The paper on vortex-motion also deals with the modification of Green's celebrated theorem of analysis, which, it was pointed out by Helmholtz, was necessary to adapt it to a space which is multiply continuous. The theorem connects a certain volume-integral taken throughout a closed space with an integral taken over the bounding surface of the space. This arises from the fact noticed above that in multiply continuous space (for example, the space within an endless tube) the functions which are the subject of integration may not be single valued. Such a function would be the velocity-potential for fluid circulating round the tube--cyclic motion, as it was called by Thomson. If a closed path of any form be drawn in such a tube, starting from a point P, and doubling back so as to return to P without making the circuit of the tube, the velocity-potential will vary along the tube, but will finally return to its original value when the starting point is reached. And the circulation round this circuit will be zero. But if the closed path make the circuit of the tube, the velocity-potential will continuously vary along the path, until finally, when P is reached again, the value of the function is greater (or less) than the value assumed for the starting point, by a certain definite amount which is the same for every circuit of the space. If the path be carried twice round in the same direction, the change of the function will be twice this amount, and so on. The space within a single endless tube such as an anchor-ring is doubly continuous; but much more complicated cases can be imagined. For example, an anchor-ring with a cross-connecting tube from one side to the other would be triply continuous. Thomson showed that the proper modification of the theorem is obtained by imagining diaphragms placed across the space, which are not to be crossed by any closed path drawn within the space, and the two surfaces of each of which are to be reckoned as part of the bounding surface of the space. One such diaphragm is sufficient to convert a hollow anchor-ring into a singly continuous space, two would be required for the hollow anchor-ring with cross-connection, and so on. The number of diaphragms required is always one less than the degree of multiplicity of the continuity. The paper also deals with the motion of solids in the fluid and the analogous motions of vortex-rings and their attraction by ordinary matter. These can be studied with vortex-rings in air produced by the apparatus described above. Such a ring made to pass the re-entrant corner of a wall--the edge of a window recess, for example--will appear to be attracted. A large sphere such as a large terrestrial globe serves also very well as an attracting body.