242 Prof. J. W. Gibbs's Comparison of the Electric Theory 



which have been used to establish equation (6), except that 

 the ordinary law of electrodynamic induction had the place of 

 the new law of elasticity. ^ is a complex vector representing 

 the electrical displacement as a harmonic function of the time ; 

 $ is a complex linear vector-operator, such that 47r$^ repre- 

 sents the electromotive force necessary to keep up the vibra- 

 tion %. Q is a complex scalar representing the electrostatic 

 potential ; VQ the vector of which the three components are 



dQ, dQ, dQ, 



dx ' dy ' dz ' 



Pot denotes the operation by which, in the theory of gravi- 

 tation, the potential is calculated from the density of matter*. 

 When it is 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 electro- 

 static force, and — Pot g the electrodynamic force. In 

 establishing the equation, it was not assumed that the elec- 

 trical motions are solenoidal, or such as to satisfy the so-called 

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

 assumption, 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) hy taking 

 the curl. This gi/es 



- curl Pot § = 47r curl <I)g (9) 



Since curl curl and j- Pot are inverse operators for sole- 

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

 the curl again. We thus get 



-g= curl curing (10) 



The considerations for the motion at the boundary between 

 different media are easily obtained from the following consi- 

 derations. Pot % and Q are evidently continuous at the 

 interface. Therefore the components parallel to the interface 

 of VQj 3,nd, by (8), of <E>^, will be continuous. Again, 

 curl Pot § is continuous at the interface, as appears from the 

 consideration that curl Pot § is the magnetic force due to the 

 electrical motions §. Therefore, by (9j, curl <I>5 is con- 



* The symbol —Pot is therefore equivalent to 47rv-2, as used by Sir 

 William Thomson (with a happy economy of symbols) at the last meeting 

 of the British Association to express the same law of electrodynamic 

 induction, except that the symbol is here used as a vector operator. See 

 * Nature,' vol. xxxviii. p. 571, sub. init. 



