1885.] Researches on the Theory of Vortex Rings. 447 



following conclusion: that, provided the oophore generation of 

 Phylloglossum (which has never yet been observed) corresponds in its 

 more important points to that of Lycopodium, we may regard 

 Phylloglossum as a form which retains and repeats in its sporophore 

 generation, the more prominent characteristics of the embryo as seen 

 in Lycopodium cernuum : it is a permanently embryonic form of a 

 lycopodiaceous plant. 



XI. " Researches on the Theory of Vortex Rings. II." By 

 W. M. Hicks, M.A., F.R.S., Principal and Professor in 

 Mathematics in Firth College, Sheffield. Received June 13, 

 1885. 



(Abstract.) 



The communication forms a continuation of some researches the 

 first part of which was published in Part I of the Transactions for 

 1884. In that paper was considered the case of a circular hollow 

 with cyclic motion through it. In the present the more general case 

 is investigated where the core is of different density from the sur- 

 rounding fluid, has a hollow inside it, and circulations additional to 

 that due to the filaments of rotational fluid actually present. It does 

 not seem to have been generally noticed that even in the case of the 

 ring ordinarily considered, where the density of the core is the same 

 as that of the surrounding fluid, and there are no additional circula- 

 tions, the full theory ought to take account of the existence of a 

 hollow, for when the energy of the motion (as was pointed out by the 

 author*) is increased beyond a certain point, depending on the circu- 

 lation and the pressure of the fluid where it is at rest, a hollow will 

 necessarily begin to form. As it seems impossible to account for the 

 very great differences in the masses of the various elements on the 

 vortex theory of matter unless the cores are of different densities, the 

 investigation includes the case where the density is arbitrary. As 

 soon as the existence of a core is postulated the ring at once becomes 

 more complex, depending on the density (or even the distribution of 

 density) of its core, on its vorticity, and on the presence or absence of 

 additional circulations. The vorticity has been taken uniform ; this 

 not only greatly simplifies the mathematical methods, but is also the 

 case we should naturally choose first to investigate. In the general 

 investigation the density is taken to be different from that of the 

 surrounding fluid, the ring is supposed hollow, with an additional 

 circulation round it, and another round the outer boundary of the 

 core. It is evident that the presence of the former necessitates the 



* " On the Problem of Two Pulsating Spheres," " Camb. Phil. Proc.," iii. 



