462 ELEMENTARY ACTIONS. 



The components of this force parallel to the axes are 



II'dssmB, ll'dzdz' 



f x =fcos/3 = ds cos/* cos/3 = 2 , 



A = 0> 



> , .' ll'dssmO , . ll'dzdx' 

 f z =/sm p = ds cos fi! sin p = . 



OC 00 



If we still denote by & the angle which the right line r makes 

 with the element ds', and by w the angle of the two planes rds and 

 rds' , we have 



dz' = ds' sin 0' cos w, 



<&' = <&' cos 0', 



which gives 



f x = sin 6 sin & cos o> dsds' , 



f z = ^ sinO cos O'dsds'. 



The action of two elements of consecutive currents is evidently null. 



In fine, we have not here an equal and opposite action and 

 reaction, but there is a different action on each of the two elements, 

 directed perpendicularly to this element, and in the plane determined 

 by the other element. 



The existence of a force perpendicular to the element is incom- 

 patible with the idea of an action at a distance; but if, on the 

 contrary, we view electrodynamic forces as resulting from a modifi- 

 cation in the elastic properties of the medium, we can easily see that 

 the reaction of this medium on an element of current may be 

 perpendicular. 



477. (n.) General Formula. We may add an exact differential of 

 the co-ordinates to each of the components f x ,f y , and/ 2 without the 

 action of the closed circuit on the element ds' being modified. We 

 may then take as components of the elementary action the following 

 general expressions in which X, Y, and Z, are any given functions of 

 the co-ordinates : 



