566 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 



or 



Similarly, Y= par' /iyp' 



(J). 



These are the equations which determine the mechanical force acting on a 

 conductor carrying a current. The force is perpendicular to the current and 

 to the lines of force, and is measured by the area of the parallelogram formed 

 by lines parallel to the current and lines of force, and proportional to their 

 intensities. 



Mechanical Force on a Magnet. 



(77) In any part of the field not traversed by electric currents the dis- 

 tribution of magnetic intensity may be represented by the differential coefficients 

 of a function which may be called the magnetic potential. When there are no 

 currents in the field, this quantity has a single value for each point. When 

 there are currents, the potential has a series of values at each point, but its 

 differential coefficients have only one value, namely, 



dd> d4> dd> 



-r-=a, -j- = p, -f = y- 

 dx dy dz r 



Substituting these values of a, $, y in the expression (equation 38) for the 

 intrinsic energy of the field, and integrating by parts, it becomes 



+ + 



dy dz 



The expression 2 + - + - dV^mdV .. .. (39) 



\ dx dy dz / 



indicates the number of lines of magnetic force which have their origin within 

 the space V. Now a magnetic pole is known to us only as the origin or 

 termination of lines of magnetic force, and a unit pole is one which has 4n- 

 lines belonging to it, since it produces unit of magnetic intensity at unit of 

 distance over a sphere whose surface is 47r. 



Hence if m is the amount of free positive magnetism in unit of volume, 

 the above expression may be written i-rrm, and the expression for the energy 

 of the field becomes 



(40). 



