# Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XI

XI.

ON DOUBLE REFRACTION AND THE DISPERSION OF COLORS IN PERFECTLY TRANSPARENT MEDIA.

[American Journal of Science, ser. 3, vol. xxiii, pp. 262–275, April, 1882.]

1. In calculating the velocity of a system of plane waves of homogeneous light, regarded as oscillating electrical fluxes, in transparent and sensibly homogeneous bodies, whether singly or doubly refracting, we may assume that such a body is a very fine-grained structure, so that it can be divided into parts having their dimensions very small in comparison with the wave-length, each of which may be regarded as entirely similar to every other, while in the interior of each there are wide differences in electrical as in other physical properties. Hence, the average electrical displacement in such parts of the body may be expressed as a function of the time and the coordinates of position by the ordinary equations of wave-motion, while the real displacement at any point will in general differ greatly from that represented by such equations.

It is the object of this paper to investigate the velocity of light in perfectly transparent media which have not the property of circular polarization in a manner which shall take account of this difference between the real displacements and those represented by the ordinary equations of wave-motion. We shall find that this difference will account for the dispersion of colors, without affecting the validity of the laws of Huyghens and Fresnel for double refraction with respect to light of any one color.

In this investigation, it is assumed that the electrical displacements are solenoidal, or, in other words, that they are such as not to produce any change in electrical density. The disturbance in the medium is treated as consisting entirely of such electrical displacements and fluxes, and not complicated by any distinctively magnetic phenomena. It might therefore be more accurate to call the theory (as here developed) electrical rather than electromagnetic. The latter term is nevertheless retained in accordance with general usage, and with that of the author of the theory.

Since the velocity which we are seeking is equal to the wave-length divided by the period of oscillation, the problem reduces to finding the ratio of these quantities, and may be simplified in some respects by supposing that we have to do with a system of stationary waves. That the relation of the wave-length and the period is the same for stationary as for progressive waves is evident from the consideration that a system of stationary waves may be formed by two systems of progressive waves having opposite directions.

2. Let ${\displaystyle x,y,z}$ be the rectangular coordinates of any point in the medium, which with the system of waves we may regard as indefinitely extended, and let ${\displaystyle \xi +\xi ',\eta +\eta ',\zeta +\zeta '}$ be the components of electrical displacement at that point at the time ${\displaystyle t;}$ ${\displaystyle \xi ,\eta ,\zeta }$ being the average values of the components of electrical displacement at that time in a wave-plane passing through the point. Then ${\displaystyle \xi ,\eta ,\zeta ,\xi ',\eta ',\zeta ',x,y,z}$ are perfectly defined quantities, of which ${\displaystyle \xi ,\eta ,\zeta }$ are connected with ${\displaystyle x,y,z,}$ and ${\displaystyle t}$ by the ordinary equations of wave-motion, while each of the quantities ${\displaystyle \xi ',\eta ',\zeta '}$ has always zero for its average value in any wave-plane. We may call ${\displaystyle \xi ,\eta ,\zeta }$ the components of the regular part of the displacement, and ${\displaystyle \xi ',\eta ',\zeta '}$ the components of the irregular part of the displacement. In like manner, the differential coefficients of these quantities with respect to the time, ${\displaystyle {\dot {\xi }},{\dot {\eta }},{\dot {\zeta }},{\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}},}$ may be called respectively the components of the regular part of the flux, and the components of the irregular part of the flux.

Let the whole space be divided into elements of volume ${\displaystyle {\text{D}}v,}$ very small in all dimensions in comparison with a wave-length, but enclosing portions of the medium which may be treated as entirely similar to one another, and therefore not infinitely small. Thus a crystal may be divided into elementary parallelopipeds, all the vertices of which are similarly situated with respect to the internal structure of the crystal. Amorphous solids and liquids may not be capable of division into equally small portions of which physical similarity can be predicated with the same rigor. Yet we may suppose them capable of a division substantially satisfying the requirements.

From these definitions it follows that at any given instant the average value of each of the quantities ${\displaystyle \xi ',\eta ',\zeta '}$ in an element ${\displaystyle {\text{D}}v}$ is zero. For the average value in one such element must be sensibly the same as in any other situated on the same wave-plane. If this average were not zero, the average for the wave-plane would not be zero. Moreover, at any given instant, the values of ${\displaystyle \xi ,\eta ,\zeta }$ may be regarded as constant throughout any element ${\displaystyle {\text{D}}v,}$ and as representing the average values of the components of displacement in that element. The same will be true of the quantities ${\displaystyle {\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}}}$ and ${\displaystyle {\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}.}$

3. Since we have excluded the case of media which have the property of circular polarization, we shall not impair the generality of our results if we suppose that we have to do with linearly polarized light, i.e., that the regular part of the displacement is everywhere parallel to the same fixed line, all cases not already excluded being reducible to this. Then, with the origin of coordinates and the zero of time suitably chosen, the regular part of the displacement may be represented by the equations

 ${\displaystyle \xi }$ ${\displaystyle =\alpha \cos 2\pi {\frac {u}{l}}\cos 2\pi {\frac {t}{p}},}$ ${\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}}$ (1) ${\displaystyle \eta }$ ${\displaystyle =\beta \cos 2\pi {\frac {u}{l}}\cos 2\pi {\frac {t}{p}},}$ ${\displaystyle \zeta }$ ${\displaystyle =\gamma \cos 2\pi {\frac {u}{l}}\cos 2\pi {\frac {t}{p}},}$

where ${\displaystyle l}$ denotes the wave-length, ${\displaystyle p}$ the period of vibration, ${\displaystyle \alpha ,\beta ,\gamma }$ the maximum amplitudes of the displacements ${\displaystyle \xi ,\eta ,\zeta ,}$ and ${\displaystyle u}$ the distance of the point considered from the wave-plane which passes through the origin. Since ${\displaystyle u}$ is a linear function of ${\displaystyle x,y,}$ and ${\displaystyle z,}$ we may regard these equations as giving the values of ${\displaystyle \xi ,\eta ,\zeta ,}$ for a given system of waves, in terms of ${\displaystyle x,y,z,}$ and ${\displaystyle t.}$

4. The components of the irregular displacement, ${\displaystyle \xi ',\eta ',\zeta ',}$ at any given point, will evidently be simple harmonic functions of the time, having the same period as the regular part of the displacement. That they will also have the same phase is not quite so evident, and would not be the case in a medium in which there were any absorption or dispersion of light. It will however appear from the following considerations that in perfectly transparent media the irregular oscillations are synchronous with the regular. For if they are not synchronous, we may resolve the irregular oscillations into two parts, of which one shall be synchronous with the regular oscillations, and the other shall have a difference of phase of one-fourth of a complete oscillation. Now if the mediimi is one in which there is no absorption or dispersion of light, we may assume that the same electrical configurations may also be passed through in the inverse order, which would be represented analytically by writing ${\displaystyle -t}$ for ${\displaystyle t}$ in the equations which give ${\displaystyle \xi ,\eta ,\zeta ,\xi ',\eta ',\zeta ',}$ as functions of ${\displaystyle x,y,z,}$ and ${\displaystyle t.}$ But this change would not affect the regular oscillations, nor the synchronous part of the irregular oscillations, which depends on the cosine of the time, while the non-synchronous part of the irregular oscillations, which depends on the sine of the time, would simply have its direction reversed. Hence, by taking first one-half the sum, and secondly one-half the difference, of the original motion and that obtained by substitution of ${\displaystyle -t}$ for ${\displaystyle t,}$ we may separate the non-synchronous part of the irregular oscillations from the rest of the motion. Therefore, the supposed non-synchronous part of the irregular displacement, if capable of existence, is at least wholly independent of the wave-motion and need not be considered by us.

We may go farther in the determination of the quantities ${\displaystyle \xi ',\eta ',\zeta '.}$ For in view of the very fine-grained structure of the medium, it will easily appear that the manner in which the general or average flux in any element ${\displaystyle {\text{D}}v}$ (represented by ${\displaystyle \xi ,\eta ,\zeta }$) distributes itself among the molecules and intermolecular spaces must be entirely determined by the amount and direction of that flux and its period of oscillation. Hence, and on account of the superposable character of the motions which we are considering, we may conclude that the values of ${\displaystyle \xi ',\eta ',\zeta '}$ at any given point in the medium are capable of expression as linear functions of ${\displaystyle \xi ,\eta ,\zeta }$ in a manner which shall be independent of the time and of the orientation of the wave-planes and the distance of a nodal plane from the point considered, so long as the period, of oscillation remains the same. But a change in the period may presumably affect the relation between ${\displaystyle \xi ',\eta ',\zeta '}$ and ${\displaystyle \xi ,\eta ,\zeta }$ to a certain extent. And the relation between ${\displaystyle \xi ',\eta ',\zeta '}$ and ${\displaystyle \xi ,\eta ,\zeta }$ will vary rapidly as we pass from one point to another within the element ${\displaystyle {\text{D}}v.}$

5. In the motion which we are considering there occur alternately instants of no velocity and instants of no displacement. The statical energy of the medium at an instant of no velocity must be equal to its kinetic energy at an instant of no displacement. Let us examine each of these quantities, and consider the equation which expresses their equality.

6. Since in every part of an element ${\displaystyle {\text{D}}v}$ the irregular as well as the regular part of the displacement is entirely determined (for light of a given period) by the values of ${\displaystyle \xi ,\eta ,\zeta ,}$ the statical energy of the element must be a quadratic function of ${\displaystyle \xi ,\eta ,\zeta ,}$ say

 ${\displaystyle ({\text{A}}\xi ^{2}+{\text{B}}\eta ^{2}+{\text{C}}\zeta ^{2}+{\text{E}}\eta \zeta +{\text{F}}\zeta \xi +{\text{G}}\xi \zeta ){\text{D}}v,}$
where ${\displaystyle {\text{A, B, }}}$ etc. depend only on the nature of the medium and the period of oscillation. At an instant of no velocity, when
 ${\displaystyle \sin 2\pi {\frac {t}{p}}=0,}$⁠and⁠${\displaystyle \cos ^{2}2\pi {\frac {t}{p}}=1,}$
the above expression will reduce by equations (1) to
 ${\displaystyle ({\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta )\cos ^{2}2\pi {\frac {u}{l}}{\text{D}}v.}$
Since the average value of ${\displaystyle \cos ^{2}2\pi {\frac {u}{l}}}$ in an indefinitely extended space is ${\displaystyle {\tfrac {1}{2}},}$ we have for the statical energy in a unit of volume
 ${\displaystyle {\text{S}}={\tfrac {1}{2}}({\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta ).}$ (2)
7. The kinetic energy of the whole medium is represented by the double volume-integral[1]
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle \iint {\frac {({\dot {\xi }}+{\dot {\xi '}})_{1}({\dot {\xi }}+{\dot {\xi '}})_{2}}{r}}dv_{1}dv_{2},}$
where ${\displaystyle dv_{1},dv_{2}}$ are two infinitesimal elements of volume, ${\displaystyle ({\dot {\xi }}+{\dot {\xi '}})_{1},({\dot {\xi }}+{\dot {\xi '}})_{2}}$ the corresponding components of flux, ${\displaystyle r}$ the distance between the elements, and ${\displaystyle \textstyle \sum }$ denotes a summation with respect to the coordinate axes. Separating the integrations, we may write for the same quantity
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle \int ({\dot {\xi }}+{\dot {\xi '}})_{1}\left[\int {\frac {({\dot {\xi }}+{\dot {\xi '}})_{2}}{r}}dv_{2}\right]dv_{1}.}$
It is evident that the integral within the brackets is derived from ${\displaystyle {\dot {\xi }}+{\dot {\xi '}}}$ by the same process by which the potential of any mass is derived from its density. If we use the symbol ${\displaystyle Pot}$ to express this relation, we may write for the kinetic energy
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle \int ({\dot {\xi }}+{\dot {\xi '}}){\text{ Pot }}({\dot {\xi }}+{\dot {\xi '}})dv.}$
The operation denoted by this symbol is evidently distributive, so that ${\displaystyle {\text{Pot }}({\dot {\xi }}+{\dot {\xi '}})={\text{Pot }}{\dot {\xi }}+{\text{Pot }}{\dot {\xi '}}.}$ The expression for the kinetic energy may therefore be expanded into
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi }}{\text{ Pot }}{\dot {\xi }}dv+{\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi }}{\text{ Pot }}{\dot {\xi '}}dv+{\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi '}}{\text{ Pot }}{\dot {\xi }}dv+{\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}dv.}$
But ${\displaystyle {\dot {\xi ',}}}$ and therefore ${\displaystyle {\text{Pot }}{\dot {\xi '}},}$ has in every wave-plane the average value zero. Also ${\displaystyle {\dot {\xi }},}$ and therefore ${\displaystyle {\text{Pot }}{\dot {\xi }},}$ has in every wave-plane a constant value. Therefore the second and third integrals in the above expression will vanish, leaving for the kinetic energy
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi }}{\text{ Pot }}{\dot {\xi }}dv+{\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}dv,}$ (3)
which is to be calculated for a time of no displacement, when
 ${\displaystyle {\dot {\xi }}=\pm {\frac {2\pi \alpha }{p}}\cos 2\pi {\frac {u}{l}},}$⁠${\displaystyle {\dot {\eta }}=\pm {\frac {2\pi \beta }{p}}\cos 2\pi {\frac {u}{l}},}$⁠${\displaystyle {\dot {\zeta }}=\pm {\frac {2\pi \gamma }{p}}\cos 2\pi {\frac {u}{l}}\cdot }$ (4)
The form of the expression (3) indicates that the kinetic energy consists of two parts, one of which is determined by the regular part of the flux, and the other by the irregular part of the flux.

8. The value of ${\displaystyle {\text{Pot }}{\dot {\xi }}}$ may be easily found by integration, but perhaps more readily by Poisson's well-known theorem, that if ${\displaystyle q}$ is any function of position in space (as the density of a certain mass),

 ${\displaystyle {\frac {d^{2}{\text{ Pot }}q}{dx^{2}}}+{\frac {d^{2}{\text{ Pot }}q}{dy^{2}}}+{\frac {d^{2}{\text{ Pot }}q}{dz^{2}}}=-4\pi q,}$ (5)
where the direction of the coordinate axes is immaterial, provided that they are rectangular. In applying this to ${\displaystyle {\text{Pot }}{\dot {\xi }},}$ we may place two of the axes in a wave-plane. This will give
 ${\displaystyle {\frac {d^{2}{\text{Pot }}{\dot {\xi }}}{du^{2}}}=-4\pi {\dot {\xi }}.}$ (6)
In a nodal plane, ${\displaystyle {\text{Pot }}{\dot {\xi }}=0,}$ since ${\displaystyle {\dot {\xi }}}$ has equal positive and negative values in elements of volume symmetrically distributed with respect to any point in such a plane. In a wave-crest (or plane in which ${\displaystyle {\dot {\xi }}}$ has a maximum value), ${\displaystyle {\text{Pot }}{\dot {\xi }}}$ will also have a maximum value, which we may call ${\displaystyle {\text{K}}.}$ For intermediate points we may determine its value £rom the consideration that the total disturbance may be resolved into two systems of waves, one having a wave-crest, and the other a nodal plane passing through the point for which the potential is sought. The maximum amplitudes of these component systems will be to the maximum amplitude of the original system as ${\displaystyle \cos 2\pi {\frac {u}{l}}}$ and ${\displaystyle \sin 2\pi {\frac {u}{l}}}$ to unity. But the second of the component systems will contribute nothing to the value of the potential. We thus obtain
 ${\displaystyle {\text{Pot }}{\dot {\xi }}={\text{K}}\cos 2\pi {\frac {u}{l}},}$${\displaystyle {\frac {d^{2}{\text{Pot }}{\dot {\xi }}}{du^{2}}}=-{\frac {4\pi ^{2}}{l^{2}}}{\text{K}}\cos 2\pi {\frac {u}{l}}=-{\frac {4\pi ^{2}}{l^{2}}}{\text{Pot }}{\dot {\xi }}.}$
Comparing this with equation (6), we have
 {\displaystyle {\begin{aligned}-{\frac {4\pi ^{2}}{l^{2}}}{\text{ Pot }}{\dot {\xi }}&=-4\pi {\dot {\xi }},\\{\text{Pot }}{\dot {\xi }}&={\frac {l^{2}}{\pi }}{\dot {\xi }}.\end{aligned}}} (7)
Hence, and by equations (4),
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi }}{\text{ Pot }}{\dot {\xi }}dv={\frac {l^{2}}{2\pi }}\textstyle \sum \displaystyle \int {\dot {\xi }}^{2}dv={\frac {2\pi l^{2}}{p^{2}}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\int \cos ^{2}2\pi {\frac {u}{l}}dv.}$
The kinetic energy of the regular part of the flux is therefore, for each unit of volume,
 ${\displaystyle {\text{T}}={\frac {\pi l^{2}}{p^{2}}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2}).}$ (8)
9. With respect to the kinetic energy of the irregular part of the flux, it is to be observed that, since ${\displaystyle {\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}}}$ have their average values zero in spaces which are very small in comparison with a wave-length, the integrations implied in the notations ${\displaystyle {\text{Pot }}{\dot {\xi '}},{\text{Pot }}{\dot {\eta '}},{\text{Pot }}{\dot {\zeta '}}}$ may be confined to a sphere of a radius which is small in comparison with a wave-length. Since within such a sphere ${\displaystyle {\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}}}$ are sensibly determined by the values of ${\displaystyle {\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}}$ at the center of the sphere, which is the point for which the value of the potentials are sought, ${\displaystyle {\text{Pot }}{\dot {\xi '}},{\text{Pot }}{\dot {\eta '}},{\text{Pot }}{\dot {\zeta '}}}$ must be functions—evidently linear functions—of ${\displaystyle {\dot {\xi }},{\dot {\eta }},{\dot {\zeta }};}$ and ${\displaystyle {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}},{\dot {\eta '}}{\text{ Pot }}{\dot {\eta '}},{\dot {\zeta '}}{\text{ Pot }}{\dot {\zeta '}}}$ must be quadratic functions of the same quantities. But these functions will vary with the position of the point considered with reference to the adjacent molecules.

Now the expression for the kinetic energy of the irregular part of the flux,

 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}dv,}$
indicates that the we may regard the infinitesimal element ${\displaystyle dv}$ the energy (due to this part of the flux)
 ${\displaystyle {\tfrac {1}{2}}\textstyle \sum \displaystyle {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}dv.}$
Let us consider the energy due to the irregular flux which will belong to the above defined element ${\displaystyle {\text{D}}v,}$ which is not infinitely small, but which has the advantage of being one of physically similar elements which make up the whole medium. The energy of this element is found by adding the energies of all the infinitesimal elements of which it is composed Since these are quadratic functions of the quantities ${\displaystyle {\dot {\xi }},{\dot {\eta }},{\dot {\zeta }},}$ which are sensibly constant throughout the element ${\displaystyle {\text{D}}v,}$ the sum will be a quadratic function of ${\displaystyle {\dot {\xi }},{\dot {\eta }},{\dot {\zeta }},}$ say
 ${\displaystyle ({\text{A}}'{\dot {\xi }}^{2}+{\text{B}}'{\dot {\eta }}^{2}+{\text{C}}'{\dot {\zeta }}^{2}+{\text{E}}'{\dot {\eta }}{\dot {\zeta }}+{\text{F}}'{\dot {\zeta }}{\dot {\xi }}+{\text{G}}'{\dot {\xi }}{\dot {\eta }}){\text{D}}v,}$
which will therefore represent the energy of the element ${\displaystyle {\text{D}}v}$ due to the irregular flux. The coefficients ${\displaystyle {\text{A}}',{\text{B}}',}$ etc., are determined by the nature of the medium and the period of oscillation. They will be constant throughout the medium, since one element ${\displaystyle {\text{D}}v}$ does not differ from another.

This expression reduces by equations (4) to

 ${\displaystyle {\frac {4\pi ^{2}}{p^{2}}}({\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta )\cos ^{2}2\pi {\frac {u}{l}}{\text{D}}v.}$
The kinetic energy of the irregular flux in a unit of volume is therefore
 ${\displaystyle {\text{T}}'={\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta ).}$ (9)
10. Equating the statical and kinetic energies, we have
 ${\displaystyle {\tfrac {1}{2}}({\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta )}$${\displaystyle ={\frac {\pi l^{2}}{p^{2}}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})+{\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta ).}$ (10)
The velocity (${\displaystyle {\text{V}}}$) of the corresponding system of progressive waves is given by the equation
 {\displaystyle {\begin{aligned}{\text{V}}^{2}={\frac {l^{2}}{p^{2}}}&={\frac {1}{2\pi }}{\frac {+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta }{\alpha ^{2}+\beta ^{2}+\gamma ^{2}}}\\&-{\frac {2\pi }{p^{2}}}{\frac {{\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta }{\alpha ^{2}+\beta ^{2}+\gamma ^{2}}}\cdot \end{aligned}}} (11)
If we set
 ${\displaystyle a={\frac {1}{2\pi }}{\text{A}}-{\frac {2\pi }{p^{2}}}{\text{A}}',}$⁠${\displaystyle b={\frac {1}{2\pi }}{\text{B}}-{\frac {2\pi }{p^{2}}}{\text{B}}',}$⁠etc., (12)
and
 ${\displaystyle \rho ^{2}=\alpha ^{2}+\beta ^{2}+\gamma ^{2},}$
the equation reduces to
 ${\displaystyle {\text{V}}^{2}={\frac {a\alpha ^{2}+b\beta ^{2}+c\gamma ^{2}+e\beta \gamma +f\gamma \alpha +g\alpha \beta }{\rho ^{2}}}\cdot }$ (13)
For a given medium and light of a given period, the coefficients ${\displaystyle a,b,}$ etc., are constant.

This relation between the velocity of the waves and the direction of oscillation is capable of a very simple geometrical expression. Let ${\displaystyle r}$ be the radius vector of the ellipsoid

 ${\displaystyle ax^{2}+by^{2}+cz^{2}+eyz+fzx+gxy=1.}$ (14)
Then
 ${\displaystyle {\frac {1}{r^{2}}}={\frac {ax^{2}+by^{2}+cz^{2}+eyz+fzx+gxy}{r^{2}}}\cdot }$
If this radius is drawn parallel to the electrical oscillations, we shall have
 ${\displaystyle {\frac {x}{r}}={\frac {\alpha }{\rho }},}$⁠${\displaystyle {\frac {y}{r}}={\frac {\beta }{\rho }},}$⁠${\displaystyle {\frac {z}{r}}={\frac {\gamma }{\rho }},}$
and
 ${\displaystyle {\text{V}}^{2}={\frac {1}{r^{2}}}\cdot }$ (15)
That is, the wave-velocity for any particular direction of oscillation is represented in the ellipsoid by the reciprocal of the radius vector which is parallel to that direction.

11. This relation between the wave-length, the period, and the direction of vibration, must hold true not only of such vibrations as actually occur, but also of such as we may imagine to occur under the influence of constraints determining the direction of vibration in the wave-plane. The directions of the natural or unconstrained vibrations in any wave-plane may be determined by the general mechanical principle that if the type of a natural vibration is infinitesimally altered by the application of a constraint, the value of the period will be stationary.[2] Hence, in a system of stationary waves such as we have been considering, if the direction of an unconstrained vibration is infinitesimally varied in its wave-plane by a constraint while the wave-length remains constant, the period will be stationary. Therefore, if the direction of the unconstrained vibration is infinitesimally varied by constraint, and the period remains rigorously constant, the wave-length will be stationary. Hence, if we make a central section of the above described ellipsoid parallel to any wave-plane, the directions of natural vibration for that wave-plane will be parallel to the radii vectores of stationary value in that section, viz., to the axes of the ellipse, when the section is elliptical, or to all radii, when the section is circular.

12. For light of a single period, our hypothesis has led to a perfectly definite result, our equations expressing the fundamental laws of double refraction as enunciated by Fresnel. But if we ask how the velocity of light varies with the period, that is, if we seek to derive from the same equations the laws of the dispersion of colors, we shall not be able to obtain an equally definite result, since the quantities ${\displaystyle {\text{A, B,}}}$ etc., and ${\displaystyle {\text{A}}',{\text{B}}',}$ etc., are unknown functions of the period. If, however, we make the assumption, which is hardly likely to be strictly accurate, but which may quite conceivably be not far removed from the truth, that the manner in which the general or average flux in any small part of the medium distributes itself among the molecules and intermolecular spaces is independent of the period, the quantities ${\displaystyle {\text{A, B,}}}$ etc., and ${\displaystyle {\text{A}}',{\text{B}}',}$ etc., will be constant, and we obtain a very simple relation between V and p, which appears to agree tolerably well with the results of experiment.

If we set

 ${\displaystyle {\text{H}}={\frac {{\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta }{\rho ^{2}}},}$ (16)
and
 ${\displaystyle {\text{H}}'={\frac {{\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta }{\rho ^{2}}},}$ (17)
our general equation (11) becomes
 ${\displaystyle {\text{V}}^{2}={\frac {\text{H}}{2\pi }}-{\frac {2\pi {\text{H}}'}{p^{2}}},}$ (18)
where ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}'}$ will be constant for any given direction of oscillation, when ${\displaystyle {\text{A, B,}}}$ etc., and ${\displaystyle {\text{A}}',{\text{B}}',}$ etc., are constant. If we wish to introduce into the equation the absolute index of refraction (${\displaystyle n}$) and the wavelength in vacuo (${\displaystyle \lambda }$) in place of ${\displaystyle {\text{V}}}$ and ${\displaystyle \rho ,}$ we may divide both sides of the equation by the square of the constant (${\displaystyle k}$) representing the velocity of light in vacuo. Then, since
 ${\displaystyle {\frac {\text{V}}{k}}={\frac {1}{n}},}$⁠and⁠${\displaystyle kp=\lambda ,}$
our equation reduces to
 ${\displaystyle {\frac {1}{n^{2}}}={\frac {\text{H}}{2\pi k^{2}}}-{\frac {2\pi {\text{H}}'}{\lambda ^{2}}}\cdot }$ (19)
It is well known that the relation between ${\displaystyle n}$ and ${\displaystyle \lambda }$ may be tolerably well but by no means perfectly represented by an equation of this form.

13. If we now give up the presumably inaccurate supposition that ${\displaystyle {\text{A, B,}}}$ etc., and ${\displaystyle {\text{A}}',{\text{B}}',}$ etc., are constant, equation (19) will still subsist, but ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}'}$ will not be constant for a given direction of oscillation, but will be functions of ${\displaystyle p,}$ or, what amounts to the same, of ${\displaystyle \lambda .}$ Although we cannot therefore use the equation to derive a priori the relation between ${\displaystyle n}$ and ${\displaystyle \lambda ,}$ we may use it to derive the values of ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}'}$ from the empirically determined relation between ${\displaystyle n}$ and ${\displaystyle \lambda .}$ To do this, we must make use again of the general principle that an infinitesimal variation in the type of a vibration, due to a constraint, will not affect the period. If we first consider a certain system of stationary waves, then a system in which the wave-length is greater by an infinitesimal ${\displaystyle dl}$ (the direction of oscillation remaining the same), the period will be increased by an infinitesimal ${\displaystyle dp,}$ and the manner in which the flux distributes itself among the molecules and intermolecular spaces will presumably be infinitesimally changed. But if we suppose that in the second system of waves there is applied a constraint compelling the flux to distribute itself in the same way among the molecules and intermolecular spaces as in the first system (so that ${\displaystyle \xi ',\eta ',\zeta '}$ shall be the same functions as before of ${\displaystyle \xi ,\eta ,\zeta ,}$—a supposition perfectly compatible with the fact that the values of ${\displaystyle \xi ,\eta ,\zeta }$ are changed), this constraint, according to the principle cited, will not affect the period of oscillation. Our equations will apply to such a constrained type of oscillation, and ${\displaystyle {\text{A, B, }}}$ etc., and ${\displaystyle {\text{A}}',{\text{B}}',}$ etc., and therefore ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}',}$ will have the same values in the last described system of waves as in the first system, although the wave-length and the period have been varied. Therefore, in differentiating equation (18), which is essentially an equation between ${\displaystyle l}$ and ${\displaystyle p,}$ or its equivalent (19), we may treat ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}',}$ as constant. This gives

 ${\displaystyle -{\frac {2}{n^{3}}}{\frac {dn}{d\lambda }}={\frac {4\pi {\text{H}}'}{\lambda ^{3}}}\cdot }$
We thus obtain the values of ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}'}$
 ${\displaystyle {\text{H}}'=-{\frac {\lambda ^{3}}{2\pi n^{3}}}{\frac {dn}{d\lambda }},}$⁠${\displaystyle {\text{H}}={\frac {2\pi k^{2}}{n^{2}}}-{\frac {2\pi k^{3}\lambda }{n^{3}}}{\frac {dn}{d\lambda }}\cdot }$ (20)
By determining the values of ${\displaystyle {\text{H}}}$ and ${\displaystyle {\text{H}}'}$ for diflerent directions of oscillation, we may determine the values of ${\displaystyle {\text{A, B, }}}$ etc., and ${\displaystyle {\text{A}}',{\text{B}}',}$ etc.

By means of these equations, the ratios of the statical energy (${\displaystyle {\text{S}}}$), the kinetic energy due to the regular part of the flux (${\displaystyle {\text{T}}}$), and the kinetic energy due to the irregular part of the flux (${\displaystyle {\text{T}}'}$), are easily obtained in a form which admits of experimental determination. EquationS (8) and (9) give

 ${\displaystyle {\text{T}}={\frac {\pi \rho ^{2}l^{2}}{p^{2}}},}$⁠${\displaystyle {\text{T}}'={\frac {2\pi ^{2}{\text{H}}'\rho ^{2}}{p^{2}}}\cdot }$
Therefore, by (20),
 ${\displaystyle {\frac {{\text{T}}'}{\text{T}}}={\frac {2\pi {\text{H}}'}{l^{2}}}={\frac {2\pi {\text{H}}'n^{2}}{\lambda ^{2}}}=-{\frac {\lambda }{n}}{\frac {dn}{d\lambda }}=-{\frac {d\log n}{d\log \lambda }}\cdot }$ (21)
 ${\displaystyle {\frac {\text{S}}{\text{T}}}={\frac {{\text{T}}+{\text{T}}'}{\text{T}}}=1+{\frac {{\text{T}}'}{\text{T}}}={\frac {d\log \lambda -d\log n}{d\log \lambda }}={\frac {d\log l}{d\log \lambda }}\cdot }$ (22)
 ${\displaystyle {\frac {{\text{T}}'}{\text{S}}}=-{\frac {d\log n}{d\log l}}\cdot }$ (23)
Since ${\displaystyle {\text{S, T, }}}$ and ${\displaystyle {\text{T}}'}$ are essentially positive quantities., their ratios must be positive. Equation (21) therefore requires that the index of refraction shall increase as the period or wave-length in vacuo diminishes. Experiment has shown no exceptions to this rule, except such as are manifestly attributable to the absorption of light.

14. It remains to consider the relations between the optical properties of a medium and the planes or axes of symmetry which it may possess. If we consider the statical energy per unit of volume (${\displaystyle {\text{S}}}$) and the period as constant, we may regard equation (2) as the equation of an ellipsoid, the radii vectores of which represent in direction and magnitude the amplitudes of systems of waves having the same statical energy. In like manner, if we consider the kinetic energy of the irregular part of the flux per unit of volume (${\displaystyle {\text{T}}'}$) and the period as constant, we may regard equation (9) as the equation of an ellipsoid, the radii vectores of which represent in direction and magnitude the amplitudes of systems of waves having the same kinetic energy due to the irregular part of the flux. These ellipsoids, which we may distinguish as the ellipsoids (${\displaystyle {\text{A, B, }}}$ etc.) and (${\displaystyle {\text{A}}',{\text{B}}',}$ etc.), as well as the ellipsoid before described, which we may call the ellipsoid (${\displaystyle a,b,}$ etc), must be independent in their form and their orientation of the directions of the axes of coordinates, being determined entirely by the nature of the medium and the period of oscillation. They must therefore possess the same kind of symmetry as the internal structure of the medium.

If the medium is symmetrical about a certain axis, each ellipsoid must have an axis parallel to that. If the medium is symmetrical with respect to a certain plane, each ellipsoid must have an axis at right angles to that plane. If the medium after a revolution of less than 180° about a certain axis is then equivalent to the medium in its first position, or symmetrical with it with respect to a plane at right angles to that axis, each ellipsoid must have an axis of revolution parallel to that axis. These relations must be the same for light of all colors, and also for all temperatures of the medium.

15. From these principles we may infer the optical characteristics of the different crystaDographic systems.

In crystals of the isometric system, as in amorphous bodies, the three ellipsoids reduce to spheres. Such media are optically isotropic at least so far as any properties are concerned which come within the scope of this paper.

In crystals of the tetragonal or hexagonal systems, the three ellipsoids will have axes of rotation parallel to the principal crystallographic axis. Since the ellipsoid (${\displaystyle a,b,}$ etc.) has but one circular section, there will be but one optic axis, which will have a fixed direction.

In crystals of the orthorhombic system, the three ellipsoids will have their axes parallel to the rectangular crystallographic axes. If we take these directions for the axes of coordinates, ${\displaystyle {\text{E, F, G,}}{\text{E}}',{\text{F}}',{\text{G}}',e,f,g}$ will vanish and equation (13) will reduce to

 ${\displaystyle {\text{V}}^{2}={\frac {a\alpha ^{2}+b\beta ^{2}+c\gamma ^{2}}{\rho ^{2}}}\cdot }$
If the coordinate axes are so placed that
 ${\displaystyle a>b>c,}$
the optic axes will lie in the X-Z plane, making equal angles ${\displaystyle \phi }$ with the axis of Z, which may be determined by the equation
 ${\displaystyle \tan ^{2}\phi ={\frac {a-b}{b-c}}={\frac {p^{2}({\text{A}}-{\text{B}})-4\pi ^{2}({\text{A}}'-{\text{B}}')}{p^{2}({\text{B}}-{\text{C}})-4\pi ^{2}({\text{B}}'-{\text{C}}')}}\cdot }$
To get a rough idea of the manner in which ${\displaystyle \phi }$ varies with the period, we may regard ${\displaystyle {\text{A, B, C,}}{\text{A}}',{\text{B}}',{\text{C}}',}$ as constant in this equation.

But since the lengths of the axes of the ellipsoid (${\displaystyle a,b,}$ etc.) vary with the period, it may easily happen that the order of the axes with respect to magnitude is not the same for all colors. In that case, the optic axes for certain colors will lie in one of the principal planes, and for other colors in another. For the color at which the change takes place, the two optic axes will coincide. The differential coefficient ${\displaystyle {\frac {d\phi }{dp}}}$ becomes infinitely great as the optic axes approach coincidence.

In crystals of the monodinic system, each of the three ellipsoids will have an axis perpendicular to the plane of symmetry. We may choose this direction for the axis of X. Then ${\displaystyle >{\text{F, G,}}{\text{F}}',{\text{G}}',f,g}$ will vanish and equation (13) will reduce to

 ${\displaystyle {\text{V}}^{2}={\frac {a\alpha ^{2}+b\beta ^{2}+c\gamma ^{2}+e\beta \gamma }{\rho ^{2}}}\cdot }$
The angle ${\displaystyle \theta }$ made by one of the axes of the ellipsoid (${\displaystyle a,b,}$ etc.) in the plane of symmetry with the axis of Y and measured toward the axis of Z, is determined by the equation
 ${\displaystyle \tan 2\theta ={\frac {e}{c-b}}={\frac {p^{2}{\text{E}}-4\pi ^{2}{\text{E}}'}{p^{2}({\text{C}}-{\text{B}})-4\pi ^{2}({\text{C}}'-{\text{B}}')}}\cdot }$
To get a rough idea of the dispersion of the axes of the ellipsoid (${\displaystyle a,b,}$ etc) in the plane of symmetry, we may regard ${\displaystyle {\text{B, C, E, }}{\text{B}}',{\text{C}}',{\text{E}}',}$ as constant in this equation, and suppose the axis of Y so placed as to make ${\displaystyle {\text{E}}}$ vanish.

It is evident that in this system the plane of the optic axes will be fixed, or will rotate about one of the lines which bisect the angles made by the optic axes, according as the mean axis of the ellipsoid (${\displaystyle a,b,}$ etc.) is perpendicular to the plane of symmetry or lies in that plane. In the first case the dispersion of the two optic axes will be unequal. The same crystal, however, with light of different colors, or at different temperatures, may afford an example of each case.

In crystals of the triclinic system, since the ellipsoids (${\displaystyle {\text{A, B, }}}$ etc.) and (${\displaystyle {\text{A}}',{\text{B}}',}$ etc.) are determined by considerations of a different nature, and there are no relations of symmetry to cause a coincidence in the directions of their axes, there will not in general be any such coincidence. Therefore the three axes of the ellipsoid (${\displaystyle a,b,}$ etc.), that is, the two lines which bisect the angles of the optic axes and their common normal, will vary in position with the color of the light.

16. It appears from this foregoing discussion that by the electromagnetic theory of light we may not only account for the dispersion of colors (including the dispersion of the lines which bisect the angles of the optic axes in doubly refracting media), but may also obtain Fresnel's laws of double refraction for every kind of homogeneous light without neglect of the quantities which determine the dispersion of colors.

But a closer approximation than that of this paper will be necessary to explain the phenomena of circularly polarizing media, which depend on very minute differences of wave-velocity, represented perhaps by a few units in the sixth significant figure of the index of refraction. That the degree of approximation which will give the laws of circular and elliptic polarization will not add any terms to the equations of this paper, except such as vanish for media which do not exhibit this phenomenon, will be shown in another number of this Journal.

1. The fluxes are supposed to be measured by the electromagnetic system of units. It is to be observed that the difference of opinion which has prevailed with respect to the estimation of the energy of electrical currents does not extend to such as are solenoidal, which may be regarded as composed of closed circuits.
2. See Rayleigh's Theory of Sound, voll. i, p. 84.