1879.] pulsating spheres in a fluid. ' 283 



the remark that such corpuscles by their battering would raise 

 bodies to a white heat seems to contradict this explanation. 



The foregoing theory would explain the gravitation of vortex 

 atoms, if by any means it could be shown how vortex atoms in an 

 incompressible fluid could change their volume, and pulsate in a 

 constant periodic time. That this may be so I will show from the 

 following considerations. Let us -consider, for the sake of illustra- 

 tion, a single straight vortex in an infinite fluid, or rather, cyclic 

 motion about an infinite straight line. The velocity of the fluid 

 near the axis will be very greai, and there will be therefore a 

 small vacuum along it, of the form of a cylinder, whose radius is 



'v/inr) "^^^®^® '^'^f^ i^ *^® cyclic constant, 11 the pressure of the 



fluid and p its density. This will also be the case with a vortex 

 of any form, and there inay be a vacuum along its axis. Now 

 suppose a vortex ring in a fluid, and suppose it has impressed 

 on it pulsations of this vacuum (as might be done for instance 

 by suddenly increasing the pressure 11), then it would proceed 

 to keep up these pulsations with a constant period, which 

 depends on the vortical constant and the pressure, and which is 

 probably independent of the shape of the ring. If then two such 

 vortex rings be in a fluid, and if their phases are concordant, 

 they will attract one another inversely as the square of the 

 distance. 



Returning to the case of a straight vortex, it will be found that 

 no pulsation will take place, or rather its period will be infinitely 

 great ; but this will not be the case with a finite vortex ring in 

 three dimensions. In fact, suppose we have a liquid cylinder of 

 infinite length, , under the action of no forces except a uniform 

 pressure 11 around its surface; and suppose also that in this 

 cylinder there is cyclic motion but no vortex. It will be found 

 that if the radius of the cylinder is a, that of the empty space 

 along the axis is 



and that the time of a small pulsation is 



a^b^ //log' a — loo- b 



Ztt 



A6 V V CL'-b' 



If in this we put a= x> the radius of the bore is a / ^^ , 



and the time of pulsation becomes infinitely large. This is due 

 to the fact, that in this case there is required an infinite impulse 



