2(36 Prof. R. Clausius on the Determination of the 



be taken into consideration separately, are brought into cal- 

 culation ; and hence we get, on the whole, 



k 





and since all the terms herein are differential coefficients with 

 respect to x or /, they can be collected into two differential 

 coefficients. From this equation we get the sought expression 

 of r 2 , namely 



§6. 



In the three preceding sections the x component of the 

 force exerted by a current-element ds f upon a moved unit of 

 electricity is deduced from the three fundamental laws. In 

 each of the three expressions (5), (6), and (8) there is a term 

 which is a differential coefficient with respect to /, and which 

 therefore vanishes in the integration over a closed current sf. 

 Hence the force exerted by a closed current, or even by a 

 system of closed currents, is represented by expressions of 

 simplified form, which we will now consider more closely. 



We start from the expression given in equation (5). When 

 we multiply this by ds f and then integrate it over a closed 

 current or a system of closed currents, we obtain the x com- 

 ponent of that force which, according to my fundamental law, 

 the current, or S} r stem of currents, must exert on a moved unit 

 of electricity. These components being denoted by £ ; we get 



In this equation it is tacitly presupposed that the length of 

 the closed conductor s f remains unchanged, so that those ele- 

 ments dsf which at a given time form the closed conductor 

 form it also during the succeeding time, and no element enters 

 or leaves it. In reality, however, cases may occur in which 

 the length of the conductor changes — for instance, when at a 

 place a sliding of two parts of the conductor on one another 



