Mr maxwell, ON FARADAY'S LINES OF FORCE. 68 



quantity of magnetic induction acting through a given point, and ai/3i7i are the resolved inten- 

 sities of magnetization at the same point, or, what is the same thing, the components of the 

 force which would be exerted on a unit south pole of a magnet placed at that point without 

 disturbing the distribution of magnetism. 



The electric currents are found from the magnetic intensities by the equations 



flj = —---/- &c. 

 dz ay 



When there are no electric currents, then 



a^dx + fi^dy + y^dz = dp^, 



a perfect differential of a function of *, y, z. On the principle of analogy we may call p^ the 

 magnetic tension. 



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



dpi dpi dp^ 



— m—- , — m —— , and — m -— , 

 die dy 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 



Q = fffpiPidxdydz 

 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 



Q = + — jjjifl\f^i + ^i/^i + Ciyi)d.vdydx, 



in which a, jSj 7, are the differential coefficients of p^ with respact to a?, y, x respectively. 



If we now assume that this expression for Q is true whatever be the values of oi (Bi yu 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. 



So that in the case of electric currents, the components of the currents have to be multiplied 

 by the functions aafioyo 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 oq jS^ 70 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 considering 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. 



