verted,—namely, that the velocity of wave-propagation is constant in the same homogeneous medium,-is deduced on the particular supposition, that the sphere of action of the molecules of a vibrating medium is indefinitely small compared with the length of a wave. If this restriction be removed, we have no longer any ground for concluding that the waves of different lengths will be propagated with the same velocity; and the conclusion hitherto acquiesced in must be regarded but as an approximate result. It was in this point of view that the question presented itself to M. Cauchy. Resuming the problem of wave-propagation with the more general equations, he has proved that there exists, generally, a relation between the velocity of propagation (or the refractive index in vacuo) and the length of the wave; and, therefore, that the rays of different colours will be differently refracted.

in which A and denote, as before, the wave-length and time of vibration. M. Cauchy has proved that k and s are connected by an equation of the form

s2 = a1k2 + a2k* + azk® + &c.,

in which the coefficients a1, a2, as, &c., vary with the medium. Now the velocity of wave-propagation is

Accordingly, the velocity of propagation is a function of the wave-length, and varies with the colour.

(56) In a vacuum, and in media (such as atmospheric air)

which do not disperse the light, the coefficients a2, as, &c., are insensible, and we have

that is, the velocity of propagation is independent of the wavelength, and the same for light of all colours.

In other media we may, as a first approximation, neglect the third and following terms of the series, and we have

2

V2 = a1 + a2k2.

Hence, if V1, V2 denote the velocities of propagation for two definite rays of the spectrum, and k1, k2, the corresponding values of k,

The truth of this formula has been verified by M. Cauchy, by introducing in it the values of the refractive indices and wavelengths, as determined by Fraunhofer for the seven definite rays in certain media.

(57) The general formula, above given, is unsuited to a comparison with observation in its present form, inasmuch as the variable k

=

2π λ

is not independent of V. This diffi

culty is overcome by M. Cauchy by inverting the first series. The result is of the form

k2 = A182 + A25a + A356 + &c.

M. Cauchy has shown that this series, as well as the former, is convergent, and that all the terms after the third may be neglected. Hence, since

an equation expressing the refractive index in terms of the time of vibration, or of the wave-length in vacuo.

(58) The constants in this formula, A1, A2, A3, will be determined, when we know three values of μ, with the corresponding values of s, or of the wave-length in vacuo; and the formula may be then applied to calculate the values of μ corresponding to any other values of s, which may be thus compared with the results of observation. The comparison has been made by Professor Powell, and by M. Cauchy himself, by means of the observations of Fraunhofer on the refractive indices of water and several kinds of glass, and the agreement of the calculated and observed results is within the limits of the errors of observation.

But the truth of a formula, expressing the relation between the refractive index and the wave-length in vacuo, can only be satisfactorily tested in the case of highly-dispersive media; and for such media no observations of sufficient accuracy hitherto existed. To supply this want, Professor Powell undertook the laborious task of determining the refractive indices corresponding to the seven definite rays of Fraunhofer, for a great number of media, including those of a highly dispersive power, and of comparing them with the theory of M. Cauchy. The result of the comparison is, on the whole, satisfactory.

(59) It is an interesting consequence of the preceding formula, pointed out by Professor Powell, that as s diminishes, or the wave-length in vacuo increases, the value of μ approximates to a fixed limit, given by the equation

μ2 = А1,

which, therefore, defines the limit of the spectrum on the side of the less refrangible rays. This limiting index corresponds to a point not greatly below the red extremity of the visible spectrum.

CHAPTER IV.

DOUBLE REFRACTION.

(60) IT has hitherto been assumed that, when a ray of light is incident upon the surface of a transparent medium, the intromitted portion pursues, in all cases, a single determinate direction. This is, however, very far from the fact. In many, indeed in most cases,-the refracted ray is divided into two distinct pencils, each of which pursues a separate course, determined by a distinct law.

This property is called double refraction. It was first discovered by Erasmus Bartholinus, in the well-known mineral called Iceland spar. After a long series of observations, he found that one of the rays within the crystal followed the known law of refraction, while the other was bent according to a new and extraordinary law not hitherto noticed. An account of these experiments was published at Copenhagen, in the year 1669, under the title " Experimenta Crystalli Islandici dis-diaclastici, quibus mira et insolita refractio detegitur.”

A few years after the date of this publication, the subject was taken up by Huygens. This distinguished philosopher had already unfolded the theory which supposes light to consist in the undulations of an ethereal fluid; and from that theory had derived, in the most lucid and elegant manner, the laws of ordinary refraction (33). He was, therefore, naturally anxious to examine whether the new properties of light, discovered by Bartholinus, could be reconciled to the same theory; and, in his desire to assimilate the two classes of phenomena, he was happily led to assign the true law of extraordinary refraction. The important researches of Huy

gens on this subject are contained in the fifth chapter of his "Traité de la Lumiere."

(61) The property of double refraction is possessed by all crystallized minerals, excepting those belonging to the tessular system, i. e. those whose fundamental form is the cube. It belongs likewise to all animal and vegetable substances, in which there is a regular arrangement of parts; and, in fine, to all bodies whatever, whose parts are in a state of unequal compression or dilatation. The separation of the two refracted pencils is in some cases considerable, and the course of each easily ascertained by observation; but it is generally too minute to be directly observed, and its existence is only proved by the appearance of certain phenomena, which are known to arise from the mutual action of two pencils. In Iceland spar, the substance in which the property was first discovered, the separation of the pencils is very striking: and, as this mineral is found in considerable masses, and in a state of great purity and transparency, it is well fitted for the exhibition of the phenomena.

A

B

(62) Carbonate of lime, of which Iceland spar is a variety, crystallizes in more than 300 different forms, all of which may be reduced by cleavage to the rhombohedron, which is accordingly the primitive form. The angles of the bounding parallelograms, CAB and ABD, in the rhombohedron of Iceland spar, are 101° 55′ and 78° 5'. Two of the solid angles, at A and O, are contained by three obtuse angles; while the remaining four are bounded by one obtuse and two acute angles. The line AO, joining the summits of the obtuse solid angles, is called the axis of the rhombohedron, and is equally inclined to the three faces which meet there. The angles at which the faces themselves are mutually inclin ed are 105° 5' and 74 55'.

C