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

which has a very simple geometrical signification. If we consider the ellipsoid

$ax^{2}+by^{2}+cz^{2}+eyz+fzx+gxy,$

(24)

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

${\begin{aligned}2ax_{1}x_{2}+2by_{1}y_{2}+2cz_{1}z_{2}&+e(y_{1}z_{2}+z_{1}y_{2})\\&+f(z_{1}x_{2}+x_{1}z_{2})+g(x_{1}y_{2}+y_{1}x_{2})=0\end{aligned}}$

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

$a\alpha _{1}^{2}+b\beta _{1}^{2}+c\gamma _{1}^{2}+e\beta _{1}\gamma _{1}+f\gamma _{1}\alpha _{1}+g\alpha _{1}\beta _{1}={\text{U}}_{1}^{2}(\alpha _{1}^{2}+\beta _{1}^{2}+\gamma _{1}^{2}),$

(25)

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

$a\alpha _{2}^{2}+b\beta _{2}^{2}+c\gamma _{2}^{2}+e\beta _{2}\gamma _{2}+f\gamma _{2}\alpha _{2}+g\alpha _{2}\beta _{2}={\text{U}}_{2}^{2}(\alpha _{2}^{2}+\beta _{2}^{2}+\gamma _{2}^{2}).$

(26)

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

${\text{V}}={\frac {l}{p}}\cdot$

(27)

By equations (22), (25), and (26), equations (18) and (20) are reduced to the form

${\text{U}}_{1}^{2}\rho _{1}^{2}\pm {\frac {\phi }{\text{V}}}\rho _{1}\rho _{2}={\text{V}}^{2}\rho _{1}^{2},$${\text{U}}_{2}^{2}\rho _{2}^{2}\pm {\frac {\phi }{\text{V}}}\rho _{1}\rho _{2}={\text{V}}^{2}\rho _{2}^{2},$

(28)

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