1S5 



ately connected with the quantity of electric currents, it follows that 

 the presence of paramagnetic bodies, like iron, will, by diminishing 

 the total resistance to magnetic induction while the total intensity 

 is constant, increase its quantity. Hence the increase of external 

 effect due to the introduction of a core of soft iron into an electric 

 helix. 



From the researches of Faraday into the induction of electric cur- 

 rents by changes in the magnetic field, it appears that a conductor, 

 in cutting the lines of magnetic force, experiences an electromotive 

 force, tending to produce a current perpendicular to the lines of 

 motion and of magnetic force, and depending on the number of lines 

 cut by the conductor in its motion. 



It follows that the total electromotive force in a closed circuit is 

 measured by the rate of change of the number of lines of magnetic 

 force which pass through it ; and it is indifferent whether this change 

 arises from a motion of this circuit, or from any change in the mag- 

 netic field itself, due to changes of intensity or position of magnets 

 or electric currents. 



This law, though it is sufficiently simple and general to render 

 intelligible all the phenomena of induction in closed circuits, con- 

 tains the somewhat artificial conception of the number of iinec vass- 

 ing through the circuit, exerting a physical influence on it. It 

 would be better if we could avoid, in the enunciation of the law, 

 making the electromotive force in a conductor depend upon lines of 

 force external to the conductor. Now the expressions which we 

 obtained for the connexion between magnetism and electric currents 

 supply us with the means of making the law of induced currents 

 depend on the state of the conductor itself. 



We have seen that from certain expressions for magnetic intensity 

 we could deduce those for the quantity of currents, so that the cur- 

 rents which pass through a given closed curve may be measured by 

 the total magnetic intensity round that curve. Here we have an 

 integration round the curve itself instead of one over the enclosed sur- 

 face. In the same way, if we assume the mathematical existence of 

 a state, bearing the same relation to magnetic quantity that mag- 

 netic intensity bears to electric quantity, we shall have an expression 

 for the quantity of magnetic induction passing through a closed cir- 

 cuit in terms of quantities depending on the circuit itself, and not 

 on the enclosed space. 



Let us therefore assume three functions of xy z, a j3 y , such that 

 a x b x c x being the resolved parts of magnetic quantity, 



dz dy ' dx dz ' dy dx' 



then it will appear that if we assume -—2, —^-, —~- as the expres- 

 ™ dt dt dt r 



sions for the electromotive forces at any point in the conductor, the 

 total electromotive force in any circuit will be the same as that ex- 

 pressed by Faraday's law. Now as we know nothing of these in- 

 ductive effects except in closed circuits, these expressions, which are 



