CONNECTION OF ELECTRICITY AND MAGNETISM, 249 



Ampere has derived bis well-known law of the attracting or repelling 

 force between two linear elements of electric currents, not, however, 

 without introducing into his reasoning a hypothesis. He assumed, 

 namely, that between every pair of such elements there acts oul3^ one 

 force, not a couple of forces, and that the direction oi this force is the 

 straight line joining the center of the elements. 



Another elementary law was derived from the phenomena of induced 

 currents at first by Gauss, as early as 1835, but not published, and 

 afterward by Prof. F. E. Neumann, (senior,) of Konigsberg, in 1845. 

 The mathematical expression of this law was based on the value of the 

 mechanical work which could be done by Ampere's forces, or, as it was 

 called in analogy with the nomenclature applied by Green and Gauss to 

 magnetic and electro-static forces, on the value of the electro-dynamic 

 potential of two currents. This valae again was determined completely 

 only for closed circuits, but its mathematical expression led to a value 

 of the same quantity also for linear elements, which is much less com- 

 plicated than that of Ampere's forces. To calculate it, take the product, 

 1, of the intensities of the two currents; 2, of the length of both the 

 linear elements; 3, of the cosine of the angle between the directions of 

 the latter, and divide by their distance. 



Taken negatively, this potential expresses the potential energy of the 

 pondero motoric forces which is spent when two currents move without 

 altering their intensity, and the forces can be calculated from the value 

 of this energy by well-known methods. Taken positively, the same, 

 potential gives the value of the energy, which is equivalent to the exist- 

 ing motion of electricity, and which is spent in induced currents if the 

 intensity of the currents or their position in space is altered. In this 

 way this whole chapter of physics, containing the greatest variety of 

 new and surprising phenomena, has been brought under one most 

 simple law. 



As has been said already, the value of the electro-dynamic potential is 

 completely determined for closed circuits, but not for linear elements. 

 To the latter, certain arbitrary functions may be added without altering 

 the potential of closed circuits. Neumann had already remarked this 

 ambiguity. In my first paper I have treated this question, and have 

 striven to find out such consequences of the theory as might lead 

 to an experimental decision of the problematical point. I limited the 

 arbitrariness of the unknown function by the assum^jtion that the 

 law according to which the unknown part of the potential depends 

 on distance is the same as for the known part. Then the Avhole 

 uncertainty is reduced to the value of one unknown constant, which 

 plays in the theory of electricity nearly the same part as the second 

 constant of elasticity in the theory of elastic solids. I was able to 

 decide one important point at least, namely, that this new constant 

 cannot have a negative value without producing an unstable equilib- 

 rium of electricity in conducting bodies. Under such an assumption 



