458 ELEMENTARY ACTIONS. 



It follows from this that 



hii'ds' . hii'ds' . 



dn -- G sm a = -- - Jb sin a = a<b . 

 2 2! 



The action of the closed circuit on the element is therefore 

 perpendicular to the force F and to the element ds' that is to say, 

 Ampere's directive plane and proportional to the surface of the 

 parallelogram constructed on the force F, and the element ds'. 



If the force of the field F = IG, at the point where is the element 

 ds\ was produced by a magnetic system, the action in like manner 

 would be along the y axis, and its value would be I'ds'F sin a, where 

 I' is the electromagnetic intensity of the current which traverses the 

 element. 



The two actions are in the same direction, and they are pro- 

 portional; if we assume that they are identical, it will follow 

 that 



A H/ 



h 2 . 

 It 



As the numerical expression of a magnitude is inversely as the 

 unit with which it is measured, it will be seen that the constant h is 

 equal to twice the square of the ratio arbitrarily chosen to measure 

 the strength of the current in electromagnetic units. 



472, If we suppose that the currents have, in the first place, 

 been determined in electromagnetic units, we have then 



we thus arrive at Ampere's formula, which we had already obtained 

 (459), 



(15) d^ = sin sin 6' cos w cos 6 cos 0' , 



(16) d <2 "d/ = -cose cos cos & , 



7-2 [_ 2 J 



and which we may write in the more symmetrical form (351) 



J r dsds' 



