PRINCIPLE OF SINUOUS CURRENTS. 



45 1 



Let us consider two elements a and b (Fig. 109) in any position ; 

 let ds and ds' be their lengths, i and t' the intensities of the two 

 currents referred to a given unit, 6 and & the angles of their 

 directions with the right line OO' joining their centres, r the distance 

 OO', lastly co the angle of the planes drawn through the right line 

 OO' and the two elements. 



Fig. 109. 



Let us take the plane which passes through the element ds and 

 the right line OO' as plane of the figure, and replace each of these 

 elements by its projections on three rectangular axes ; one of these 

 axes is the right line OO', the other a right line in the plane of the 

 figure, and the third a perpendicular to this plane. The element a 

 has only two projections 



a' = ds cos 0, 

 a" = ds sin B ; 



the three projections of the element b are 



b" ds' sin & cos w , 

 b'" = ds' cos 0' sin w. 



The total action consists of the actions of each of these elements 

 a' and a" on each of the elements b', b"> and b'". 



Of these six actions, four are null from the principle of symmetry, 

 that of a' on b" and b'" t and that of a" on b' and b'". 



There only remains to be examined the action of a' on &', and 

 that of a" on b". 



The former is exerted between two elements directed along the 

 same right line; it might be represented by 



ii'dsds cos cos 0'F(r). 

 G G 2 



