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

(63) In a vacuum, and in media (such as atmospheric air) which do not disperse the light, the coefficients a2, A3, &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 ki, 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 wave-lengths in certain media, as determined by Fraunhofer for the seven definite rays.

(64) The general formula, above given, is unsuited to a comparison with observation in its present form, inasmuch as the variable k is not independent of V. This diffi

2π
λ

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

k2 = A1 s2+A2 s1+A3 86 + &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.

(65) 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 refrac tive 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.

(66) It is an interesting consequence of the preceding for

mula, 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

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.

ABSORPTION AND EMISSION.

(67) ALL bodies absorb a portion of the light which enters their substance; and this absorption is different for rays of different refrangibilities, so that white light always becomes coloured after passage through a medium of sufficient thickness. There is, accordingly, no body perfectly transparent. On the other hand, no known body is perfectly opaque. Gold itself, which is, the densest of the metals, transmits a faint greenish light, when in the state of gold leaf.

The phenomena of absorption are all explained on the supposition that light, which has traversed a given thickness of any medium, loses by absorption the same fractional part of its intensity, in passing through any given thickness of the medium. According to this hypothesis, the intensity of the light must diminish in geometrical progression, as the thickness increases in arithmetical. For if i denote the intensity of the light incident on the medium, and ia the intensity after it has traversed the unit of thickness, the intensity, after passing through two, three, &c., units, will be ia, i a3, &o. And generally, the intensity of the light, I, after traversing the thickness 0, will be

I = i ao.

The quantity a is the coefficient of transmission; and it is different for the light of different colours. Hence, when the incident light is white, the elements of which it consists will be absorbed in different proportions, and the emergent light

will no longer contain the rays of different colours in the same proportion as originally, and will, consequently, be coloured.

(68) In general the colour of the emergent light is the same for all thicknesses traversed, the tint only becoming deeper as the thickness is augmented. But there are media in which the colour itself changes with the thickness. This will be seen by the foregoing formula. For if i and i' be the original intensities of the two colours, which we shall suppose to predominate in the transmitted light, and a and a' their coefficients of transmission, we have only to suppose i>i', and a <a'. Under such circumstances i a° will be at first greater than i' a", when the thickness is small; but it will diminish more rapidly with an increase of thickness, and finally become smaller. It is easily seen that the thickness at which the change occurs is given by the formula

0 = log. i - log. i'
log. a' - log. a

(69) The complete analysis of the phenomena of absorption is effected by interposing the absorbing medium in a pencil of rays of solar light which has traversed a prism, and receiving the emergent rays on a screen. For the most part the spectrum of the light, which has passed through absorbing media, thus exhibits a single maximum of absorption. There are, however, media whose action upon light is more complex. The blue glass which is coloured by cobalt, has three such maxima, dividing the spectrum into four portions.* The solution of chlorophyll (the colouring matter

* This affords an easy method of obtaining a red light which is nearly homogeneous. We have only to combine a plate of this glass with one of the red glasses which transmit only the less refrangible rays of the spectrum, and we shall obtain a red ray of considerable purity.