ON FARADAY'S LINES OF FORCE. 203 



a perfect differential of a function of x, y, z. On the principle of analogy we 

 may call p l the magnetic tension. 



The forces which act on a mass m of south magnetism at any point are 



dp l dp l dp l 



m -P . m -f- , and m -4- . 

 ax ay dz 



in the direction of the axes, and therefore the whole work done during any 

 displacement of a magnetic system is equal to the decrement of the integral 



throughout the system. 



Let us now call Q the total potential of the system on itself. The increase 

 or decrease of Q will measure the work lost or gained by any displacement 

 of any part of the system, and will therefore enable us to determine the 

 forces acting on that part of the system. 



By Theorem III. Q may be put under the form 



= + ITT 1 1 I ( aA 



in which Oj&yj are the differential coefficients of p t with respect to x, y, z 

 respectively. 



If we now assume that this expression for Q is true whatever be the 

 values of Oj, /? y lt we pass from the consideration of the magnetism of permanent 

 magnets to that of the magnetic effects of electric currents, and we have then 

 by Theorem VII. 



Q = 1 1 I Wi - -^ (o, s + A + r c s ) j- dxdydz. 



So that in the case of electric currents, the components of the currents have 

 to be multiplied by the functions a,, ft,, y respectively, and the summations of 

 all such products throughout the system gives us the part of Q due to those 

 currents. 



We have now obtained in the functions a,, /? / the means of avoiding 

 the consideration of the quantity of magnetic induction which passes through 

 the circuit. Instead of this artificial method we have the natural one of con- 

 sidering the current with reference to quantities existing in the same space 

 with the current itself. To these I give the name of Electro-tonic functions, or 

 components of the Electro-tonic intensity. 



2G 2 



