232 
MR. G. H. LIVENS ON THE 
on- each other, ])eiiig connected by the equation given above, and there will be a 
relation l)etween the forms for tvm different theories. In fact if S and T are the 
forms corresponding to any one mode of separation and if we write 
O/ Q I 
o = >5 + —^ 
dt 
where U is any arbitrary A'ector function we shall have 
where 
and S' and T' are appropriate forms for a new mode of separation. In this way, hy 
assigning convenient values for U, we might tentatively construct a number of 
interesting fonmdm. 
The last result also slnnvs why it is that the particular form chosen for the kinetic 
energy is ii-relevant to the genei-al dynamical discussion of paragraph 7. In fact, 
if, instead of the form T used on tliat occasion, we had employed the general value 
derived above 
T- [divUdr 
the part of the variation depending on this energy becomes the time integral of 
flT — j div dU dyr, 
and the latter integral reduces to a surface integral over the intinitely distant boundary 
and cannot therefore contribute anything in this general variational equation. 
Of course, from another point of view, the various forms of the theory here under 
review, differ merely in assigning different distributions to the magnetic energy in the 
held, each of these distributions being ultimately consistent with the same proper total 
for this quantity ; and the fact that they all lead to the same dynamical equations, 
merely verihes a well-known result of analytical dynamics that the particular form of 
expression for the energies of the system is immaterial to the ultimate dynamical 
equations for the field inside a continuous medium. Of course the solutions of 
boundary problems such as are, for example, involved in a specification of the energy 
flux, depend essentially on the particular form assumed for the energy distribution ; 
