Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/244

connected with the period), because the wave-length is enormously great compared with the size of the molecules and the distances between them.

The potential energy on the elastic theory must be increased by a term of the form $b_{\text{D}}h^{2},$ where $b_{\text{D}}$ is a quadratic function of the direction-cosines of the displacement. For the ponderable particles must oppose a certain elastic resistance to the displacement of the ether, which in æolotropic bodies will presumably be different in different directions. The potential energy on the electric theory will be represented by a single term of the same form, say ${\text{G}}_{\text{D}}h^{2},$ where a quadratic function of the direction-cosines of the displacement, ${\text{G}}_{\text{D}},$ takes the place of the constant ${\text{G}},$ which was sufficient when the ponderable particles were absent. Both ${\text{G}}_{\text{D}}$ and $b_{\text{D}}$ will vary to some extent with the period, like ${\text{A}}_{\text{D}}$ and $f_{\text{D}},$ and for the same reason.

In regard to that potential energy, which on the elastic theory is independent of the direct action of the ponderable molecules, it has been supposed that in sdolotropic bodies the effect of the molecules is such as to produce an asolotropic state in the ether, so that the energy of a distortion varies with its orientation. This part of the potential energy will then be represented by ${\text{B}}_{\text{ND}}{\frac {h^{2}}{l^{2}}},$ where ${\text{B}}_{\text{ND}}$ is a function of the directions of the wave-normal and the displacement. It may easily be shown that it is a quadratic function both of the direction-cosines of the wave-normal and of those of the displacement Also, that if the ether in the body when undisturbed is not in a state of stress due to forces at the surface of the body, or if its stress is uniform in all directions, like a hydrostatic pressure, the function ${\text{B}}_{\text{ND}}$ must be symmetrical with respect to the two sets of direction-cosines.

The equation of energies for the elastic theory is therefore

${\text{A}}_{\text{D}}{\frac {h^{2}}{l^{2}}}={\text{B}}_{\text{ND}}{\frac {h^{2}}{l^{2}}}+b_{\text{D}}h^{2},$

(5)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {{\text{B}}_{\text{ND}}}{{\text{A}}_{\text{D}}-b_{\text{D}}p^{2}}}\cdot$

(6)

The equation of energies for the electrical theory is

${\text{F}}l^{2}{\frac {h^{2}}{p^{2}}}+f_{\text{D}}{\frac {h^{2}}{p^{2}}}={\text{G}}_{\text{D}}h^{2},$

(7)

which gives

${\text{V}}^{2}={\frac {l^{2}}{p^{2}}}={\frac {{\text{G}}_{\text{D}}}{\text{F}}}-{\frac {f_{\text{D}}}{{\text{F}}p^{2}}}\cdot$

(8)

It is evident at once that the electrical theory gives exactly the form that we want. For any constant period the square of the wave-velocity is a quadratic function of the direction-cosines of the displacement. When the period varies, this function varies,