276 Prof. R. Clausius on the Determination of the 



§9. 



We now turn to the determination of the electromotive force 

 which is induced in an element of the conductor by a closed 

 current or system of currents. 



For it we have only to determine the component of the force 



force also, the consideration of the subject could have been simplified. 

 That is to say, we obtain for the ponderomotive force, even with single 

 current-elements acting upon one another, expressions which contain not 

 the velocities of the positive and the negative electricity as quantities to 

 be separately dealt with, but only the current-intensity on the whole. 

 According to my fundamental law the expressions for this case have the 

 very same form as for the case in which the current exerting the force is 

 closed. If the potential of the two current-elements ds and ds' upon each 

 other is denoted by uds ds', and the x component of the force which ds 

 suffers from ds' by ^dsds', then we can put 



r ds ds' 

 £_ du _ o / ou \ 

 ox os\^dxJ 



According to Riemann's fundamental law the same expression holds good 

 for the potential ; but the operation to be employed for the derivation of 

 the force-component is somewhat more complicated, namely 



t _ dw_ d / dw \ , d / dw \ 

 ox b s L. (W oV L ox' J 



OS OS 



Finally, according to Weber's fundamental law, for the potential, which 

 in this case may be denoted by u 2 dsds', the equation 



r E 



\ r °s OS' OS os 1 J 



is valid ; and for the derivation of the force- component the same operation 

 as with Riemann's law is to be employed, namely 



ox d«Ud*y + d«' L.dW 



OS OS 



According to this the ponderomotive force can be deduced from the po- 

 tential of each two current-elements upon one another ; but this potential, 

 notwithstanding its partially correspondent form, is clearly to be distin- 

 guished from the quantity which is obtained when, of Neumann's poten- 

 tial of two closed currents upon one another, the part corresponding to 

 two single current-elements ds and ds' is taken. For Neumann's potential 

 is the magnetic potential, and consequently a potential of the same sort as 

 Green's, while the thing in question here is the electro dynamic potential, 

 on which account also an operation quite other than with Greens potential 

 is requisite in order to derive the force-components. 



