Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/234

It is important to observe that the existence of molecular vibrations of ponderable matter, due to the passage of light through the medium, will not affect the reasoning by which this equation has been established, provided that the nature and intensity of these vibrations in any small part of the medium (as measured by a wave-length) are entirely determined by the electrical forces and motions in that part of the medium. But the equation would not hold in case of molecular vibrations due to magnetic force. Such vibrations would constitute an oscillating magnetization of the medium, which has already been excluded from the discussion.

The supposition which has sometimes been made,^[1] that electricity possesses a certain mass or inertia, would not at all affect the validity of the equation.

10. The equation may be reduced to a form in some respects more simple by the use of the so-called imaginary quantities. We shall write $\iota$ for ${\sqrt {(-1)}}.$ If we differentiate with respect to the time, and substitute $-{\frac {4\pi ^{2}}{p^{2}}}[{\mathsf {U}}]_{\text{Ave}}$ for $[{\ddot {\mathsf {U}}}]_{\text{Ave}},$ we obtain

${\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}[{\dot {\mathsf {U}}}]_{\text{Ave}}-\nabla [{\dot {q}}]_{\text{Ave}}=\Phi [{\dot {U}}]_{\text{Ave}}-{\frac {4\pi ^{2}}{p^{2}}}\Psi [{\mathsf {U}}]_{\text{Ave}}.$

If we multiply this equation by $\iota ,$ either alone or in connection with any real factor, and add it to the preceding, we shall obtain an equation which will be equivalent to the two of which it is formed. Multiplying by $-{\frac {p\iota }{2\pi }}$ and adding, we have

${\begin{aligned}{\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}\left([{\dot {\mathsf {U}}}]_{\text{Ave}}-\iota {\frac {p}{2\pi }}[{\dot {\mathsf {U}}}]_{\text{ave}}\right)&\nabla \left([q]_{\text{Ave}}-\iota {\frac {p}{2\pi }}[{\dot {q}}]_{\text{Ave}}\right)\\&=\left(\Phi +\iota {\frac {2\pi }{p}}\Psi \right)\left([{\mathsf {U}}]_{\text{Ave}}-\iota {\frac {p}{2\pi }}[{\dot {\mathsf {U}}}]_{\text{Ave}}\right)\cdot \end{aligned}}$

If we set

${\mathsf {W}}=[{\mathsf {U}}]_{\text{Ave}}-\iota {\frac {p}{2\pi }}[{\dot {\mathsf {U}}}]_{\text{Ave}},$

(9)

${\text{Q}}=[q]_{\text{Ave}}-\iota {\frac {p}{2\pi }}[{\dot {q}}]_{\text{Ave}},$

(10)

$\Theta =\Phi +\iota {\frac {2\pi }{p}}\Psi ,$

(11)

our equation reduces to

${\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}{\mathsf {W}}-\nabla {\text{Q}}=\Theta {\mathsf {W}}.$

(12)

In this equation $\Theta$ denotes a complex linear vector function, i.e., a vector function of which the X-, Y-, and Z-components are expressed in terms of the X-, Y-, and Z-components of the independent variable by means of coefficients of the form $a+\iota b.$ ${\mathsf {W}}$ is a bivector of which

1. See Weber, Abhandl. d. K. Sächs. Gessellsch. d. Wiss., vol. vi, pp. 593–597; Lorberg, Crelle's Journal, vol.lxi, p. 66.