ELECTRO-MAGNETISM. 



the direction of the long branches until 

 it is stopped by the end of the vessel *. 

 This repulsion is exerted, not only at the 

 indefinitely small distance that occurs 

 between the parts immediately in suc- 

 cession, but also at finite distances be- 

 tween all the parts of the same current. 

 (188.) Since it appears, from this and 

 other experiments, that portions of the 

 same, or of different currents, moving in 

 the same continuous line, or at oblique 

 angles, repel one another, Ampere found 

 it necessary to introduce another term in 

 the formula. Since the action due to 

 this cause is greatest when the two cur- 

 rents are in the same continuous line, 

 as A B,/g-. 104, and vanishes when the 

 medial line is at right angles to both of 

 them, as in the positions A and C, he in- 

 ferred that in all intermediate positions, as 

 at D, for instance, the action would be 

 proportional to the cosine of the angle 

 BAD, between the direction of the cur- 

 rent A, and the medial line A D, which 

 we have already expressed by the sym- 

 bol . The same reasoning applies to 



Fig. 104. 



C 



the current D, when the variations of 

 its position are taken into account. The 

 term to be added to the force, as for- 

 merly determined, must therefore be 

 some function of the cosines of the an- 

 gles a. and ft, and may be expressed by 

 prefixing to them the indeterminate co- 

 efficient k. So that we now have 



ab 

 f = - (sin . . sin. /j. cos. ^ + k cos. . cos. /s). 



(189.) In his earlier speculations, Am- 

 pere had regarded the value of h as so 

 small that it might safely be neglected ; 

 but subsequent researches led him to the 

 conclusion that it is in reality equal to 

 } 2 , so that the whole of this second 

 term of the formula is negative, when 

 the cosines of the two angles are them- 

 selves positive. 



(190.) The electro-dynamic forces 

 which are thus called into action by vol- 

 taic currents, and of which the intensi- 

 ties and directions are determined ac- 

 cording to the laws above defined, arc 

 subject to the same laws of composition 

 and resolution as all other mechanical 

 forces, and present the same facilities 

 for the application of mathematical rea- 

 soning. It would of course be impos- 

 sible to attempt giving, within the limits 

 of this treatise, even an outline of the 

 analytical investigations by which various 

 important conclusions have been de- 

 duced. We must content ourselves with 

 stating merely results, referring such of 

 our readers as desire further information 

 on the subject to the works of Ampere, 

 Biot, and Savary. 



(191.) It follows evidently from what 

 has been said, that when two electric 

 currents, situated in the same plane, are 

 inclined to one another at any angle, they 

 are always mutually repulsive when one 

 of them approaches to, and the other 

 recedes from, the summit of the angle ; 

 and, on the contrary, they attract one 

 another when they both approach to, or 

 both recede from, 'that angular point. 

 When the intensities and positions of the 

 currents are the same in the two cases, 

 and the only difference is in the change 

 of the direction of one of the currents, 

 then the attractive force in the latter 

 case will be found to be precisely equal 

 to the repulsive force in the former case. 

 And, universally, whatever be the action 

 of a system of fixed conductors upon a 

 moveable conductor, it is immediately 

 changed into an equal and contrary 

 action, by reversing the direction of the 

 current, through either the moveable or 

 the fixed part of the system. If the di- 

 rection of the current through both the 

 parts be reversed at the same time, the 

 original action is reproduced. In this 

 way we obtain a criterion by which the 

 mutual actions of different parts of any 

 electro-dynamical apparatus may be dis- 

 tinguished from those depending on the 

 influence of the earth. The former ef- 

 fects are permanent, but the character 

 of the latter is changed by reversing the 

 direction of the current throughout the 

 whole system. 



(192.) It is hardly necessary to enter 

 into any minute description of the appa- 

 ratus by which these conclusions may 

 be experimentally verified. The modes 

 of suspension adapted to the particular 

 object of the experiment will readily 

 occur to any one vvho has made himself 



