454 Messrs. Watson and Burbury on the Law 



the potential of two elementary currents ids and i'ds' the form 



... f . cos e , _, cos 6 cos 6' I 7 , , 

 n'l A — — + B ■ >dsds', 



where A and B are any constants, we should for any pair of 

 closed circuits obtain as j the potential an expression propor- 

 tional to 



>s 



COS 6 7 , , 



as as ,- 



r 



that is, practically identical with that obtained from F. E. 

 Neumann's law. But the force exerted by a closed circuit on 

 a separate element of another would not be identical in the 

 two cases ; and it may by proper choice of the constants A 

 and B be made always normal to the element. 



5. Ampere's solution of the problem is as follows, assuming 

 a force and not a potential between two elementary currents. 

 Let i cos 6 ds, i! cos 6 r aY, according to the usual notation, be 

 the component parts of the elements resolved in r, the line 

 joining their centres ; and i sin 6 ds, i' sin & cos <fi ds' ', the com- 

 ponent parts perpendicular to r in the plane of r and ds. Then 

 it is assumed that the two radial components attract each 

 other with a force in the direction of r varying inversely as 

 the square of the distance, viz. 



-s cos 6 cos Q'ii' ds ds' 



and the two transverse components attract each other with a 

 force in the direction of r, viz. 



^2 sin 6 sin 6 / cos <f>ii' ds ds' \ 



where a and b are constants. It is then shown that if the rela- 

 tion between a and b be 2a + b = 0, the desired result will follow 

 — namely, that the force exerted by any closed circuit on any 

 element of a current is always normal to the element. This 

 relation, then, satisfies Professor Tait's canon III. above given, 

 and also satisfies VI. It will be found also to lead to the 

 correct expression for the mutual potential of two closed 

 circuits, and therefore satisfies all experimental conditions. 



Ampere's results may be concisely expressed as follows ; 

 viz. the action of ids upon i! ds' is a force in the direction of 

 the line, r, which joins them, and whose intensity is 



y/r as ds' 



