96 M. E. Edlund on the Nature of Electricity. 



elements along their line of junction. If the two currents fol- 

 low the same direction^ and consequently have the same sign, 



the elements attract each other as long as the term -cos^^< 1. 



But if they go in opposite directions and therefore have contrary 

 signs^ repulsion takes place as far as that limit. If, now,//- and 

 fj denote the quantities of electricity in the unit of length of the 

 two circuits, we shall have /jLh = i and /Jh — i', h denoting the 

 velocity of the current. Now fxds and fjlds' correspond to what 

 in the theoretic formula were denoted by m and in'. Ampere's 

 formula may therefore be written 



kmmlh^ f 3 „ A ,„. 



+ -72— (1-2'''''"^; ^^ 



Making cos ^ = 0, we obtain, by comparison w^ith formula (4), 



o- 



kli- 



whence we derive, on replacing h^ by A^(i — cos^ 6) : — 



2^|r r- [1 - cos^ 6^\ = W{1 - cos^ 6) . . (7) 



Making, in formula (3), cos^=l, the value of the function 

 ^/r becomes =0. In this case the two current-elements are in 

 one and the same line, by which their relative velocity becomes 

 constant and =0. Formula (3) thus becomes 



+ ^ [</>( + *) +<^(-^)] (8) 



Putting, in the same way, cos ^= 1 in the empiric formula (6), 

 and comparing it with formula (8), we obtain 



<^[ + h)+<i>[-h) = -\kh\ 

 from which, substituting hcos6 for h, we get 



(P{ + h. cos 0)+(f)(-h. cos 6) = — ^kh^.coi^ e. . (9) 

 Introducing now into the theoretic formula (3) the found 

 values of the function -v/r and the sum (/>( + ^. cos ^) +(/>( — ^. cos 0), 

 we obtain 



which is identical with the formula derived directly from the ob- 

 servations. 



Formula (9) determines the sum of the two functions (f>. This 

 sum is always negative. Of course we cannot immediately con- 

 clude from this the form of the function itself, since a term may 

 have vanished in the addition. We know, from the preceding, that 

 4>[ — h) must always be negative, but, joer contra, (f>{ + h) always 



