1891.] On the Theory of Electrodynamics. 527 



It is to be observed that the form of 0(r) affects only the forces (iv) 

 in this scheme of attraction, as one would expect from the fact that 

 0(r) disappears if either current flows round a complete circuit. 



Not to refer to (ii) and (iii), we notice from (iv) that two changing 

 electrifications attraet each other with a force involving a term which 

 is constant at all distances, unless a special form of (r) be assigned 

 differing from any of the values which occur in. the sequel. It is 

 difficult to imagine- the mechanical basis of such an action ; the 

 remarks of von Helmholtz in justification (against Bertrand) may, 

 however, be referred to.* 



This investigation of the mechanical forces is equivalent to von 

 Helmholtz's with the exception .that he takes at the beginning 0(r) 

 to be proportional to r, on the general grounl that the potential 

 energy of two elements in all natural actions involves only the 

 inverse first power of the. distances The validity of this consideration 

 seems to be weakened by the fact noticed above that 0(r) occurs only 

 in the force (iv). For what follows it will not be necessary to restrict 

 the form of 0(r). 



To discuss the propagation of electrical action in continuous media, 

 we have to trans-late T from the -form suitable to linear -distributions 

 to the form suitable to volume distributions. Following the method 

 first developed by Kirchhoff, and -for this case the analysis of von 

 Helmholtz, the energy function for any field of currents- is to be 

 obtained by summation of the energy functions of all the pairs of 

 elementary filaments of currents that compose it, care being taken 

 that no pair is counted twice over. The proper form will be a 

 volume integral ; instead of ds, ds',the elements of the filament, it will 

 involve dr, d-r\ the elements of volume, and instead of ,.', the 

 resultant currents, it will involve their components per unit sectional 

 area uvw and u'v'w' ' . 



mi m f ids ids' cos e if d , d ,, >. , , / 



I hus T ---- j_ | t ' . 0(r) ds ds 



JJ r jj ds as 



r r * 



=-i| I - (uu +vv'''-\-ww) d-rd-r 

 JJ r 



the factors \ being inserted because the volume integrals, being 

 extended all over the system, take each pair of elements twice over. 

 Hence 



* ' Wissen. Abbandl.,' I, p. 708. 



