496^498] Mechanical Forces 429 



Squaring and adding, the force per unit length is seen to be a force of 

 amount iHsin 6 per unit length in a direction perpendicular to the directions 

 of i and H, 6 being the angle between these two directions. 



498. If the force on the circuit is to be capable of explanation in terms 

 of action at a distance, it is clear that this force must be the resultant of 

 forces between the element ds and the different magnetic poles. 



If the magnetic system is a single pole of unit strength, the force becomes 

 one of amount i sin 0/r 2 per unit length, in a direction perpendicular to that 

 containing the pole and the element of the circuit, this being the reaction 

 corresponding to the action already found in 496. 



This system of forces is not the only one which can explain the observed 

 forces, for there are other ways of distributing the resultant force given by 

 expression (421) over the different elements of the circuit. The most general 

 way of distributing the forces is to assign to the element ds a force 



dx n dy w dz d<f> 

 -j- + tr j^ + -" -j -j~ 

 ds ds ds ds 



where </> is any single- valued function of position, so that l-^ds = Q. In 



J ds 



order that this may represent a force in direction I, m, n it must be of the 

 form 



IE, + raH + nZ. 



The first three terms are already each of this form, so that the term -^ 



OS 



must also be of this form. In order further that the force may be inde- 

 pendent of changes of axes, < must be of the form 



+n 



)^, 

 dzJ T 



dy 



where ty does not depend on the particular axes which are in use. Thus 

 can only depend on the direction of ds and the distance r. 



The components of force are now seen to be 



The first terms compound to give the force already found, which is per- 

 pendicular to r and ds. The last terms give the force arising from a potential 



~ . Since ty can depend only on r and ds, this latter force must necessarily 





