212 
MR. G. H. LIVENS ON THE 
the vector I, as the vector P, is attached to the matter and moves with it. The last 
two equations contain, therefore, an explicit expression of the etfect of the motion so 
that they are in a sense more convenient than the equations 
D = —E + P, H = B-47rI, 
477 
defining merely the values of D and H at any point: they are, of course, ultimately 
consistent with these equations for, taking the second one as an example, we have 
div H = -^ div B—477 div —477 div Curl [!»'„], 
dt dt dt 
or 
^div (H-B+47rI) = 0. 
With the possible exception of the equation defining dUfdt it is now generally 
agreed that the scheme here presented provides a completely eftective specification of 
the kinematical connexions in the electromagnetic field. 
To obtain some idea of the effect of these connexions on the dynamical processes 
operative in the field a further assumption is necessary, and this may take one of 
several forms which will be reviewed in the sequel. For the present we are concerned 
merely with these equations as effective representatives of the electromagnetic 
processes. They are sometimes given another form, by the introduction of a scalar 
potential <p and a vector potential A, these being such that 
B = Curl A, E = - - ^ -grad 
0 ctz 
with the other two equations 
Curl H = — C, div E = 477^. 
c 
The first two of these equations are equivalent to the remaining fundamental 
equation of Maxwell’s theory which they replace, but they suffer from the serious 
disadvantage that the quantities A and <p specified in them are not completely 
defined by the equations as given and require additional data to fix them. 
4. We have just noticed that an additional assumption of a dynamical character is 
necessary to render the Maxwellian electromagnetic scheme completely effective as an 
electrodynamic theory. The simplest and most direct form of this assumption may be 
taken to be that expressing the force of electrodynamic origin acting on an arbitrarily 
moving element of charge, this force being, per unit charge equal to 
F= E+-[.B] 
c 
