Me maxwell, ON FARADAY'S LINES OF FORCE. 57 



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 



da^ db, dc-i 

 dx dy dz ' 

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

 the present limited to closed currents; and in fact 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 



d^V dW d'V 

 d^v^ + df^^^'^'P''' 

 (where V and p are functions of oryx 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. 



Theoeem II. 



The value of V which will satisfy the above conditions is found by integrating the expression 



pd.vdydx 



ffh 



(x — x'l^ "^ y — y\ "^ ^ — «' 1') 



where the limits of xyz are such as to include every point of space where p is finite. 



The proofs of these theorems may be found in any work on attractions or electricity, and 

 in particular in Green's Essay on the Application of Mathematics to Electricity. See Arts. 

 18, 19 of this Paper. See also Gauss, on Attractions, translated in Tayl.'s-.'s Scientific Memoirs. 



Theorem III. 



Let U and V be two functions oi xyz, then 



rrr..ldPV dW d'V\ , , , rrrldUdV dUdV dU dV\ ^ , , 



jjJ''[dx^'-df^d^r'^^'='-jjJ[-d.^v-'^ 



where the integrations are supposed to extend over all the space in which U and V have values 



differing from (Green, p. 10.) 



This theorem shews that if there be two attracting systems the actions between them are 

 equal and opposite. And by making (7 = F we find that the potential of a system on itself is 

 proportional to the integral of the square of the resultant attraction through all space; a 

 Vol. X. Part I. 8 



