that we have introduced this calculation, but in order to apply it. to the explanation, which we shall attempt to give, of the very interesting phenomena described at large in Mr. Biot's memoir.

Mr. Arago had discovered, in 1811, that polarised light was resolved, by passing through thin plates of mica or sulfate of lime, or thicker plates of rock crystal, and of some kinds of flint glass, into two portions differently coloured. Mr. Biot has experimentally investigated the law, according to which these phenomena take place, and has reduced the results of his experiments into such a form, as to enable us to calculate from them, what colours will be exhibited by a plate of sulfate of lime, of a given thickness, and in a given situation with respect to the incident light.

The axis of the crystals of sulfate of lime is, either accurately or very nearly, in the plane of the plates which they afford, and makes an angle of 16° 13′ with one of the natural lines of fracture of the plates; while that of rock crystal is nearly parallel to the longitudinal surfaces of the crystal. Mr. Biot's method of exhibiting the colours in question, is to take a thin and smooth plate of sulfate of lime or Muscovy talc, or a well polished plate of rock crystal, cut as thin as possible, which affords no appearance of colour in the open air, except when some of the incident light has been polarised by reflection from the blue atmosphere, and to place it horizontally on a black substance; then allowing the white light of the clouds to fall on .it, at an inclination of about 35°, to receive this light, when reflected from it, on a black glass, making an equal angle with the reflected rays, in a plane perpendicular to the first plane of reflection; so that the plate may be visible by reflection in the black glass. In this manner the plate appears to be very brilliantly illuminated by the light of the colour which it is calculated to exhibit: when its axis coincides with the plane of incidence, no colour is visible; and the appearance becomes most distinct when the axis makes an angle of 45° with that plane. In this situation of the axis, Mr. Biot finds that the colour reflected by talc, and by rock crystal, is precisely the same as if the incidence were perpendicular, and the same as is transmitted by the extraordinary refraction; while the light transmitted by the ordinary refraction exhibits the complementary colour, as in the case of the ordinary colours of thin plates: these transmitted colours being separable, as Mr. Arago had found, by means of any doubly refractive substance, or by oblique reflection. In Mr. Biot's arrangement the light reflected from the upper surface of the plate is polarised according to the general law, and is therefore not reflected by the black glass, but absorbed: and the same is true of the light reflected from the lower surface of the plate, and then transmitted back by the ordinary refraction: but that which has been

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.

Y

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 pressure 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 talc 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--.195c2H)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 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 explanation 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

4

160; consequently the difference of the times is to twice the whole time in the ordinary refraction as I to 320, and to the time in a 1 plate of glass of which the refractive density is 1.55, as I to 318. In 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 suf ficient 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 incidence s, 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://(1-5) ——-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

rr

199

Now since, in the substances which we are considering, n is little more than 1, we may put n=1+1, n2 = +2l, and n2=1+47; then

=

nn +qq

41

21

1

which will also be the value

of

14-99

rr

and if for

-(1+2k kl) ss

n(rr- (1+2kklyss

we write k2, the excess will become 1+99

=✓ (r2 (1+2k) s: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 (r2s ) =l

and the sum of these differences is l

rr-hh ss

√(rr-ss)'

rr-ss

(rss)

rr-(1-kk)ss
(rr-ss)

h being the cosine of the inclination of the plane 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.

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, rr-hh s being 1, the expression becomes l and its value varies √(rr−1)2 in the ratio of r2 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 rr- - Ess 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.

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

nns

rr

and the tangent of refraction (rr-nnss)' the expression for the

excess of time becomes r/(1-n

nnss

√(rr-nnss)

=√(r2 — n2s2),

while the excess for the ordinary refraction is✔ (r2 — s2) as before;

and the difference becomes

Iss
√(rr-ss)'

,which vanishes with the angle