Ponderomotive and Electromotive Forces. 275 



If we now add the expressions referring to the two electri - 

 cities, we get for the total current-element ds: — 



~dx 

 Potential = hds(c + c^SH^ ^~> 



Force-component = hds(c + c x ) ^- SH^ ^-* ), 



QX qs qs J 



or, if i denotes the product /i(c-fci), which signifies the cur- 

 rent-intensity in ds, 



~dx 

 Potential ~ids2H x ~r- > 



Force-component = ids I ^— XH Z ^- ^-^ ) • 



"We will now denote the potential of the closed current upon 

 the current-element ds by JJds, and the x component of the 

 force undergone by the current-element by ads ; we have 

 then, for the determination of U, taking also into account 

 equations (16), to put 



U-fflB.g-*^^^, . . • (32) 



and regarding this quantity U as a function of x, y, z, 

 Q-) z^-> ^-, we can give to the expression of S the following 

 form : — 



B-^HK-^) (33) 



If, instead of the potential function II corresponding to my 

 fundamental law, the potential-function IJ 1 = U + Gr x or 

 n 2 = II + G-2, corresponding to the fundamental laws of Rie- 

 mann or Weber, be employed, we have only to take also into 

 account separately the additional term G x or Gr a . But this, 



rj rp fill (1 ¥ 



since it is independent of the velocity-components -=-> -£, —-, 



is equal for both the electricities flowing in ds, and hence, after 

 multiplication by lids and —lids, is cancelled in the addition. 

 Accordingly, in regard to the potential of a closed current 

 upon a current-element, and in regard to the ponderomotive 

 force exerted by a closed current upon an element of the cur- 

 rent, there is no difference between the three laws ; equations 

 (32) and (33) hold in all three cases*. 



* I will here incidentally remark that if we had been treating only of 

 the ponderomotive force, and not at the same time of the electromotive 



