Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/250

The presence of ponderable matter disturbs the motions of the ether, and renders them too complicated for us to follow in detail. Nor is this necessary, for the quantities which occur in the equations of optics represent average values, taken over spaces large enough to smooth out the irregularities due to the ponderable particles, although very small as measured by a wave-length.^[1] Now the general principles of harmonic motion^[2] show that to maintain in any element of volume the motion represented by

${\mathfrak {E}}={\mathfrak {A}}e^{2\pi \iota {\frac {t}{p}}},$

(4)

${\mathfrak {A}}$ being a complex vector constant, will require a force from outside represented by a complex linear vector function of ${\mathfrak {E}}$ that is, the three components of the force will be complex linear functions of the three components of ${\mathfrak {E}}.$ We shall represent this force by

${\text{B}}\Psi {\ddot {\mathfrak {E}}}\,dx\,dy\,dz,$

(5)

where $\Psi$ represents a complex linear vector function.^[3]

If we now equate the force required to maintain the motion in any element to that exerted upon the element by the surrounding ether, we have the equation

$\Psi {\ddot {\mathfrak {E}}}=-{\text{curl curl }}{\mathfrak {E}},$

(6)

which expresses the general law for the motion of monochromatic light within any sensibly homogeneous medium, and may be regarded as implicitly including the conditions relating to the boundary of two such media, which are necessary for determining the intensities of reflected and refracted light.

For let	$u,v,w$	be the	components of	${\mathfrak {E}},$
	$u',v',w'$	"	"	${\text{curl }}{\mathfrak {E}},$
	$u'',v'',w''$	"	"	${\text{curl curl}}{\mathfrak {E}},$

so that

$u'$	$={\frac {dw}{dy}}-{\frac {dv}{dz}},$	$v'$	$={\frac {du}{dz}}-{\frac {dw}{dx}},$	$w'$	$={\frac {dv}{dx}}-{\frac {du}{dy}},$
$u''$	$={\frac {dw'}{dy}}-{\frac {dv'}{dz}},$	$v''$	$={\frac {du'}{dz}}-{\frac {dw'}{dx}},$	$w''$	$={\frac {dv'}{dx}}-{\frac {du'}{dy}},$

and let the interface be perpendicular to the axis of Z. It is evident

1. This is in no respect different from what is always tacitly understood in the theory of sound, where the displacements, velocities, densities considered are always such average values. But in the theory of light, it is desirable to have the fact clearly in mind on account of the two interpenetrating media (imponderable and ponderable), the laws of light not being in all respects the same as they would be for a single homogeneous medium.
2. See Lord Rayleigh's Theory of Sound, vol. i, chapters iv, v.
3. It amounts essentially to the same thing, whether we regard the force as a linear vector function of ${\mathfrak {E}}$ or of ${\ddot {\mathfrak {E}}},$ since these di£fer only by the constant factor $-{\frac {4\pi ^{2}}{p^{2}}}$ But there are some advantages in expressing the force as a function of ${\ddot {\mathfrak {E}}}$ because the greater part of the force, in the most important cases, is required to overcome the inertia of the ether, and is thus more immediately connected with ${\ddot {\mathfrak {E}}}.$