495, 496] Mechanical Forces 427 



where (equations (377)) 



.(420). 



Here x, y, z are the cartesian coordinates of any point on the circuit, 

 r the distance from this point to the particle, and Z, m, n the direction 

 cosines of the axis of the particle. If n is the potential of the shell at 

 x ', y', z, a second expression for the mutual potential of the shell and the 

 magnetic particle is, by formula (353), 



f, an an an 



I* ^ -r + m^t + n 

 V dx dy 



This expression is accordingly equal to expression (419). 



The component of force at x', y, z' in the direction (I, m, n) produced by 

 the shell under consideration is 



/ 7 an , an an\ 



[I -$-, + m =-; ; + n -x-, 

 V dx dy dz J 



and this, by what has just been said, is equal to 



ds ds 



On substituting for F, G, H from equations (420), etc., this becomes 



. /T 7 ra /i\a* a /i\ay] , f 1 f n , 



-0 Ho" ~U--^- -U^^ + m^...V+7i^...M^5. 

 ^L la^/ Vr/ 85 dz \rj clz] ( j ( jj 



This expression will also give the force at x', y , z' in direction I, m, n from 

 a current </> flowing in a circuit coinciding with the boundary of the shell. 

 On examining the rules of signs given in 438 and 483, it will be found that 

 the direction of this current must be the same as that in which the inte- 

 gration round the circuit is taken, provided that right-handed axes are in use, 

 but must be the opposite direction if we are using left-handed axes. 



We accordingly see that the force from a current of strength i can be 

 regarded as made up of a contribution of amount 



.r, fa n\dz a ma*/) f i f n 



-* IM5T I 51 Vl-).a r+^l ^r + n v-r 



L [dy \rjds dz \rjds) ( j ( JJ 



per unit length from each element of the circuit. This again can be written 

 in the form 



IX + mY + nZ, 



i { y v dz z z d 

 = - -T ^- - --- -^ 



r 2 I r ds r ds 



^ 



where X = - 



