Light and the Theory of a Quasi-labile Ether. 131 



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 rectan- 

 gular components of the first, and u', v', w l , those of its 

 curl, 



, _ dw _ dv i __ du dw , _ dv du /,,. 

 dy dz' dz dx dx dy 



where x, y, z are rectangular coordinates. With this under- 

 standing, if the displacement of the ether is represented by 

 the vector (5, the force exerted upon any element by the sur- 

 rounding ether will be 



— B curl curl (g dx dy dz, (2) 



where B is a scalar (the so-called rigidity of the ether) having 

 the sanie constant value throughout all space, whether ponder- 

 able 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, 



A(g = — B curl curl (£, (3) 



where A denotes the density of the ether. 



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.* ISTow the general principles of harmonic mo- 

 tionf show that to maintain in any element of volume the 

 motion represented by 



2tu- 



&=%e p , (4) 



91 being a complex vector constant, will require a force from 

 outside represented by a complex linear vector function of (§, 

 that is, the three components of the force will be complex 



* This is in no respect different from what is always tacitly understood in the 

 theory of sound, where the displacements, velocities, densities considered are al- 

 ways such average values. But in the theory of light, it is desirable to have the 

 fact clearly in mind on accouut 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. 



f See Lord Rayleigh's Theory of Soimd, vol i, chapters iv, v. 



