BY A STATIONARY ELECTRIC CURRENT. 205 



dq during its motion along the path s. The component of the 



accelerating force in the direction of its motion, will for every 



dY 

 point of its path be represented by -j-, and hence the com- 



dY 

 ponent of the moving foffce acting on dq by dq -r-* The work 



produced by the force during the motion along the element ds 

 will therefore be 



^dq,^.d», (3) 



and consequently, the work produced along the way from Sq to 

 s, will be 



/' 



=dgldV as={Y,-yo)dg;. ... (4) 



ds 



wherein Vj, and Vj denote the values of V corresponding to Sq 

 and s^. 



From this it is at once evident, that this quantity of work is 

 completely defined by the values of the potential function at the 

 extreme points of the path described, without its being neces- 

 sary to know the path followed between these points. Further, 

 the product V. dq is the potential * of the free electricity upon 

 the element dq, so that the foregoing expression represents the 

 increase of the potential between Sq and s^ ; and as a like ex- 

 pression applies to every other element of electricity, and can 

 therefore be extended to a finite quantity of electricity, we can 

 deduce the following theorem : — 



TTie quantity of work produced by a force in the conductor 

 during any definite motion of a quantity of electricity, is equal to 

 the corresponding increase of the potential of this quantity of elec- 

 tricity, and of the free electricity upon each other. 



In this development we have conceived the motion of the 

 electricity to be such, that one and the same quantity of elec- 

 tricity traverses the whole path under consideration ; the actual 

 motion of the electricity, however, may be of a quite different 

 character. For instance, let us assume each particle of mass to 



* I here make the distinction between the terms potential and potential 

 function, introduced in my former memoir (Pogg. Ann. vol. Ixxxvi., pp. 163 

 and 342. Scien. Mem. 2nd Scr. vol. i. part 1). 



