476 



REPORT 1880. 



After that continue the process further, and the result will be that each time the 

 containing canister is allowed to go round freely in space, the fluid will tend to a 

 condition in which a certain portion of the original vortex core gets filtered into a 

 position next to the boundary, and the fluid within it tends to a more and more 

 nearly uniform mixture of vortex with irrotational fluid. This central vortex-sponge, 

 on repetition of the process of preventing the canister from going roimd, and again 

 leaving it free to go round, becomes more and more nearly irrotational fluid, and the 

 outer belt of pure vortex becomes thicker and thicker. The final condition towards 

 which the whole tends is a belt constituted of the original vortex core now next the 

 boundary ; and the fluid which originally revolved irrotationally round it now 

 placed at rest within it, being the condition (VIII, above) of absolute mmimum 

 energy. Begin once more with the condition (VII. above) of absolute maximum 

 energy, and leave the fluid to itself, whether mth the canister free to go round 

 sometimes, or always held fixed, provided only it is ultimately held from going 

 round in space ; the ultimate condition is always the same, viz., the condition (VIII.) 

 of absolute minimum energy. 



XI. That there may be an infinite number of configurations of stable motions, 

 each of them having the energy of a thorough minimum as said in IV. above, we 

 see, by considering the case in which the cylindric boundary of the containing can- 

 ister consists of two wide portions communicating by a narrow passage, as shown in 

 the sketch. If such a canister be completely filled with irrotationally moving fluid 

 of uniform vorticity, the stream luies must be something like those indicated in the 

 sketch. 



Hence if a small portion of the whole fluid is irrotational, it is clear that there 

 may be a minimum energy, and therefore a stable configuration of motion, with the 

 whole of this in one of the wide parts of the canister ; or the whole in the other ; 

 or any proportion in one and the rest in the other ; or a small portion in the elliptic 

 whirl in the connecting canal, and the rest divided in any proportion between the 

 two wide parts of the canister. 



•5. On Inverse Figures in Geoiietry. Bij Professor H. J. S. Smith, M.A., 



F.K.8. 



6. On a Mathematical Solution of a Logical Problem. By Professor 

 H. J. S. Smith, M.A., F.B.8. 



7. On the Bistrihution of Circles on a SpJiere. By Professor H. J. S. 



Smith, M.A., F.B.8. 



€. Notes on Non-Euclidian Geometry. By Robert S. Ball, LL.D., F.B.S. 



The problem I propose to consider relates to the kinematics of a rigid body in 

 non-Euclidian space. I can hardly say that the commimication is exactly novel, as 



