Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/242

vacuum. If we write $h$ for the amplitude measured in the middle between two nodal planes, the velocities of displacement will be as ${\frac {h}{p}},$ and the kinetic energy will be represented by ${\text{A}}{\frac {h^{2}}{p^{2}}},$ where ${\text{A}}$ is a constant depending on the density of the medium. The potential energy, which consists in distortion of the medium, may be represented by ${\text{B}}{\frac {h^{2}}{l^{2}}},$ where ${\text{B}}$ is a constant depending on the rigidity of the medium. The equation of energies, on the elastic theory, is therefore

${\text{A}}{\frac {h^{2}}{p^{2}}}={\text{B}}{\frac {h^{2}}{l^{2}}}$

(1)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {\text{B}}{\text{A}}}\cdot$

(2)

In the electrical theory, the kinetic energy is not determined by the simple formula of ordinary dynamics from the square of the velocity of each element, but is found by integrating the product of the velocities of each pair of elements divided by the distance between them. Very elementary considerations suffice to show that a quantity thus determined when estimated per unit of volume will vary as the square of the wave-length. We may therefore set ${\text{F}}l{\frac {h^{2}}{p^{2}}}$ for the kinetic energy, ${\text{F}}$ being a constant. The potential energy does not consist in distortion of the medium, but depends upon an elastic resistance to the separation of the electricities, which constitutes the electrical displacement, and is proportioned to the square of this displacement. The average value of the potential energy per unit of volume will therefore be represented in the electrical theory by ${\text{G}}h^{2}$ where ${\text{G}}$ is a constant, and the equation of energies will be

${\text{F}}l{\frac {h^{2}}{p^{2}}}={\text{G}}h^{2}$

(3)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {\text{G}}{\text{F}}}\cdot$

(4)

Both theories give a constant velocity, as is required. But it is instructive to notice the profound difference in the equations of energy from which this result is derived. In the elastic theory the square of the wave-length appears in the potential energy as a divisor; in the electrical theory it appears in the kinetic energy as a factor.

Let us now consider how these equations will be modified by the presence of ponderable matter, in the most general case of transparent and sensibly homogeneous bodies. This subject is rendered much more simple by the fact that the distances between the ponderable molecules are very small compared with a wave-length. Or, what