ON FARADAY'S LINES OF FORCE. 195 



These equations enable us to deduce the distribution of the currents of 

 electricity whenever we know the values of a, ft, y, the magnetic intensities. 

 If a, /8, y be exact differentials of a function of x, y, z with respect to x, y 

 and z respectively, then the values of a 2 , 6 3> c a disappear ; and we know that the 

 magnetism is not produced by electric currents in that part of the field which 

 we are investigating. It is due either to the presence of permanent magnetism 

 within the field, or to magnetizing forces due to external causes. 



We may observe that the above equations give by differentiation 



= 

 dx dy dz ~ 



which is the equation of continuity for closed currents. Our investigations are 

 therefore for the present limited to closed currents ; and we know little of the 

 magnetic effects of any currents which are not closed. 



Before entering on the calculation of these electric and magnetic states it 

 may be advantageous to state certain general theorems, the truth of which may 

 be established analytically. 



THEOREM I. 

 The equation 



<FV d'V d'V 



(where V and p are functions of x, y, z never infinite, and vanishing for all points 

 at an infinite distance), can be satisfied by one, and only one, value of V. See 

 Art. (17) above. 



THEOREM II. 



The value of V which will satisfy the above conditions is found by inte- 

 grating the expression 



pdxdydz 



f/f_ 



JJJ(x-x 



where the limits of x, y, z are such as to include every point of space where p 

 is finite. 



252 



