Ponder omotive and Electromotive Forces. 279 



ties contained in the sums (H y and H z ) their values deter- 

 mined by equations (16), we get 



@= 4 W 2 1£ |^_* | f £ 2 |f |* *,. . (38 ) 

 os J r Qt os ot J r ds os 



This is the most convenient form of the expression of © result- 

 ing from my fundamental law ; and the product (&ds is the 

 electromotive force induced in a conductor-element ds by a 

 closed current or system of currents. 



To obtain the corresponding expressions for Eiemann and 

 Weber's fundamental laws, we need only in the formulae (28) 

 and (29), representing the potential function, to take sepa- 

 rately into consideration the added terms Gi and G 2 , which 



do not contain the velocity-components ~y ~j ~, and there- 

 fore are to be differentiated only with respect to x 3 y, z. As 

 we can now again form for Grj the equation 



oGio^ [ oGioy agijfr^agk 



~ox ~bs By o« 0-2 os os 



and for G 2 the corresponding equation, we obtain, denoting 

 the electromotive force according to Eiemann and Weber's 

 fundamental laws by ©i and @ 2 : — 



©i=©+^p (39) 



®» = @+^T (40) 



These expressions represent very clearly the difference be- 

 tween the electromotive forces resulting from the three fun- 

 damental laws. 



From the developments carried out in the last two sections 

 it will, I think, be sufficiently apparent how much the intro- 

 duction of the electrodynamic potential function of closed 

 currents contributes to giving to the entire department of 

 electrodynamics with which we are concerned a uniform cha- 

 racter — the knowledge of that one quantity being sufficient, 

 without any accessory assumption, for the derivation of every 

 thing further by simple analytical operations. 



