234 COMPARISON OF THE ELECTRIC THEORY OF LIGHT 



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 prin- 

 ciples of harmonic motion! show that to maintain in any element of 



volume the motion represented by 



2 1 



<g = 8le p , (4) 



' / 



21 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 linear functions of the 

 three components of (. We shall represent this force by 



W&dxdydz, (5) 



where "^ represents a complex linear vector f unction. J 



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 _ ^ , ~ 



<!>($ = curl curl (5, (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 (, 



u' y v', w' curl (, 



u", v", w" curl curl (, 



so that , dw dv , du dw , dv du 



ni /}) nil 



U> ^j 7 , V 7 7 , W ^T^ ' 



dy dz dz dx dx 



dw' dv' du' dw' dv' du' 



qj " . _ n\ _ f )l) __ 



dy dz' ~ dz dx ' dx dy ' 



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



* 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 homo- 

 geneous medium. 



tSee Lord Rayleigh's Theory of Sound, vol. i, chapters iv, v. 



: It amounts essentially to the same thing, whether we regard the force as a linear 



* 2 



vector function of (5 or of (5, since these differ only by the constant factor - -^ . But 



there are some advantages in expressing the force as a function of (5, 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 (S. 



