206 CLAUSIUS ON THE WORK PERFORMED 



be provided with a certain quantity of electricity, and conceive 

 a number of such particles, 1, 2, 3, 4, &c., situated in a row ; 

 the motion of electricity may take place in such a manner, that 

 a small quantity will pass over from 1 to 2, an equal but dif- 

 ferent quantity from 2 to 3 ; again, an equal but still a different 

 quantity from 3 to 4, and so forth. For the validity of the fore- 

 going theorem, however, it is of no importance which of these 

 two kinds of motion we assume, for the theorem merely requires 

 that all parts of the entire path be traversed by an equal, but 

 not by the same quantity of electricity. 



By this theorem, it is now easy to determine the work pro- 

 duced within any portion of a conductor traversed by a sta- 

 tionary current during the unit of time. 



Let a closed surface be given, bounding a part of the space 

 filled by the conductor, then we have merely to determine the 

 increase of the potential for every particle of electricity traversing 

 this enclosed space during the unit of time, or, in other words, 

 to multiply the element of electricity by the values of the poten- 

 tial function corresponding to the points of entrance and exit, 

 and then to take the difference of these two products. The sum 

 of all these differences, which gives the required quantity of 

 work, can be conveniently represented in the following manner. 

 Let dw be an element of the surface of the space enclosed, and 

 idw the quantity of electricity passing through the same in the 

 unit of time, which must be taken as positive or negative, accord- 

 ing as it is leaving or entering the space in question ; then, if 

 W represent the work produced within the space, 



W=y\idw, (I) 



where the integration is to be extended over the entire surface. 

 If herein we set, according to equation (1), 



'-^dN' 

 whereby the external direction of the normal is to be considered 

 as positive, then equation (I) can be also written thus : 



dY 



w^kTw 



Immediately connected with these equations are those which 

 express the quantity of heat generated within the enclosed sur- 

 face. 



