790 
MR. J. LARMOR ON A DYNAMICAL THEORY OF 
occupied by possibly vacuous cores of the vortex atoms. Its motion is partly hydro- 
dynamical and irrotational, and is partly of rotational elastic quality. Its equations 
of motion are, for the averaged displacements which represent the general circum¬ 
stances of crystalline quality, 
P 
P 
lY? , dcVi cWg . dp 
dp ^ d)j dz ^ d.v 
Td-n da~f dcVi , dp 
dt' dz ~ do: ^ “ 
P Tfl + W 
drrf dp 
~dy ~dz 
= 0, 
where t], Q is the linear displacement, {/, g, h) is its vorticity or curl, and is a 
hydrostatic pressure in the medium, the symbol denoting the acceleration of a 
moving particle as contrasted with the rate of change of velocity at a fixed point. 
98. These equations represent the general circumstances of the propagation of 
radiation through the medium ; and in them the velocity of translation of the 
medium due to vortices in it has been averaged. But if we desire to investigate in 
detail the motion and vibrations of a single vortex-ring or a vortex-systein in a 
rotationally elastic fluid medium, it is of course not legitimate to average the motion 
of translation near the ring. The determination of the circumstances of the influence 
of a moving medium on the radiation also requires a closer approximation. Con¬ 
sidering therefore the free cether, which is devoid of crystalline quality, and 
substituting 
(^, 77 , i) = {u -f Wi, V -p Vi, w -f 
so as to divide the velocity into two parts one of which represents the translation of 
the medium and the other its vibration, we have 
^ = 4 + (“ + "1' !; + (’’ + ”1) ^ X > 
dt dt 
dz 
SO that 
D 
, , . hi Sin du dio , dii 
dt = * + rf? + rfv + - 
very approximately where 
S 
dt 
,d d d d 
represents — d- •?(-—\- v-— lo 
dt dx dy 
dz 
Hence separating the hydrodynamical part in the form 
[u, V, iv) 
(± A A\„ 
\dx ’ dtj ’ d: J 
