190 ON FARADAY'S LINES OF FORCE. 



In order to express mathematically the electrical currents in any conductor, 

 we must have a definition, not only of the entire flow across a complete section, 

 but also of the flow at a given point in a given direction. 



DEF. The quantity of a current at a given point and in a given direction 

 is measured, when uniform, by the quantity of electricity which flows across 

 unit of area taken at that point perpendicular to the given direction, and when 

 variable by the quantity which would flow across this area, supposing the flow 

 uniformly the same as at the given point. 



In the following investigation, the quantity of electric current at the point 

 (xyz) estimated in the directions of the axes x, y, z respectively will be denoted 

 by a,, 6,, c t . 



The quantity of electricity which flows in unit of time through the ele- 

 mentary area dS 



= dS (la t + mb t + nc t ), 

 where I, m, n are the direction-cosines of the normal to dS. 



This flow of electricity at any point of a conductor is due to the electro- 

 motive forces which act at that point. These may be either external or internal. 



External electro- motive forces arise either from the relative motion of currents 

 and magnets, or from changes in their intensity, or from other causes acting 

 at a distance. 



Internal electro-motive forces arise principally from difference of electric 

 tension at points of the conductor in the immediate neighbourhood of the point 

 in question. The other causes are variations of chemical composition or of tem- 

 perature in contiguous parts of the conductor. 



Let p t represent the electric tension at any point, and X v Y t , Z t the sums 

 of the parts of all the electro-motive forces arising from other causes resolved 

 parallel to the co-ordinate axes, then if a,, /3,, y a be the effective electro-motive 

 forces 



dx 



(A). 



