XV. Researches on the theory of vortex rings.—Part II

Abstract

The present 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 round it. In the following pages the more general case is investigated where the core is of different density from that of the surrounding fluid, has a hollow inside it, and circulations additional to that due to the rotational filaments actually present. The investigation is not merely one of mathematical interest, for the vortex atom theory of matter has—so far as it has yet been developed—shown such claims on our consideration that anything throwing light on it will be of value. The supposition of a dense core may possibly be necessary to account for the different masses of the yarious elements. As soon as the existence of a core is postulated the ring at once becomes more complex, depending on the density (or even the arrangement of density) of its core, on its vorticity, and on the presence or absence of additional circulations. In what follows 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 additional circulation round the outer boundary of the core. It is evident that the presence of the former circulation necessitates the perpetual existence of the hollow. It is shown that the presence of the latter circulation is necessary to render the ring stable when its density is greater than that of the rest of the fluid.