of Light and the Theory of a Quasi-labile y^ther. 243 



tinuous. The solenoidal condition requires that the component 

 of 5 normal to the interface shall be continuous. 



The following quantities are therefore continuous at the 

 interface : — 



] 



the components parallel to the interface of ^%, 



the component normal to the interface of %, Y (11) 



all components of curl <J>5. 



Of these conditions, the two relating to the normal compo- 

 nents of ^ and curl $5 are easily shown to result from the 

 other four conditions, as in the analogous case in the elastic 

 theory. 



If we now compare in the two theories the differential 

 equations of the motion of monochromatic light, for the in- 

 terior of a sensibly homogeneous medium, (6) and (10), and 

 the special conditions for the boundary between two such 

 media as represented by the continuity of the quantities (7) 

 and (11), we find that these equations and conditions become 

 identical, if 



8=^(?, (12) 



(S=n, (13) 



^ = ^-' (14) 



In other words^ the displacements in either theory are subject 

 to the same general and surface conditions as the forces 

 required to maintain the vibrations in an element of volume 

 in the other theory. 



To fix our ideas in regard to the signification of ^ and <I>, 

 we may consider the case of isotropic media, in which these 

 operators reduce to ordinary algebraic quantities, simple or 

 complex. Now the curl of any vector necessarily satisfies the 

 solenoidal condition (the so-called " equation of continuity "), 

 therefore by (6) "^(^ and ® will be solenoidal. So also will ^ 

 and O^ in the electrical theory. Now for solenoidal vectors, 



d^ d^ d^ 

 - curl curl = ^,+^^, + ^-,; ;. . (15) 



so that the equations (6) and (10) reduce to 



*^=(£ + $ + £)«. • • • • (i«) 



112 



