Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/216

according as a revolution from $\rho _{1}$ to $\rho _{2}$ appears clockwise or counter-clockwise to one looking in the direction of the wave-normal. Since $\rho _{1}$ and $\rho _{2}$ are determined by displacements in planes one-quarter of a wave-length distant from each other, and the plane to which the latter relates lies on the side toward which the wave-normal is drawn, it follows that $\Theta$ is positive or negative according as the combination of displacements has the character of a right-handed or a left-handed screw.

10. The kinetic energy of the medium, which is to be estimated for an instelnt of no displacement, may be shown as in § 7 of the former paper (page 185 of this volume) to consist of two parts, of which one relates to the regular flux (${\dot {\xi }},{\dot {\eta }},{\dot {zeta}}$), and the other to the irregular flux (${\dot {\xi '}},{\dot {\eta '}},{\dot {zeta'}}$). The first, in the notation of that paper, is represented by

${\tfrac {1}{2}}\int ({\dot {\xi }}{\text{ Pot }}{\dot {\xi }}+{\dot {\eta }}{\text{ Pot }}{\dot {\eta }}+{\dot {\zeta }}{\text{ Pot }}{\dot {\zeta }})dv,$

which reduces to

${\frac {l^{2}}{2\pi }}\int ({\dot {\xi }}^{2}+{\dot {\eta }}^{2}+{\dot {\zeta }}^{2})dv.$

By substitution of the values given by equations (1), we obtain for the kinetic energy due to the regular flux in a unit of volume

${\text{T}}={\frac {\pi l^{2}}{p^{2}}}(\alpha _{1}^{2}+\beta _{1}^{2}+\gamma _{1}^{2}+\alpha _{2}^{2}+\beta _{2}^{2}+\gamma _{2}^{2}).$

(10)

11. The kinetic energy of the irregular part of the flux is represented by the volame-integral

${\tfrac {1}{2}}\int ({\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}+{\dot {\eta '}}{\text{ Pot }}{\dot {\eta '}}+{\dot {\zeta '}}{\text{ Pot }}{\dot {\zeta '}})dv.$

Now, since ${\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}}$ are everywhere linear functions of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and diff. coeff. (see § 4), and since the integrations implied in the notation $Pot$ may be confined to a sphere of which the radius is small in comparison with a wave length,^[1] and since within such a sphere ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and diff. coeff. are sufficiently determined (in a linear form), by the values of the same twelve quantities at the center of the sphere, it follows that ${\text{Pot }}{\dot {\xi '}},{\text{Pot }}{\dot {\eta '}},{\text{Pot }}{\dot {\zeta '}}$ must be linear functions of the values of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and diff. coeff. at the point for which the potential is sought. Hence,

${\tfrac {1}{2}}({\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}+{\dot {\eta '}}{\text{ Pot }}{\dot {\eta '}}+{\dot {\zeta '}}{\text{ Pot }}{\dot {\zeta '}})$

will be a quadratic function of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and diff. coeff. But the seventy-eight coefficients by which this function is expressed will vary with the position of the point considered with respect to the surrounding molecules.

1. See § 9 of the former paper, on page 187 of this volume.