674 Prof. J. J. Thomson on the Magnetic Properties of 



and for values of r small compared with the wave-length, 

 27rY/p, of the vibrations, they give the same magnetic force 

 as that w^hich would be produced if the particle was moving 

 uniformly with the velocity it possessed at the instant under 

 consideration . 



This derivation of the magnetic force when the velocity is 

 variable from the solution obtained on the assumption that 

 the velocity is uniform may be extended to cases more 

 complicated than that of a single particle; if we have a 

 collection of particles separated from each other by distances 

 small compared with the wave-lengths of their vibrations, and 

 if in the expression for the magnetic force due to these 

 particles, calculated on the assumption that the velocity is 

 uniform, there is the term 



^^^ dx^ dy-"' dz"" 'r' 



r being the distance from a point in the midst of the particles, 

 then if <j)(t) is proportional to e^^\ the solution, when we take 

 into account the variability of the motion, will have for the 

 corresponding term 



''.r 



^,. d^ d^' d"" e y 



9(0 



dcc^ dy'"" dz"" r 



2. We shall first find the magnetic field due to a particle 

 with charge e describing with uniform velocity a circular 

 orbit in the plane xy. Let be the centre of the orbit, 

 Q the position of the particle, P the point, not necessarily in 

 the plane of xy^ at which the magnetic force is to be deter- 

 mined ; then if a is the radius of the orbit, « the angular 



Fig.l. 



velocity of rotation, the angle OQ makes with the axis of ;r, 

 w= — ao) sin ^, i'=aft)cos^, i^ = 0. 



