Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/220

204 which has a very simple geometrical signification. If we consider the ellipsoid and especially its central section by a plane parallel to the planes of the wave-system which we are considering, it will easily appear that the equation  will hold of any two points $$x_{1}, y_{1}, z_{1}$$ and $$x_{2}, y_{2}, z_{2}$$ which belong to conjugate diameters of this central section. Therefore equation (23) expresses that the displacements $$\alpha_{1}, \beta_{1}, \gamma_{1}$$ and $$\alpha_{2}, \beta_{2}, \gamma_{2}$$ parallel to conjugate diameters of the central section of the ellipsoid (24) by a wave-plane. But since the displacements $$\alpha_{1}, \beta_{1}, \gamma_{1}$$ and $$\alpha_{2}, \beta_{2}, \gamma_{2}$$ are also at right angles to each other, it follows that they are parallel to the axes of the central section of the ellipsoid (24) by a wave-plane. That is:—The axes of the displacement-ellipse coincide in direction with those of a central section of the ellipsoid (24) by a wave-plane.

16. If we write $$\text{U}_{1}, \text{U}_{2}$$ for the reciprocals of the semi-axes of the central section of the ellipsoid (24) by a wave-plane, $$\text{U}_{1}$$ being the reciprocal of the one to which the displacement $$\alpha_{1}, \beta_{1}, \gamma_{1}$$ is parallel, we have as is at once evident if we substitute the coordinates of an extremity of the axis for the proportional quantities $$\alpha_{1}, \beta_{1}, \gamma_{1}.$$ So also  If we write $$\text{V}$$ for the velocity of propagation of the system of progressive waves corresponding to the system of stationary waves which we have been considering, we shall have  By equations (22), (25), and (26), equations (18) and (20) are reduced to the form  where we are to read $$+$$ or $$-$$ according as the disturbance has the character of a right-handed or a left-handed screw. In a progressive system of waves, when the combination of displacements has the character of a right-handed screw, the rotations will be such as appear clockwise to the observer, who looks in the direction opposite to that of the propagation of light. We shall call such a ray right-handed.

We may here observe that in case $$\phi = 0$$ the solution of these