132 J. W. Gitibs — Comparison of the Electric Theory of 



linear functions of the three components of (g. We shall 

 represent this force by 



B ¥(k dx dy dz, (5) 



where W 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 sur- 

 rounding ether, we have the equation 



W(k= — curl curl (g, (6) 



which expresses the general law for the motion of monochro- 

 matic 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 m, 



v, w 

 v', w' 



be the components 



u u 



of (g, 



curl (g, 



u", 



-inf 



, v", w" 



i( H 



curl curl (g," 



lac 



, dw 

 dy 



dv 



~ d? 



, du dw 

 ~ dz dx' 



, dv du 

 dx dy' 



' ,, dw' 



dv' 

 ~ dz' 



,, du' dw' 

 ~ dz dx' 



„ dv' du' 

 dx dy 



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

 evident that if u' or v' is discontinuous at the interface, the 

 value of u" or v" becomes in a sense infinite, i. e., curl curl (g, 



and therefore by (6) ¥(&, will be infinite. Now both (g and W 



are discontinuous at the interface, but infinite values for ¥<& 

 are not admissible. Therefore u' and v' are continuous. 

 Again, if u or v is discontinuous, u' or v' will become infinite, 

 and therefore u" or v". Therefore u and v are continuous. 

 These conditions may be expressed in the most general manner 

 by saying that the components of (g and curl (g parallel to the 

 interface are continuous. This gives four complex scalar con- 

 ditions, or in all eight scalar conditions, for the motion at the 

 interface, which are sufficient to determine the amplitude and 



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

 linear vector function of S or of S, since these differ only by the constant factor 



47T 2 



. But there are some advantages in expressing the force as a function of 



£, 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. 



