[ 314 ] 

 XLV. Intelligence and Miscellaneous Articles. 



ON THE ACTION OP TWO ELEMENTS OF A CURRENT. 

 BY J. BERTRAND. 



" nPWO parallel currents attract one another when they have the 

 ■J' same direction ; they repel each other when their directions 

 are opposite." After enunciating this rule, Ampere believed he 

 could immediately generalize it by extending it to the elements of 

 the currents, to which he applies, whatever may be their relative 

 direction, the idea of a course in the same or in opposite directions. 

 Two currents are said to be in the same direction when they both 

 increase their distance from the foot of the common perpendicular, 

 or when they both approach it ; in the contrary cases they have 

 different directions. Adopting this language, it is not accurate to 

 say that two elements having the same direction attract one the 

 other ; it is not accurate even for parallel elements. As the asser- 

 tion has been reproduced in all the treatises on physics, and serves 

 as a basis for several important explanations, I have thought it 

 would be important to show that it is inconsistent with Ampere's 

 law itself, and to solve the following problem : — 



Given an element of a current, to find in a point M of space the 

 direction which must be assigned to another element in order that 

 their mutual action may be attractive, repellent, or nil. 



Suppose the element ds placed at the origin of the coordinates 

 and directed along the axis of the X's, let us seek the condition on 

 which an element whose coordinates are as', y', z' will be without 

 action on ds. Naming the angles formed by the two elements with 

 the straight line which joins them 6 and 6', and the angle which 

 they make with one another e, according to the law of Ampere the 

 condition is, 



cose = |cos0cos0' (l) 



But, naming the radius vector r, and the attracting element ds, we 

 have 



A x' n , dr dx' 



cos0=-, COS0SS — , cos e= -=-r. 

 r as as 



Equation (1) becomes 

 of which the integral is 



S- € ^=2 C ^, (2) 



r ds ds 



r*=Ax\ (3) 



the equation of a surface of revolution whose axis is the axis of X, 

 and of which the meridian curve has for its equation, in polar co- 

 ordinates, 



r==Aeos 2 (4) 



"Whatever the form and direction of a current enveloping such a 



