240 
MR. G. H. LIVENS ON THE 
The vector Ag chosen in this way is, practically speaking, the sethereal 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,,, appears as the velocity. 
We need not, however, take the quantities in this way. We might take 
div Ao = - — div [ I dh 
“ c dt J 
and then we should have 
W. = i 1 
with 
V‘k', = 4"^+4ir^th4irrClulC„* 
c* dt dt J 
where we have used 
A'2 = Ag—T tt 1 I dt. 
Tn this case 02 is the scalar potential of the magnetic distribution, whilst Ah belongs 
to the current distribution. With these differential equations the general values 
of 02 ^2 ii^i regular fields are such that 
0 
, = rLftiAUrf,. 
c .! r 
whilst 
dA2 Att\ Att 
dt" 
c- J 
Curl CuT 
dv 
the square brackets in the integrands denoting that their values are taken at each 
point for the time ( t—- ). 
The.se are the most interesting cases of the solutions for 02 and A2, hut we may 
construct any number of othei-s. It must he noticed, however, that the equation 
p_ 1 dA^.j 
'c dt 
+ grad 02, 
does not imply that the vector B is derived from a potential in steady fields, for it is 
impossible to satisfy the e(.]uations with A2 independent of the time; we may 
have dA^fdt constant in time but not Aj. 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 
