1884.] equations of the electromagnetic field. 123 



the action of currents, F 2 , etc. from the direct action of magnetic 

 bodies. If we adopt Ampere's hypothesis that magnetism is due 

 to molecular electric currents, the terms in F 2 , etc. disappear. 



The electromotive force at the point parallel to the axis of x 



is — -j- , which is equal to jr jf • According to Helmholtz 



the part of the electromotive force which arises from the action of 



the currents is — /j, -rr , so that 



<^=J-E etc (2) 



dt * dt ,ctc {Z) - 



Again, let F 2 arise from magnetism distributed throughout 

 space in such a way that its components at x, y', z are A, B, G, 

 let p be the reciprocal of the distance between two points whose 

 co-ordinates are x, y, z and x, y', z'. Then (Maxwell, II. § 405) 



-j—, — C -^-, J dx'dy'dz', etc. 



Hence if we put L =fffAp dx'dy'dz', M = etc., iV= etc., 

 , ^ dM dN 



WGhaVe F ^-dz-lTy> 



and _^ = 1{^_^1 



dt dt\dy dz )' 



This agrees with Helmholtz's expression just referred to for the 

 part of the E. m. f. which depends on magnets. 



Again, to find the magnetic force due to this let a, /3, y be the 

 components of the magnetic force, a, b, c of the magnetic induction 

 and ^ the magnetic potential. 



Then, we have 



>dL dM dN\ 



* \dx dy dz , 



A dH dG 



and m= a = -= r - 



dy dz 



= dE 1 _dG 1+ dE 2 _dG 2 

 dy dz dy dz 



= dR i _dG 1 d/dL dM oW\_ 

 dy dz dx\dx dy dz) ^ 



dy dz dx 

 dH x dG, 



= ^y"dz~ + ^ ( 3 >> 



etc 



