On the mutual threading of vortex rings

Although the fact that two circular vortices are capable of threading one another in permanent succession has long been known, I do not know that any attempt has been made at a discussion of the general conditions. The following pages contain a contribution to such a discussion for the case of thin circular filaments. The nature of a ring is given when its circulation and volume are known. Its configuration at any time is given when its aperture or the radius of the cross-section of the filament is known. In the case of two rings, the nature of their combination is defined if their two apertures at the instant when they are co-planar are each known. This configuration will be called their standard position. With two such rings, the mean area of the two apertures, supposed weighted with their circulations, remains constant throughout the motion. They can therefore also be defined by the radius of this mean area and the ratio of the two apertures in the standard position. This latter method is the more convenient. It is found possible to determine the relative paths of one as seen from the other, and to obtain the conditions of permanent union for any two given rings, but the complete solution in terms of the time, and the actual paths in the fluid, are not expressible in general terms. Such can only be arrived at by numerical quadrature for cases in which the various coefficients involved have their numerical values given. Any special case can thus be solved, but naturally the process is very laborious. In the corresponding problem in two dimensions, however, a complete solution is attainable. For this reason, and because it illustrates the general method, the problem of two like and equal two-dimensional pairs (rectilinear filaments) is first touched upon. The more complete discussion for all the possible combinations is extremely interesting, especially when the pairs have opposite circulations, but is here omitted, in view of the greater physical importance of the theory of the ring.