FUNDAMENTAL PRINCIPLES OF MATHEMATICS. Ill 



indeed, be shown that no reversible process can be derived from 

 Weber's law by which work could be done without expenditure of 

 energy. Besides the hypothesis of Weber, based on the action between 

 electrically- charged points, there was the older one of F. E. Neumann, 

 which considered not the action of one charged point upon another, but 

 of one linear current element on another. This was regarded by Helin- 

 holtz as one of the happiest and most fruitful conceptions which the 

 newer mathematical physics has produced. The law which can be 

 deduced from the hypothesis of Weber for the mutual action of two 

 linear current elements differs from the Neumann potential law, and 

 Helmholtz found himself, in the course of his investigations, confronted 

 with the question whether this hypothesis did in fact represent the 

 true state of affairs and how both the law of Weber and that of Neu- 

 mann were related to the laws of Maxwell, which I shall soon have 

 occasion to mention. 



He found that all these laws may be reduced to a common form and 

 differ then only in the values of a constant which appears. All phe- 

 nomena which are presented by the circulation of closed currents 

 through metallic circuits can be accounted for equally well under any 

 of the several hypotheses; but in the case of incomplete circuits they 

 lead to considerably different consequences. For the purpose of decid- 

 ing between these hypotheses, Helmholtz developed, with the help of 

 his generalized induction law, the equations of motion of electricity in 

 an extended conducting solid. He found that for a negative value of 

 the undetermined constant, such as is required by the assumptions of 

 Weber, a condition of neutral equilibrium of the electricity results, and 

 thus there may be generated currents of infinite strength and the con- 

 dition of infinite electrical density. The value zero required by Max- 

 well and the positive value required by Neumann for this constant do 

 not, on the contrary, lead to these difficulties. These conclusions were 

 subjected to many attacks, and their opponents sought both theoretic- 

 ally and experimentally to show that the hypothesis proposed by E. 

 Neumann and extended by Helmholtz to form the fundamental law of 

 electro- dynamic phenomena was incompatible with observation. There 

 is, in fact, a difference between the potential law of E. Neumann for 

 closed circuits when applied to incomplete circuits and the form of the 

 induction law which Helmholtz had earlier derived. For the potential 

 law ascribes electro-dynamic effect only to currents of electricity and 

 their action at a distance, and not to the electrical charges put in 

 motion with the conductors. Experiments show that this assumption 

 is contradicted by the fact. 



With wonderful acuteness of perception Helmholtz had from the 

 start seen that the solution of all these questions could only be accom- 

 plished by the very difficult experimental investigation of incomplete 

 circuits, and he was indeed uncertain that a solution would ever be 

 obtained, since there might be no incomplete circuits, as the insulator 



