219, 220] ELECTROMAGNETISM. 429 



The second factor in the second integral may be written 



and we may then perform the integration around the circuit 1, 

 integrating by parts, obtaining 



r 

 I 



^ cj 



= (c?# 2 &PI + dy 2 oyi + 

 T 



' . -L- 



The integrated part vanishes, for the factors 8^, 8y 1? 8^ are the 

 same for the beginning and end of the circuit. Accordingly the 

 expression for the work becomes 



denoting the change made by changing x^y-^z^ keeping x^y^z^ 

 constant. We have accordingly obtained the work as the change 

 due to the motion in the value of a line integral around both 

 circuits. Consequently the mechanical forces are derivable from a 

 force-function, and the integral represents the negative mutual 

 potential energy due to the magnetic forces acting between the 

 two currents. 



( 1 1 ) - W = /!/ 2 I I - (dxidx* + cfa/icfa/ 2 + dz l 



r T [ f 

 J J 



cos 



This form of the integral was given by Franz Emil Neumann* in 

 1845 and is generally known in Germany by the name of the 

 Electrodynamic Potential of the currents 1 and 2. 



* Neumann, "Allgemeine Gesetze der inducirten Strome." Abh. Berl. ATcad. 

 1845. 



