transmitted [back from] the lower surface by the extraordinary refraction, [not to it, as Mr. Biot's words imply,] has acquired a contrary character, and when it arrives at the black glass, it is partially reflected. On the other hand, a black glass, of which the plane of incidence coincides with that of the plate, reflects the complimentary tint, afforded by the light which had been reflected by the lower surface of the plate, and transmitted back by the ordinary refraction, but exhibits the colour more faintly, because it is mixed with the whole light reflected from the upper surface. A similar arrangement may also be very conveniently applied to the observation of the colours of natural bodies, independently of the glare occasioned by their superficial reflection.

The colours dependent on the extraordinary refraction Mr. Biot found to agree exactly with the colours of thin plates of glass as seen by reflection, and those which are derived from the ordinary refraction with the colours seen by transmission in the Newtonian experiments, supposing the thickness of the plate to be reduced in the ratio of 360 to 1; this ratio being constant for the same specimen of the talc, although the number varied in different specimens from 333 to 395. For mica, it appeared to be 450, but was liable to still greater variation: for rock crystal, it was exactly 360, at least in several plates cut out of the same piece. The measurements of the thickness of the plates were executed with the greatest care by Mr. Cauchoix's spherometer, which appears to be capable of great precision, although the pres'sure exerted by a fine screw, which is the immediate instrument of examination, must be a cause of considerable uncertainty, where the objects to be measured are extremely minute.

Mr. Biot observed, that when the axis of the crystal approached to the plane of incidence, the colours ascended in the scale of Newton's measures, as if the thickness were diminished; and that they descended when the plate was turned in a contrary direction. The difference thus produced appeared to be greater in plates of rock crystal and of mica than in those of talc; but the comparative measures have not been detailed; and it may be remarked, that the greater thickness of the plates of rock crystal employed may possibly have made the difference more apparent. When the axis made an angle of 45° with the plane of incidence, the change of the inclination of the incident light had no effect on the colour exhibited either by tale or by rock crystal: but mica, probably from the oblique situation of the axis of refraction, did not observe the same law. Mr. Biot has expressed the thickness corresponding to the tint, exhibited under these different circumstances, by the formula 1 +(.065 ç2 H--.195ç2H)s2; while in another series of experiments the coefficients appeared to be .00959 and .1428; H

being the angle formed by the axis with the plane of incidence, and s the sine of the angle of incidence: so that the greatest possible variation must have been from .87 to 1.26, or from .867 to 1.152. Mr. Biot has also improved Mr. Malus's expressions for the intensity of the light under different circumstances; but as the colour is wholly independent of the intensity, we omit to mention these expressions more particularly.

This intricate and laborious investigation appears to have been conducted with much patience, and with minute attention to the strictest accuracy; nor does the present memoir by any means exhaust the whole of the experiments which Mr. Biot has promised to the public. Dr. Brewster has remarked that he has the undivided merit of having generalised the facts,' and of having 'discovered the law of these remarkable phenomena.' This 'law' however is merely an expression of the facts considered as insulated from all others; and not an explanatio by which they are reduced to an analogy with any more extensive class of phenomena; and we have no doubt that the surprise of these gentlemen will be as great as our own satisfaction, in finding that they are perfectly reducible, like all other cases of recurrent colours, to the general laws of the interference of light, which have been established in this country, and of which we have given an account in our sixth number (p. 475.); and that all their apparent intricacies and capricious variations are only the necessary consequences of the simplest application of these laws. They are, in fact, merely varieties of the colours of mixed plates,' in which the appearances are found to resemble the colours of simple thin plates, when the thickness is increased in the same proportion, as the difference of the refractive densities is less than twice the whole density: the colours exhibited by direct transmission,' corresponding to the colours of thin plates seen by reflection, and to the extraordinary refraction of the crystalline substances, and the colours of mixed plates exhibited by indirect light' to the colours transmitted through common thin plates, and to those produced by the ordinary refraction of the polarising substances. The measures, which Mr. Biot has obtained, differ much less from the calculation derived from these principles only, than they differ among themselves; and we cannot help thinking such a coincidence sufficient to remove all doubts, if any existed, of the universality of the law on which that calculation is founded; notwithstanding the difficulty of explaining the production of the different series of colours by the different refractions. (See our No. XVII. p. 124.)

In the first place, it appears from Mr. Malus's experiments, that the extraordinary and ordinary refractive densities of the rock crystal, in a plane perpendicular to the axis, are in the ratio of 159 to

VOL. XI. NO. XXI.

D

160; consequently the difference of the times is to twice the whole time in the ordinary refraction as 1 to 320, and to the time in a plate of glass of which the refractive density is 1.55, as 1 to 318. În Mr. Biot's experiments on this substance, the proportion of the thicknesses appeared to be 1 to 360, while in the sulfate of lime, the number varied from 333 to 395; and it must be observed that any accidental irregularities, or foreign substances adhering to the plate, would tend, in Mr. Biot's mode of measurement, to make the thickness appear greater: while, on the other hand, an error of a single unit in the third place of decimals of the index of refractive density, as determined by Mr. Malus, would be sufficient to make the coincidence perfect: and a greater degree of accuracy can scarcely be expected in experiments of this kind.

We have next to inquire what must be the effect of the obliquity of the incident light according to the general law of periodical colours; and we shall here find the agreement of the experiments with the theory equally striking. We must compare the excesses of the times occupied in the transmission of light by the respective refractions, above the time required for its simple reflection from a point in the upper surface, exactly opposite to the respective point of reflection in the lower; and the difference between these excesses will give the interval required for determining the colour. Calling the thickness unity, and the sine of incidences, the excess for the ordinary refraction will be represented by the time within the plate, which is as the secant of refraction, diminished by the difference of the times without the plate, which is as its tangent, and as the sine of incidence jointly, (see Ph. Tr. 1802. pl. 1 fig.3,) or byr:√(1) - —ss : r√✓ (1 — — ) = √(r2 — s2): and for the extraordinary refraction, when the axis is parallel to the surface, the former part will be inversely as, and will be expressed by r: n/

ss nn+qa

rr

(1. + and the latter by

whence the whole becomes (r2.

Now since, in the substances which we are considering, n is little

more than 1, we may put n=1+l, n2 = 1 + 2l, and n4=1+47; then

; and if for

(1+2k kl) ss

n(rr-(1+2kkl)ss

1+99

̧=√ (r2 − (1+2k2l) s2: n. Now the difference

between √ (r2 - s2) and √/(r2 —(1+2k2l)s2) is

kklss

✔(rr-ss)

; and the

difference between the latter root, and the same quantity divided

by n, isl✔(r2 - (1+2k2ls), or very nearly l✅(r2 −s )=l

and the sum of these differences is l

rr- -(1-kk)ss
√(rr→ss)

rr-ss

√(rr-ss)'

or if 1 k2 =

rr-hh ss h being the cosine of the inclination of the plane (rr-ss)' of incidence to the axis: nor will the result be sensibly affected by taking into account the deviation of the refracted ray from this plane in oblique situations.

'√(rr—¡)'

This expression will be found to include all the effects of a change of inclination observed by Mr. Biot, and to agree sufficiently well with the formula which he has deduced from his measurements. When the light falls perpendicularly on the surface, s=0, and the difference becomes ir; when its obliquity is the utmost possible, and its value varies , being 1, the expression becomes-hh in the ratio of 2 to r2-1, according to the position of the axis. Thus in the sulfate of lime, r being 1.525, according to Dr. Wollaston's table, the utmost possible variation is in the ratio of 2.326 to 1.326, and the equivalent thickness for perpendicular rays being called 1, the extremes will become .755 and 1.325, instead of.87, and 1.26 or 1.152, which are the results of Mr. Biot's different formulas and the difference between these is as great as the variation of the first of them from our calculation. With respect to the singular fact of the indifference of the angle of incidence, when the inclination of the plane of incidence to the axis is 45°, our expression agrees exactly with Mr. Biot's observations: for when h2 = }}, =r, very nearly: thus if s=1, it only becomes √(rr-ss) 1.586 instead of 1.525, and does not vary sensibly while s remains small.

:

rr- - ss

any

In a similar manner the result may be determined for other relative situations of the axis and the refracting surface: if, for instance, they are perpendicular to each other, being√(1—n2

√(rr-nnss)'

rry,

the expression for the

nns

and the tangent of refraction

excess of time becomes r :/(1-n)

=√(r2 — n2s2),

nnss

rr] √(rr—nnss)

while the excess for the ordinary refraction is (r-s2) as before;

Iss

and the difference becomes

√(rr-ss)'

,which vanishes with the angle

thinking ourselves justified in looking forwards to a perfect coincidence between this formula and the promised experiments of Mr. Biot on substances placed in these circumstances. We understand that Dr. Brewster has lately made some observations of a nature nearly similar; but we doubt whether he has determined the refractive powers of his crystals with sufficient accuracy to allow of the application of our calculations with perfect precision.

A singular confirmation of the mode of explaining the colours of thin plates, which we have adopted, is afforded by the experiments of Mr. Arago, who found that the light forming the transmitted rings appeared to be polarised in the same direction with the reflected light, while the rest of the transmitted light was polarised in a contrary direction. It is a necessary assumption in the theory of periodical colours, that the rings seen by transmission actually depend on light twice reflected within the plate, and which must therefore be polarised like the rest of the reflected light; although, without these experiments of Mr. Arago, it would have been difficult to obtain so direct a demonstration of the fact.

The colours exhibited by thick pieces of rock crystal, cut, as in Mr. Biot's unpublished experiments, perpendicularly to the axis, might be expected to afford some explanation of those which Dr. Seebeck has observed in large cubes or cylinders of glass, placed between two oblique reflecting surfaces, or between two piles composed of thirty pieces of glass each, which produced the effect of complete polarisation on light transmitted at the appropriate angle. If, however, Dr. Seebeck's observations are correct, the analogy can be only superficial; for the effects of these pieces of glass seem to depend on their entire magnitude and outward form, without any particular relation to an axis of extraordinary refraction. Thus in the perpendicular transmission of the polarised light through any points in the diagonals of the surfaces of the cubes, or in the diameters parallel to their sides, the rays of different colours appeared to be differently affected according to the part of the glass on which they fell, and to exhibit one or the other only of the two images, which would have been visible through a piece of doubling spar, if the glass had not been interposed; so that when the whole cube was viewed at once under these circumstances, it afforded an appearance of diversified colours, arranged in very singular forms, which Dr. Seebeck compares to the figures assumed by sand on vibrating pieces of glass, and discovered some time since by Professor Chladni; but which appear to have a still nearer resemblance to those which Comparetti has described, as produced by the ad