On the Fundamental Law of Electrodynamics, 219 



alone satisfy that principle, since the material medium also 

 takes part in the action. For the actions exerted upon one 

 another by closed galvanic currents, however, we may, in ac- 

 cordance with their known laws, expect to find that principle 

 satisfied even without taking account of the presence of a ma- 

 terial substance between them. 



If now the above expressions of X, Y, and Z be multiplied 



successively by -j-y ~> and -^-. and likewise the expressions 



which are formed in a corresponding manner for the compo- 

 nents X', Y / , and Z' of the forces acting on the particle e be 



d 7j du dz 



multiplied by -j-> -j-, and-^-, and then the whole be added 



and the total multiplied by the product ee' and the time-element 

 dt, we shall obtain the expression of the work done during this 

 time-element by the two forces. This expression, if for the 

 present the terms affected by the factor n be omitted, can be 

 brought into the following form, 



-d y [l-k(v 2 + v f2 -vv' cos e)] - ~ d(v 2 + v f2 ). 



The first term in this is a complete differential, thus corre- 

 sponding with the principle of the Conservation of Energy, 

 while the second term does not fulfil this condition. 



But let us now consider two galvanic current-elements, which 

 may move in any way whatever and can be variable in their 

 intensity, we have then to admit that in each of these elements 

 an equal quantity of positive and negative electricity is present. 

 Let us denote these quantities by + e, —e, +e f , —e r , and com- 

 bine + £with +e f , +e with — e', — ewith +/, and — e with 

 — e r ; we have then for each of these four combinations to con- 

 struct an expression of the foregoing form, and to add up the 

 four expressions. Thereby we obtain from the last term (which 

 by taking away the brackets divides into two), in the whole, 

 eight terms, each two of which are equal and opposite, so that 

 collectively they annul one another. The sum then consists 

 therefore of only the four terms corresponding to the first term 

 of the foregoing expression, which, as before said, satisfies the 

 principle of the Conservation of Energy. 



In regard to the terms affected by the factor n, above omitted, 

 these likewise, in the expression of the work which refers 

 to two single current-elements, are in part eliminated; and 

 those that remain become nil by integration with respect to 

 closed currents. 



Accordingly the above equations are, as required by the 



