802 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
It is of necessity postulated tliroughoiit that {/, g, li) is circuital, for it is the curl 
of (f, ri, Q; that is, the proper current sheet must always be taken to exist at the 
surface of the conductor in order to complete the electric displacement in the medium. 
It follows as in § 57, but only under this proviso, that the magnetic force is the curl 
of Maxwell’s vector potential (F, G, H) of the current-system. 
Tlie transformation of the kinetic energy T to the directly elastic co-ordinates 
{/> fh thus established ; and the dynamical ecjuation of the medium is 
8 [(T - W)f/^ = 0 
in which the time is to remain unvaried. In order however to obtain equations wide 
enough to allow of the restriction of {f, g, It) to circuital character, which is now no 
longer explicitly involved, we must incorporate this restriction in the variational 
equation after the manner of Euler and Lagrange, and so make 
4-8 f - w) + 8 f dt f + I + f ) = 0 . 
and restrict the function of position i/y subsequently so as to satisfy the circuital 
relation. Thus 
8 dl 
i(Ff+ 0 
% 
(It 
, , , /(If 
(ly 
+ 
dh\ 
(Iz) 
dr = 0. 
Now in all cases in which the kinetic energy of a dynamical system involves the 
velocities but not the co-ordinates, the result of its variation is the same as if the 
momenta, such as F, G, H, in the expression in terms of momenta and velocities, 
were unvaried, and the result so obtained were doubled. Thus we have here 
dr=:\dl 8W; 
or, integrating by parts and omitting the boundary terms for the reasons above given. 
dt 
f - 1) + 2) s;,. ^aw. 
Therefore throughout the system the forcive corresponding to the displacement 
(/> is 
(P, Q K) = 
dG dik\ 
dt dt / ■ 
