45 2 ELEMENTARY ACTIONS. 



The second is exerted between two elements parallel to each 

 other, and perpendicular to the right line joining their centres ; we 

 might represent it by 



ii'dsds' sin sin 0' cos wf(r) , 



the two functions of the distance being different since the conditions 

 are not the same. 



The action d^ will then be expressed by the formula 



( 9 ) d^ = ii'dsds' [cos cos & F (r) + sin 9 sin & cos (o/(r)] . 

 If e be the angle of the two elements, we have 



cos e = cos cos & + sin sin & cos w, 

 and we may write 



(i o) d*$ = ii'ds ds' [cos 6 cos & [F (r) -f(r)] + cos e/(r)] . 



469. DETERMINATION OF THE FUNCTIONS F(r) and /(/). To 

 determine the functions F(r) and f(r\ we must have recourse to 

 experiment, and may employ very different methods, according to 

 the phenomenon to which we apply ourselves. We shall adopt the 

 following course, which is not perhaps the most rigorous from the 

 mathematical point of view, but which leads most rapidly to the final 

 formula. 



We start from the two following experiments devised by Ampere. 



V. When the homologous dimensions of three similar currents of 

 the same intensity are in geometrical progression that is to say, as i, m, 

 ;/z 2 , and are moreover similarly placed the actions of the extreme 

 currents on the intermediate current are equal and of opposite sign. If 

 this latter is movable along a line passing through its centre of 

 similitude, and if it is disturbed from its position of equilibrium, 

 it returns to it of itself that is to say, that the equilibrium is 

 stable. 



Ampere made this experiment with three circular currents situate 

 in the same plane, the intermediate circuit being movable about an 

 axis perpendicular to this plane. 



