240 ME. G. H. LIVENS ON THE 



The vector A 2 chosen in this .way is, practically speaking, the aethereal displacement 

 vector employed by LARMOR in his mechanical model of the electric and luminiferous 

 medium. The curl of this vector is the electric force, or at least as regards its rate 

 of change, whilst the magnetic induction B, which is proportional to the time rate of 

 change of A 2 , appears as the velocity. 



We need not, however, take the quantities in this way. We might take 



div A 2 = *q2-< 

 c dt 



and then we should have 



c a dt 

 with 



i a j\. > cti i s~* i f^ -i, 

 = -5 T^ + 47T + 47r Curl C dt 

 c 1 dr dt J 



where we have used 



A' 2 = A a 47T I dt. 



In this case fa is the scalar potential of the magnetic distribution, whilst A' 2 belongs 

 to the current distribution. With these differential equations the general values 

 of fa and A 2 in regular fields are such that 



47T f [dlV I] , 



= *= J dv 



c J ) 



whilst 



1 dA a _47rl _ 4ff f __] 

 c dt c c 



the square brackets in the integrands denoting that their values are taken at each 



/ r\ 

 point for the time (t - ). 



\ C/ 



These are the most interesting cases of the solutions for fa and A 2 , but we may 

 construct any number of others. It must be noticed, however, that the equation 



does not imply that the vector B is derived from a potential in steady fields, for it is 

 impossible to satisfy the equations with A 2 independent of the time ; we may 

 have dA-i/dt constant in time but not A,. This is the origin of the difficulty in 

 LARMOR'S mechanical model which seems to necessitate the piling up of sethereal 

 displacement in a steady magnetic field. 



14. We have determined the complete expression for the forcive per unit volume 

 on the media occupying the electromagnetic field. The next step in the general 



