218, 219] ELECTROMAGNETISM. 427 



Consequently 



dx = I(N m d yi -M m dz,\ 



(4) dU = I (L m dz, - N m dx,), 



and the whole force due to the presence of any number of 

 magnetic bodies producing a field L, M, N is the resultant of all 

 the individual actions 



(5) 



That is : the mechanical force on the element is the vector 

 product of the current element Ids 1 and of the magnetic field 

 where it is situated*. 



Suppose that the magnetic field is due to a second element ds 2 

 of strength 7 2 at a distance r from ds^. Then since by (i i) 217, 

 putting ds 2 for ds 1} x l ,y l) ^ for a?, y, -z, 



dL = - 2 [dy z (z l - z 2 ) 



dN = ^ {dx z (y - 7/ 2 ) - cfa/ 2 (^ - a? a )}, 

 we have for the mechanical force acting on ds l , by (5), 



(6) &Z= 1 ^[dy 1 {dx 2 (y l -y 2 )-dy 2 (x l -x 2 )} 



dz-L [dz 2 (^ a? a ) 



Adding and subtracting the term dx l dx^(x l x^/r* this maybe 

 written 



(7) + dx, dx, + - dy, + dz, 



* = - 2 x - (cos (rx) cos (ds-ids*) cos (rfs^) cos (rc?^)}, 



r being drawn from ds^ to c?5 2 - 



* It would be hard to devise a simpler rule for remembering the direction of the 

 force than the one given on p.' 12. 



