134 J. W. Gihbs — Comparison of the Electric Theory of 



since it is the extreme case of the electric theory which we 

 are to compare with the extreme case of the elastic. 



It results from the definitions of curl and p that curl/rQ^O. 

 "We may therefore eliminate Q from equation (8) by taking the 

 curl. This gives 



— curl Pot g=47T curl $$, (9) 



Since curl curl and — Pot are inverse operators for solen- 



oidal vectors, we may get rid of the symbol Pot by taking the 

 curl again. We thus get 



— fj = curl curl $%. (1 0) 



The conditions for the motion at the boundary between dif- 

 ferent media are easily obtained from the following considera- 

 tions. Pot g= and Q are evidently continuous at the interface. 

 Therefore the components parallel to the interface of pQ, and 



by (8) of 0$, will be continuous. Again, curl Pot ^ is con- 

 tinuous at the interface, as appears from the consideration that 



curl Pot $ is the magnetic force due to the electrical motions gv 

 Therefore, by (9), curl <P$ is continuous. The solenoidal con- 

 dition requires that the component of ^ normal to the inter- 

 face 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 ^, > (11) 



all components of curl $$. ) 



Of these conditions, the two relating to the normal components 

 of $ and curl 0$ 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 



5 = ^e, (12) 



e = $%, • (is) 



W = $~ l . (14) 



In other words, the displacements in either theory are subject 

 to the same general and surface conditions as the forces re- 



