1883.] On the Steady Motion of a Hollow Vortex. 305 



Society (Part II, 1882), and in -which is considered the vibrations in 

 the form of the axis of a solid core, and the action of two vortices on 

 one another. In the present communication I have confined myself to 

 the case of a single hollow vortex in an infinite fluid, and to vibrations 

 symmetrical about the straight axis. The reason why I have chosen 

 to begin with the hollow vortex is given below. 



The vortex-atom theory, as presented by Sir W. Thomson, has 

 always seemed to me to labour under two difficulties. It does not 

 explain the gravitation of the atoms, nor does it afford, so far as one 

 can see, any means of explaining the different densities of the various 

 elements. When the exceedingly small density of the ether com- 

 pared with what we call ordinary matter is considered, it is clear that 

 the supposition that matter is composed of vortices of the same 

 density as the ether is surrounded with great difficulties, and we are 

 driven to the conclusion that, if a vortex-ring theory be the true one, 

 the cores of the vortices must be formed of a denser material than 

 the surrounding ether, and that probably this core has rotational 

 motion. The theory of gravitation propounded by me in the " Pro- 

 ceedings of the Cambridge Philosophical Society "* only necessitates 

 that the circulation or cyclic constants of the vortex-atoms shall 

 exceed a certain amount which depends directly on the mean pressure 

 and density of the ether. It needs, therefore, no additional hypo- 

 thesis to the theory of Sir W. Thomson, but flows naturally from it. 

 This is not the case with the explanation of difference of density here 

 offered, as the simplicity of the theory is to some extent lost by 

 having two elementary matters in the place of one. 



With these views on the probable constitution of matter I have 

 attempted the problem of determining mathematically the properties 

 of a hollow vortex in an incompressible fluid, lined with an interior 

 layer of a different density from the surrounding fluid. When the 

 density is the same as the surrounding fluid, and the interior hollow 

 vanishes, we have the vortex-ring of the ordinary theory as a parti- 

 cular case. An extreme case in the opposite direction is when there 

 is no internal layer and no rotational motion in the fluid at all; 

 merely the cyclic motion about a ring-shaped hollow. This is the 

 case considered in the present communication. 



If we can argue from the case of two spheres, two vortices making 

 forced pulsations of the same period will attract one another with a 

 mean force whose principal part depends on the inverse square of the 

 distance, when they are in the same phase, and repel one another 

 according to the same law when in opposite phases. When the 

 periods are not the same they will alternately repel and attract, and 

 the mean effect, so far as the forces depend on the inverse square, will 



# " On the Problem of Two Pulsating Spheres in a Fluid," " Proc. Canib. Phil. 

 Soc.," vols, iii and iv. 



VOL. XXXV. X 



