248 The Mathematical Electricians of the 



or, integrating by parts, 



(!, . grad <) dx dy dz, or - J (H . I ) dx dy dz. 



Since B = yu,H + 47rI , this expression may be written in the 

 form 



- (H. 

 offJJJ 



but the former of these integrals is equivalent to 



>fff 

 (B . grad <) dx dydz, or - < div B dx dy dz, 



which vanishes, since B is a circuital vector. The energy of the 

 field, therefore, reduces to 

 1 

 BIT, 



integrated over all space; which is equivalent to Thomson's 

 form.* 



In the same memoir Thomson returned to the question of the 

 energy which is possessed by a circuit in virtue of an electric 

 current circulating in it. As he remarked, the energy may 

 be determined by calculating the amount of work which 

 must be done in and on the circuit in order to double the 

 circuit on itself while the current is sustained in it with 

 constant strength; for Faraday's experiments show that a 

 circuit doubled on itself has no stored energy. Thomson found 

 that the amount of work required may be expressed in the form 

 \Li*, where i denotes the current strength, and L, which is 

 called the coefficient of self-induction^ depends only on the form of 

 the circuit. 



It may be noticed that in the doubling process the inherent 



* The form actually given by Thomson was 



fff (*E? lA d-d 

 Sir}}} \:-.*) 



which reduces to the above when we neglect that part of I 2 which is due to the 

 permanent magnetism, over which we have no control. 



