Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/249

but its position in that plane is different, being perpendicular to the ray instead of to the wave-normal.^[1]

It is the object of this paper to compare this new theory with the electric theory of light. In the limiting cases, that is, when we regard the velocity of the missing wave in the elastic theory as zero, and in the electric theory as infinite, we shall find a remarkable correspondence between the two theories, the motions of monochromatic light within isotropic or aeolotropic media of any degree of transparency or opacity, and at the boundary between two such media, being represented by equations absolutely identical, except that the symbols which denote displacement in one theory denote force in the other, and vice versâ.^[2] In order to exhibit this correspondence completely and clearly, it is necessary that the fundamental principles of the two theories should be treated with the same generality, and, so far as possible, by the same method. The immediate consequences of the new theory will therefore be deduced with the same generality and essentially by the same method which has been used with reference to the electric theory in a former volume of this Journal [page 211 of this volume].

The elastic properties of the ether, according to the new theory, in its limiting case, may be very simply expressed by means of a vector operator, for which we shall use Maxwell's designation. The curl of a vector is defined to be another vector so derived from the first that if $u,v,w$ be the rectangular components of the first, and $u',v',w',$ those of its curl,

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

(1)

where $x,y,z$ are rectangular coordinates. With this understanding, if the displacement of the ether is represented by the vector ${\mathfrak {E}},$ the force exerted upon any element by the surrounding ether will be

$-{\text{B curl curl }}{\mathfrak {E}}\,dx\,dy\,dz,$

(2)

where ${\text{B}}$ is a scalar (the so-called rigidity of the ether) having the same constant value throughout all space, whether ponderable matter is present or not.

Where there is no ponderable matter, this force must be equated to the reaction of the inertia of the ether. This gives, with omission of the common factor $dx\,dy\,dz,$

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

(3)

where ${\text{A}}$ denotes the density of the ether.

1. Sir William Thomson, loc. citat. R. T. Glazebrook, Phil. Mag., Deoember, 1888.
2. In giving us a new interpretation of the equations of the electric theory, the author of the new theory has in fact enriched the mathematical theory of physics with something which may be compared to the celebrated principle of duality in geometry.