Intelligence and Miscellaneous Articles. 453 



be carried through that point, the name stream of force (or number 

 of lines of force) passing through that element may designate the 

 product of its surface by the normal component of the force. It 

 is readily seen that the stream of force of a mass m in a cone of 

 angular opening w is equal to mw. 



The energy of a magnetic mass with respect to a current is there- 

 fore equal to the product of the intensity by the stream of force 

 emanating from that mass and passing through the circuit. Let 

 us designate this stream of force by $, and agree to consider it po- 

 sitive when the forces enter the circuit through the negative surface 

 of the current — that is, through the negative side of the equivalent 

 magnetic plate ; the energy of the mass m will have for its expres- 

 sion 



lmw=— I(p. 



Similarly, the energy with respect to a current of any magnetic 

 system whatever is equal to the sum of the energies of all the masses 

 — that is to say, to the product of the intensity of the current by 

 the total stream of force of the system which passes through the 

 circuit. This energy diminishes when the system is left to the ac- 

 tion of the current ; and for a slight displacement the work of the 

 electromagnetic forces is equal to +Icfy. 



If the displacement is effected during the time dt and a liquid- 

 pile is employed, the energy derived from the chemical actions must 

 heat the circuit and furnish the electrodynamic work Id(f> corre- 

 sponding to the increment dcj> of the stream of force, which gives, 

 calling the electromotive force E and the total resistance E, 



mdt=I*Rdt+Id<j>, (1) 



from which is deduced 



IK = E-^ (2) 



dt v ' 



The intensity of the current is the same as if there existed in the 

 circuit a new electromotive force (induction) having the value 



d/f> 



dt 



The electromotive force of induction is therefore equal and of 

 contrary sign to the derivate with respect to the time from the 

 stream of force which emanates from the magnetic system and 

 passes through the circuit. This expression, here deduced from the 

 principle of the conservation of energy, is equivalent to that which 

 Neumann obtained by starting from Lenz's law. 



We shall assume as a general rule that the electromotive force of 

 induction in a circuit is always expressed by equation (3) as a func- 

 tion of the stream of force which passes through it, whatever may 

 be the causes which make the forces vary, such as the displace- 

 ment or modification of a magnetic system, change of form, inten- 

 sity, or position of an exterior current, deformation of the circuit 

 itself, or variation of the current already passing through it. 



