behave in respect of colour exactly as did the images O and E, when Q P was alone used. We may, in fact, form a table thus :

Image. 00+ EO OE+ EE

Colours extinguished.

B+ A + B + A = B + B′ + A = A
B' + A + B + A' = B + B + A′ = A'

And since the tints B, B' have disappeared from each of these formulæ, it follows that the second analyser P may be turned round in any direction without altering the tints of the overlaps in question.