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

234 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. Now the general principles of harmonic motion show that to maintain in any element of volume the motion represented by $$\mathfrak{A}$$ 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  where $$\Psi$$ represents a complex linear vector function. 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 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. so that

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