534 M. Helmholtz on the Theory of Electrodynamics. 



Under tliese circumstances^ however,, the values of the forces 

 must necessarily contain second diirerential quotients of the co- 

 ordinates, because the sum of the force-components correspond- 

 ing to the individual points and axes of coordinates, each mul- 

 tiplied by the corresponding component of the velocity, must be 

 equal to the differential quotient of the potential energy, taken 

 according to the time ; and the latter, under the condition pre- 

 supposed, necessarily contains also the second differential quo- 

 tients of the coordinates according to the time. 



In relation to complete cyclical processes, M. "W. Weber* has 

 proved that his assumption concerning the value of the electric 

 forces admits no production of work without a corresponding 

 expenditure of forces capable of producing it. 



In another place, applications v.^hich I endeavoured to make 

 of the differential equations deduced by Kirchhoff from Weber^s 

 assum^ption had led me to the discovery that they corresponded 

 to a state of unstable equilibrium of the electricity in conductors, 

 and that, according to them, currents might be developed vvhich 

 v>^ould lead to infinite current-intensities and infinite electric 

 densities. 



Replies by MM. Y/. Weber and C. Neumann have induced me 

 to resume and generalize these investigations, the results of 

 which 1 will here briefly lay before the Academy-]-. 



If we have any number, however great, of mass-points, the 

 inert mass of which ^may be denoted by /x„, and all or some of 

 which contain quanta of electricity, which, measured according 

 to electrostatic measure, may be denoted by Cn — i^ further, rnm 

 is the distance between the points n and m, and q^ the resulting 

 velocity of the point w, ^nm the angle which it makes wdth the 

 direction of the line rnm prolonged beyond w, then the value 



(1) of the electrostatic potential 



L ^ 7im J 



(2) of the electrodynamic potential 



Q= ~ S2 [^ .?„.?„. cos (»„,„) . cos(S,„)l • 



CC \_ I nm -J 



We put, further, the quantity 



* " Electrodynamische Maassbestimmungen, insbesondere iiber das 

 Princip der Eriialtung der 'Enevgie," Abh. d. math.-phys. Classe der Konigl. 

 Sdchsischen Ges. der Wissenscliaften, 1871. The value of the potential was 

 given by the same author in Pogg. Ann. 1848, vol. Ixxiii. p. 229. 



t The investigation will be published entire in the Journal fur reine und 

 angewandte Mathematik, 



