Middle of the Nineteenth Century. 249 



electrodynamic energy is being given up, and yet the operator is 

 doing positive work. The explanation of this apparent paradox 

 is that the energy derived from both these sources is being 

 used to save the energy which would otherwise be furnished by 

 the battery, and which is expended in Joulian heat. 



Thomson next proceeded* to show that the energy which is 

 stored in connexion with a circuit in which a current is flowing 

 may be expressed as a volume-integral extended over the whole 

 of space, similar to the integral by which he had already 

 represented the energy of a system of permanent and temporary 

 magnets. The theorem, as originally stated by its author, 

 applied only to the case of a single circuit; but it may be 

 established for a system formed by any number of circuits in 

 the following way : 



If N 8 denote the number of unit tubes of magnetic induction 

 which are linked with the & h circuit, in which a current i s is 

 flowing, the electrokinetic energy of the system is JSJV,^; which 



may be written |2/ r , where / r denotes the total current flowing 



through the gap formed by the r th unit tube of magnetic induc- 

 tion. But if H denote the (vector) magnetic force, and H its 

 numerical magnitude, it is known that (l/4?r) J Hds, integrated 

 along a closed line of magnetic induction, measures the total 

 current flowing through the gap formed by the line. The 

 energy is therefore (l/8?r)S jffds, the summation being extended 

 over all the unit tubes of magnetic induction, and the integra- 

 tion being taken along them. But if dS denote the cross-section 

 of one of these tubes, we have BdS = 1, where B denotes the 

 numerical magnitude of the magnetic induction B : so the energy 

 is (1 1 'Sir) SBdS / Hds ; and as the tubes fill all space, we may 

 replace 'S.dSjds by ^dxdydz. Thus the energy takes the form 

 (l/8?r) JJf BHdxdydz, where the integration is extended over the 

 whole of space ; and since in the present case B = pH, the energy 

 may also be represented by (Il8v)ffffjjrdxdydz. 



* Nichols* Cyclopaedia, 2nd ed., 1860, article " Magnetism, dynamical 

 relations of; " reprinted in Thomson's Papers on Elect, and Mag., p. 447, and 

 his Math, and Phys. Papers, p. 532. 



