THE ELECTEIC AND LUMINIFEROUS MEDIUM. 
791 
which represents irrotational motion except in the vortices, there I’emain vibrational 
equations of the type 
, du , dai , dio 
It + dx + * 
+ «= 
dj^ 
dy 
(i£\ 
dz) 
= 0 . 
lu a region in which the velocity of translation {u, v, iv) is uniform, the radiation 
is thus simply carried on by the motion of the medium. 
99. The vibrational motion which is propagated from an atom is interlinked with 
the motion of translation of the medium, only through the hydrostatic pressures 
which must be made continuous across an interface; the form of the free surface has 
ill fact to be determined so as to adjust these pressures at each instant. To fix 
our ideas, let us consider for a moment the problem of the vibrations of a single ring 
with vacuous core, moving by itself through the medium, in the direction of its axis, 
with a given atomic electric charge on it. To obtain a solution we assume that the 
radius vector of the cross section of the core varies with the time according to the 
harmonic function suitable to its types of simple vibration'; and we determine the 
irrotational motion in the medium that is produced by this motion of the surface of 
the core, and calculate the pressure 2^0 at the free surface. Next we determine the 
vibrational rotation [f, g, h) that is conditioned by the same vibratory movement of 
the surface of the core, while it is independent of the inertia of the hydrodynamical 
motion in the medium ; this has also to satisfy the condition that the tangential 
components of the rotation are null all over the surface, so that there itiay be no 
electromotive tangential traction on it. In order to satisfy all these surface conditions 
it will usually be necessary to introduce an electromotive pressure into the 
equations of vibration, although this was not required in the problem of reflexion at 
a fixed interface; in other words the pressure in that problem was quite unaffected 
and therefore left out of account. The magnitude of this pressure is then to be 
calculated from the solution; and the condition that it is equal and opposite at the 
free surface, to the pressure Pq of hydrodynamical origin, gives an equation for the 
period of the vibrations of the type assumed. If on the other hand the core is taken 
to consist of spinning fluid devoid of rotational elasticity, instead of vacuum, the 
conditions at its surface will be modified. 
100 . If the form of the ring is such tlrat the period of its hydrodynamic vibration 
is large compared with that of the corresponding electric vibration, an approximate 
solution is much easier; it is now only necessary to suppose that on each successive 
configuration of the core there is a distribution of static electricity in equilibrium, and 
to allow for the effect of this distribution on the total pressure which must vanish 
at a free surface. 
In this case the electric vibrations will continue for a comparatively long time, 
until all the energy of the disturbance in the molecule is radiated away, but they 
will be of very small intensity. The vibrations of an electric charge over a con- 
