380 M. Ampere's Memoir 



the integral of which, between the limits /3" and /3' is 

 - i?/'ds'[sin*/3"- sin3/3']. 

 Replacing ds' b}' the values of this differential found above, 

 and integrating again, we have between the limits determined 

 by the two extremities of the rectilinear conductor, 



I ii' [a" (cos /3„" — cos j3/') — a! (cos i3„' — cos /3/)]. 

 If we suppose that this conductor extends indefinitely in the 

 two directions, we must give to |3/, /3/', /3,/, /3„" the values 

 which we have already assigned them in this case, and we 

 shall have — iv [a"— a') 



for the value of the momentum of rotation, which will conse- 

 quently be proportionate to the length a"— a' of the moveable 

 conductor, and will not change, so long as that length remains 

 the same, whatever may be the distances of the extremities of 

 this latter conductor from that which is considered fixed. 



It is easy to see that this value is that of the momentum of 

 rotation which the fixed conductor imparts to the portion of 

 another rectilinear conductor, situated on a right line which 

 intersects the direction of the former at right angles, for the 

 purpose of making it turn round the point of intersection of 

 the directions of the two conductors. If we descend from the 

 top of the right angle thus formed, to their intersection, by 

 the direction of the two currents from the perpendiculars upon 

 the four right lines which join, two by two, the extremities of 

 these currents, and if we represent these perpendiculars by 

 P'> Pn Pin Pii'y ^'^ shall have 



pI = ± a' cos fij, p," = + a" cos /S/', j?/ = ± a' cos |3/', 

 pj' = ± a" cos /3,/', 

 according as the current from the conductor which has been 

 considered as fixed, is approaching or withdrawing from the 

 point where the direction of this conductor meets that of the 

 other ; and tlie value of the momentum of rotation, with which 

 it tends to revolve round this point of the moveable conductor, 

 becomes consequently' 



±i^npn"-Pl'-P,! + Pl), 

 that is to say, precisely the same as if it were produced by four 

 forces equal to | i i' ; of whicli two would be attractive and in 

 the direction of the right lines which join the extremities of 

 the same name of the two conductors, and the two others re- 

 pulsive and acting in the direction of the right lines which join 

 the extremities of diffei'ent names of the same conductors. 



If the currents extend to the point of intersection of the di- 

 rections of the two conductors, three of these four perpendi- 

 culars will be null, and the momentum of rotation will be sim- 

 ply proportional to the height of the right-angled triangle of 



which 



