Middle of the Nineteenth Century. 243 



tion, for the permanent magnets only, and $ denotes the magnetic 

 potential.* 



Helmholtz, moreover, applied the principle of energy to 

 systems containing electric currents. For instance, when a 

 magnet is moved in the vicinity of a current, the energy taken 

 from the battery may be equated to the sum of that expended 

 as Joulian heat, and that communicated to the magnet by the 

 electromagnetic force : and this equation shows that the current 

 is not proportional to the electromotive force of the battery, 

 i.e. it reveals the existence of Faraday's magneto-electric 

 induction. As, however, Helmholtz was at the time un- 

 acquainted with the conception of the electrokinetic energy 

 stored in connexion with a current, his equations were for the 

 most part defective. But in the case of the mutual action of 

 a current and a permanent magnet, he obtained the correct 

 result that the time-integral of the induced electromotive 

 force in the circuit is equal to the increase which takes 

 place in the potential of the magnet towards a current of a 

 certain strength in the circuit. 



The correct theory of the energy of magnetic and electro- 

 magnetic fields is due mainly to W. Thomson (Lord Kelvin). 

 Thomson's researches on this subject commenced with one or 

 two short investigations regarding the ponderomotive forces 

 which act on temporary magnets. In 1847 he discussed t the 

 case of a small iron sphere placed in a magnetic field, showing 

 that it is acted on by a ponderomotive force represented by 

 - grad cR~, where c denotes a constant, and R denotes the magnetic 

 force of the field ; such a sphere must evidently tend to move 

 towards the places where E' is greatest. The same analysis 

 may be applied to explain why diamagnetic bodies tend to 

 move, as in Faraday's experiments, from the stronger to the 

 weaker parts of the field. 



* We suppose all transitions to be continuous, so as to avoid the necessity for 

 writing surf ace -integrals separately. 



tCamb. and Dub. Matb. Journal, ii (1847), p. 230; W. Thomson's Papers 

 on Electrostatics and Magnetism, p. 499; cf. also Phil. Mag. xxxvii (1850), 

 p. 241. 



R 2 



