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

200 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 which reduces to  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  11. The kinetic energy of the irregular part of the flux is represented by the volame-integral 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, 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, 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.