236 COMPARISON OF THE ELECTRIC THEORY OF LIGHT 



applied as here to a vector, the three components of the result are to 

 be calculated separately from the three components of the operand. 

 VQ is therefore the electrostatic force, and Pot jf the electro- 

 dynamic force. In establishing the equation, it was not assumed that 

 the electrical motions are aolenoidal, or such as to satisfy the so-called 

 " equation of continuity." We may now, however, make this assump- 

 tion, 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 V that curl VQ = 0. We 

 may therefore eliminate Q from equation (8) by taking the curl. 



This gives 



- curl Pot g = 4<rr curl $$. (9) 



Since curl curl and -r Pot are inverse operators for solenoidal vectors, 



47T 



we may get rid of the symbol Pot by taking the curl again. We 



thus get 



-g = curl curl $$. (10) 



The conditions for the motion at the boundary between different 

 media are easily obtained from the following considerations. Potg- 

 and Q are evidently continuous at the interface. Therefore the 

 components parallel to the interface of VQ, and by (8) of $$, will be 

 continuous. Again, curl Pot is continuous at the interface, as 

 appears from the consideration that curl Pot J is the magnetic force 

 due to the electrical motions g. Therefore, by (9), curl $g i s con- 

 tinuous. The solenoidal condition requires that the component of $ 

 normal to the interface shall be continuous. 



The following quantities are therefore continuous at the interface : 



the components parallel to the interface of 3?$, "j 



the component normal to the interface of fj, (11) 



all components of curl i^.J 



Of these conditions, the two relating to the normal components of 5 

 and curl 3?^ 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 interior 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 



(12) 



(14) 



