284 ilf?^ Hicks, Oil two pulsating spheres in a fluid. [Oct. 27, 



to change the bore of the vacuum by a finite amount, since if a 

 small change be produced in it, the energy of the fluid within a 

 cylinder of radius r depends on log r and is therefore infinite when 

 r is so. But in the case of a finite body in space of three dimen- 

 sions the energy within a sphere of radius ?- depends on - , and is 

 therefore finite when r is infinite. 



In a fluid in which there are a very large number of vortex 

 atoms, the pulsations of any one will be to some extent affected 

 by the presence of the others. But this effect will be diminished 

 indefinitely if the pressure of the fluid and the vortical constants 

 be very large. It seems likely that their mutual action would 

 tend to keep up the same time of pulsation, if any tended slightly 

 to depart from it. 



It is worth noticing that if there are a number of individual 

 vortices in an incompressible fluid, then the fluid may not behave 

 rigorously as such, because the vortices in it may be elastic, as is 

 evident from what has gone before. In such a fluid pressure will 

 not be transmitted instantaneously, and it might be thought that 

 in this case the action between pulsating bodies might not be the 

 inverse square. But though the action would be diminished at 

 the same distances, yet here also at least, unless the fluid were 

 easily compressible, it would vary as the inverse square. We may 

 gather this from the fact that an incompressible fluid is the limit 

 of a compressible one whose compressibility tends to zero. The 

 most interesting point to be noted about such a fluid is that 

 gravitation would take time for its full effect to travel any 

 distance. It is to be clearly understood that it is not asserted 

 that a vacuum mtist exist in every vortex atom, but only that the 

 cyclic irrotational motion, connected with the vortex, may be so 

 large as to produce one. To make the case clear, suppose a 

 vortex ring in a fluid with cyclic motion through the ring. Let 

 the pressure on the fluid at the boundary be gradually diminished ; 

 then there will be a certain point at which the pressure at the 

 vortex will be insufiicient to keep the fluid continuous, and a 

 vacuum will arise. For instance, in the case of the cylinder 

 considered above, we have supposed no rotational motion in the 

 axis. There is therefore always a vacuum whatever the finite 

 pressure on the boundary may be. 



To get some idea of the magnitudes involved I have calculated 

 the following. Suppose there are two pulsating spheres of the 

 size of hydrogen atoms, and that the fluid has the density of the 

 ether as estimated by Sir W. Thomson, viz. 9 . 86 x 10"^". And 

 suppose also that the amplitude of pulsation is n times the radius. 



