FORMULAE OF GAUSS AND WEBER. 603 



From this is deduced 



dr ^rds ^r ds' ^r <V- 



, 



~7^ = v ^2 +2VV ^^~> + v T^- 

 dt 2 ^>s 2 2lw to 2 



If we suppose that the two curves s and / convey currents of 

 strengths I and I', then from Ampere's hypothesis (351), and 

 considering the force to be repulsive, the action of two elements 

 ds and ds' is 



2ll'dsds f 3 ar3r-] _ zlTdsds' | 



C + ~ ~~~ 



621. Suppose now that the element ds contains electrical masses 

 m and m v moving respectively with the velocities v and v v and 

 that in like manner the element ds' contains masses m' and m\, with 

 the velocities if and if v If we evaluate the actions of the masses 

 m and ;;/ x on the masses m' and m' v from the formulae (i) and (2), 

 the resultant should reproduce Ampere's laws in one or other of 

 the two forms, and therefore only contains terms in which the 

 product vv' of the velocities is a factor. 



The terms containing the squares of the velocities are, to within 

 a factor, 



(tntf + a*ifj) (m f + m\) t and (wV 2 + m\v'\) (m + m^. 

 In order that these terms may be null, we must have 

 (4) wz> 2 + #*!#! = , or 



These two conditions are realised simultaneously, if we assume 

 with Weber that an electrical current of strength I is formed by 

 two currents of contrary electricities, moving with the same velocity 

 v in opposite directions, and each having one-half the intensity. It 

 is even necessary to assume that the algebraical sum of the electrical 

 masses which exist in each element of current in the permanent 

 state is null, if the condition is to be satisfied (203) that the density 

 in the conductor is null; but, without stating anything definite 

 on the ratio of electrical masses of opposite signs, it would be 

 sufficient that the sum mv 2 + m^v\ were null in each element that 

 is to say that there were electrical masses of contrary signs with 

 the same vis viva. 



