071 a 71610 Etect)0-dy7ia7nic Expcfitnent. 379 



Whe n we want the value of this force in a case where the 

 rectilinear conductor extends indefinitely in the two directions, 

 we must make /3/ = /3/' = 0, and ^J = /3/ = tt : it seems' at first 

 sight that then it becomes null, which would be contrary to 

 experience ; but we easily see that the part of the integral in 

 which are the cosines of these four angles, is the only one which 

 vanishes in this case, and that the rest of the integral 



..,r^ tang \ ar _ J tangX/3/ -, 

 L tang i (i/ tang i /3/' J 



tang ^/5/^ cot J; /S/ 



^ tangJ/3/coti/S/" 



becomes ^ .., ^ t ^ngHUl' ^ .., ^ tang^g/ ^ .., J _a-_ . 

 ^ tang'Ji/5/ tangi/3/ a' 



This value shows that the force sought for then only de- 

 pends on the relation of the two perpendiculars a' and a' low- 

 ered on the rectilinear and indefinite conductor of the two ex- 

 tremities of that portion of the conductor on which it acts; that 

 it is also independent of the form of this portion, and only be- 

 comes null, as it ought, when the two perpendiculars are equal 

 to themselves. 



In order to have the distance of this force from the rectilinear 

 conductor, the direction of which is parallel to its own, we 

 must multiply every one of the elementary forces of which it 

 is composed by its distance from the conductor, and integrate 

 the result with reference to the same limits ; we shall thus 

 have the momentum to be divided by the force in order to 

 obtain the distance sought for. 



We easily find, after the above values, that the value of the 

 elementary momentum is 



I ii' d s' r s\xi^<\ ( —^ — ) • 



This value cannot be integrated but by substituting for one 

 of the variables r or ^ its value in the function of the other, 

 drawn from the equations which determine the form of the 

 moveable portion of the conductor. It becomes very simple 

 when this portion is found on a right line elevated on some 

 point of the rectilinear conductor, which is considered as pei'- 

 pendicularly fixed in its direction, because in taking this point 

 as the origin of 5*, we have 



s 

 "* — ~ COS /3' 



because s' is a constant relatively to the dilFerential 



I cos- /3 

 r 



the value of the elementary momentum therefore becomes 

 i / /' d 5* -^^^ d (cos^ /3) = - ] «• i ' d s' sin' /3 cos /3 d ^, 



3 B 2 the 



