202 ON FARADAY'S LINES OF FORCE. 



For, if we put a, in the form 



dz dy dx ' 



and treat 6, and c, similarly, then we have by integration by parts through 

 infinity, remembering that all the functions vanish at the limits, 



+$-)}*** 



or Q= 



and by Theorem III. 



/// Vp' dxdydz = \\\ppdxdydz, 

 so that finally 



Q = Hl^irpp - (a^a, + /S.6, + y c,)} dxdydz. 



If a,6,c, represent the components of magnetic quantity, and aj8,y, those 

 of magnetic intensity, then p will represent the real magnetic density, and p 

 the magnetic potential or tension. a,6,c, will be the components of quantity 

 of electric currents, and cg8 y will be three functions deduced from ef,6,c,, 

 which will be found to be the mathematical expression for Faraday's Electro- 

 tonic state. 



Let us now consider the bearing of these analytical theorems on the 

 theory of magnetism. Whenever we deal with quantities relating to magnetism, 

 we shall distinguish them by the suffix (,). Thus a,?>,c, are the components 

 resolved in the directions of x, y, z of the quantity of magnetic induction acting 

 through a given point, and Oj&y, are the resolved intensities 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 



When there are no electric currents, then 



