378 M. Ampere's Memoir 



Setting out from this relation between k and 7i, and naming 

 /3' and /3", 7-' and ?■", these vahies of /3 and r which correspond 

 with the two extremities of a portion of the voltaic conductor, 

 we find, for the action which it exercises on the element d 5' 

 in the direction of this element 



r cos- jS" cos- /y 1 



r cos' /3" cos'J /S' -| 



or rather iti' ag\ —pi ^ , 



since we know from other experiments that n = 2. It is suffi- 

 cient to change the sign of this expression, which is indepen- 

 dent of the form of the portion of the voltaic conductor, and 

 only depends on the situation of its two extremities with respect 

 to the element d s', in order to have the force with which the 

 same portion of the conductor is drawn in a contrary direc- 

 tion by the element following a right line parallel to the di- 

 rection of the latter ; whence it follows that if this element forms 

 a part of a fixed rectilinear conductor, we shall have the value 

 of the force which the whole conductor exercises, in order to 

 move that portion of which we ai*e speaking, in a direction 

 parallel to this conductor, by integrating between the limits 

 marked by its two extremities the value which we have just 

 found for the tangential force of the element d 5'. 



If we call a! and a" the lowered perpendiculars of the two 

 extremities of the portion of the conductor which we consider 

 as moveable, on the rectilinear conductor which we have to cal- 

 culate the action parallel to its direction, we shall have 



ds'= - 



sin g>" ' sin /3' ' 



d'r" n"djS" A' r^ a' d /S' 



cos/3'' sin- /S" cos/3' sin'' /3' ' 



and consequently, 



— — '^^" d^' _ d/3' _ 

 r" ~ sin /3" ' r' ~ sin /3' ' 



whence it is easy to conclude that the integral sought for is 



1 ■ ■' /T '^°^' ^" ^ ^" *^°^* /3' d /?' "1 



— ^11 J 1^ sin /3" sin /3' J 



= - ^ /^Tl "^"^^.w + cos /3"- cos /3' + Cl. ■ 



^ L tang^/3' ^ ^ J 



We must take this integral between the limits determined 

 by the two extremities of the rectilinear conductor; by calling 

 /3/, |3/', |3/, )3/ the values of (S' and of /3" relative to those li- 

 mits, we have immediately that of the force exercised by the 

 rectilinear conductor, and that last value evidently depends 

 only on the four angles (3/, /3,", /3,', &" 



When 



i 



