348 Mr. Megh Nad Sana on the 



rather old one, and seems to have first occurred in 1835 to 

 Gauss *, from whom the title of the paper has been borrowed. 

 Before explaining my methods, I shall give a short history 

 of the problem. 



About the year 182G Ampere published his celebrated 

 laws of electrodynamic action, which enable us to calculate, 

 with strict mathematical exactness, the action between two 

 closed electric currents. IE we assume that a current of 

 electricity consists of streams of positive and negative 

 charges moving in opposite directions, this action between 

 two closed currents is seen to be composed of the elementary 

 actions between the moving charges, taken two and two. 

 The moving charges, therefore, cannot attract or repel in 



the same manner as two stationary charges I viz. force=— s 



f*}' 



for in that case the total action would be zero. The 

 natural assumption is that the law of attraction in this 

 case is quite different, and it depends not only upon mutual 

 distance between the two electrons, but also upon their 

 velocities. This is the problem which Gauss set himself 

 to answer ; he does not of course speak of electrons, but of 

 charged particles', which mathematically amounts to the same 

 thing. 



Gauss and his followers adopted a deductive method for 

 solving this problem. Ampere had given the law which 

 should subsist between two elements of current, t, e. the 

 currents flowing through an element of length of a circuit 

 in order to account for the action between two closed 

 currents. This law was derived partly from the Geometry 

 of lines, partly from experiments, and besides, involved a 

 number of assumptions. The solution was therefore not 

 quite convincing, and, indeed, as Grassmann f and Stefan ± 

 subsequently proved, was not a unique one. Three other 

 expressions were found to be as good as Ampere's expression 

 for the action between two elements of current. Still, 

 Ampere's solution seemed to be most likely, because the 

 assumptions were simpler in this than in other cases. 



Starting with Ampere's expression for the action between 

 two elements of current, and introducing the further 

 assumption that the current consists of discrete charged 



* Much of the Introduction is taken from Maxwell's ' Electricity and 

 Magnetism,' ('haps. n. and xxin., see especially pp. 483 et seq. 

 t Loc. cit. p. 174. 

 % Populiire Schriften-Boltzmann, pp. 95 & 96. 



