which the violet is the most refracted and the blue the least. According to the figure, this ought to be followed by a region of blue-green light, for which the index of refraction is less than unity; and we must therefore suppose, that on account of the narrowness of the region, or the faintness of the light, this region has either escaped observation, or is incapable of being detected without more powerful instruments. An absorption band then follows, and is succeeded by another band of more highly refracted light, corresponding to Qq, in which the red is the least, and the orange is the most refracted. Since the value of μ, when is slightly greater than 2, is large, it follows that the dispersion is anomalous; and we thus see why it is, that when there is an absorption band in the green, orange and red are more refracted than blue light. 353. If we compare these results with the table on p. 297, it will be seen, that they give a fairly satisfactory explanation of the anomalous dispersion produced by fuchsine and cyanine. The five absorption bands produced by permanganate of potash, could be explained by taking into account some of the terms in the value of μ, which have been omitted. VON HELMHOLTZ' THEORY. 321 Von Helmholtz' Theory of Anomalous Dispersion. 354. The theory proposed by Von Helmholtz1, is a theory relating to the mutual action of ether and matter, of somewhat the same character as Lord Rayleigh's theory of double refraction; but instead of following Von Helmholtz' method, we shall give the theory in a somewhat extended form 2. Let u, v, w be the component displacements of the ether, and u1, v1, w1 those of the matter. We shall suppose, that in vacuo the ether is a medium, whose motion is governed by the same equations as those of an elastic solid. When ethereal waves pass through a material substance, the molecules of the matter will be displaced, and the matter will acquire potential energy. The proper form of the mathematical expression for this potential energy is a question of speculation; and the first hypothesis we shall make will be, that the molecular forces are proportional to the displacements of the matter, and consequently the potential energy W, of the matter will be of the form 3 2 3 W2 = { (Au22 + Bv ̧2 + Cw‚2 + 2A′v‚w1 + 2B'w1u2+2C'u‚v1) .....(1), where A, B ... are constants. The second hypothesis is, that the potential energy of the system contains a term, which depends upon the relative displacements of ether and matter. This portion of the potential energy, which we shall denote by W,, is supposed to arise from the mutual reaction of ether upon matter; and if we assume that the corresponding forces are linear functions of the relative displacements, W will be a homogeneous quadratic function of the relative. displacements, so that 2 W2 = }{A (u — u1)2 + B (v − v1)2 + C (w — w1)2 + 2A′ (v — v,) (w — w,) + 2B′ (w — w1) (u — u1) + 20′ (u — u1) (v — v1) } ... ... ... ... ... ... ... ... ... ... ... ... ..(2). The third part of the potential energy, which we shall denote by W1, is the potential energy of the ether alone, and is of the same form as that of an elastic solid. The total potential energy W of the system will therefore be 1 Pogg. Ann. vol. CLIV. p. 582; Wissen. Abhand. vol. 1. p. 213. 2 Proc. Lond. Math. Soc. vol. xxii. p. 4. B. O. 21 (3). If p, be the density of the matter, its kinetic energy will be In order to introduce Lord Rayleigh's theory, we shall suppose that the effect of the matter upon the ether, is to cause the latter to behave as if its density were æolotropic, and the third hypothesis will therefore be, that the kinetic energy T, of the ether is 355. The equations of motion of the system may now be deduced by the Principle of Least Action, viz. SSSS 8 (T1 + T2 − W1- W2- W1) dxdydzdt = 0; 1 2 A = B = C = ß2 ; A′ = B′ = C′ = 0, and the equations of motion accordingly become 356. Von Helmholtz has introduced a viscous term into the equations of motion of the matter, which has been objected to by Lord Kelvin1. I think however that this term may be justified on two independent grounds. It is an experimental fact, that when vibrations are set up in any material substance, the vibrations are gradually damped by internal friction, and the motion ultimately 1 Lectures on Molecular Dynamics, p. 149. VON HELMHOLTZ' THEORY. 323 dies away. This state of things may be represented mathematically, by the introduction of a viscous term into the equations of motion of the matter. If however we do not wish to introduce the hypothesis of internal friction into our equations, the viscous term may still be accounted for. When a molecule of matter is set into vibration by ethereal waves, the physical characteristics of the motion will be much the same as those of the spherical pendulum vibrating in air, which has been discussed at the commencement of the present chapter; consequently the vibrations of the molecules will generate secondary waves in the ether, which will carry away into space the energy, which is communicated to the molecules, and the equations of motion of the latter will therefore contain a viscous term. In the place of (9), we shall accordingly assume that the equations of motion of the matter are of the form 357. Let us now consider the propagation of waves of light, which are travelling through the medium in the positive direction of the axis of z, and let the displacements be parallel to the axis of x. Equation (8) now becomes Now if x be the free period of the matter vibrations, K From the problem of the spherical pendulum we infer, that the amplitudes of the molecules of matter are very much smaller than those of the ether; and accordingly a must be very small compared with P1 the density of the matter. If however 1 1 a2 K2 Απερι in which case the period of the waves of light would be sensibly equal to the free period of the matter, A, would be infinite in the absence of friction. This conclusion is contradicted by experience, and necessitates the introduction of a viscous term. 358. Let D denote the real part of the denominator of (14); then if we eliminate A, A, from (12) and (13), and equate the real and imaginary parts, we shall obtain Now q is the coefficient of absorption of the medium; and since the absorption is slight for most transparent substances, it follows Let P denote the coefficient of h in (16), then eliminating h between (15) and (16), we obtain the positive sign being taken, because P= a/D, when q=0. Expanding the quantity under the radical, we obtain Po Now if p. be the density of the ether in vacuo, n/V2= Poμ2, where is the index of refraction; whence (17) becomes |