Mathematical and Physical Papers, Volume 4University Press, 1910 |
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angular velocity applied approximately axis boundary calculated canal centimetres centre circular component condition constant convergent corresponding cos² curve denotes diagram direction disc displacement distance disturbance dx dy ellipsoid equal equation equilibrium finite fluid force forcive formula free surface function given gives gravity gyrostat Hence impulse incompressible infinitely small infinitesimal initial integral inviscid irrotational K₁ kinetic energy Laplace Laplace's liquid Lord Rayleigh magnetic mathematical maximum momentum moving Natural Philosophy Osborne Reynolds paper parallel period perpendicular plane portion positive pressure problem Rayleigh rear result ridge rigid ring rotation round side sinusoidal small in comparison solid solution space spherical stable steady motion suppose surface-pressure theory tides values velocity potential viscosity vortex rings vortices water-surface wave-length waves xzav zero аф
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Page 94 - ... and he cherishes the hope that his mathematical theorems on abstract hydrokinetics are of some interest in physics as illustrating the great question of the eighteenth and nineteenth centuries : — Is action at a distance a reality, or is gravitation to be explained, as we now believe magnetic and electric forces must be, by action of intervening matter?
Page 124 - I, p., and N are such as to make the shortest distances between different positions of the core or cores considerable in comparison with the core's diameter. Consider, for example, the simplest case (§ 12, fig. 5) of two simple rings linked together. 18. Go back now to the simple circular Helmholtz ring. It is clear that there must be a shape of absolute maximum energy for given vorticity and given impulse, if we introduce the restriction that the figure is to be a figure of revolution — that...
Page 72 - L, &c. ...(11). [Memorandum : — Lines of reference fixed relatively to the body.] But inasmuch as (11) remains unchanged when the lines of reference are altered to any other three lines at right angles to one another through P, it is easily shown directly from (6), (7), and (9) that if, altering the notation, we take u, v, w to denote the components of the velocity of P parallel to three fixed rectangular lines, and...
Page 6 - HYDRODYNAMICS that the dynamical explanation now suggested is mathematically demonstrated so far as to include the very approximate identity of the periods of the vibrating particles of the incandescent vapour with those of their corresponding fundamental modes at the lower temperature at which the vapour exhibits its remarkable absorbingpower for the sodium light. A very remarkable discovery made by Helmholtz regarding the simple vortex ring is that it always moves, relatively to the distant parts...
Page 105 - K', &c., are the desired equations of motion. 6. The hypothetical mode of application of K, K',...(§ 1) is impossible, and every other (such as the influence of gravity on a real liquid at different temperatures in different parts) is impossible for our ideal " liquid," that is to say, a homogeneous incompressible perfect fluid.
Page 124 - Imagine the given vorticity, instead of being distributed in a symmetrical circular ring, to be distributed in a ring still with a circular axis, but thinner in one part than in the rest. It is clear that, with the same vorticity and the same impulse, the energy with such a distribution is greater than when the ring is symmetrical. But now let the figure of the cross section of the ring, instead of being approximately circular, be made considerably oval. This will diminish the energy with the same...
Page 145 - ... the canal. The more approximately nodal character of the tides on the north coast of the English Channel than on the south or French coast, and of the tides on the west or Irish side of the Irish Channel than on the east or English side, is probably to be accounted for on the principle represented by this factor, taken into account along with frictional resistance, in virtue of which the tides of the English Channel may be roughly represented by more powerful waves travelling from west to east,...
Page 82 - ... in. per sec.) does not sensibly disturb the smoothness of the reflecting surface. A gentle zephyr flitting along the surface from point to point, may be observed to destroy the perfection of the mirror for a moment, and on departing, the surface remains polished as before ; if the air have a velocity of about a mile an hour, the surface of the water becomes less capable of distinct...
Page 5 - ... models shown to the Society, which consisted of only a single wire knotted in various ways. It seems, therefore, probable that the sodium atom may not consist of a single vortex line ; but it may very probably consist of two approximately equal vortex rings passing through one another like two links of a chain. It is, however, quite certain that a vapour consisting of such atoms, with proper volumes and angular velocities in the two rings of each atom, would act precisely as incandescent sodium-vapour...
Page 190 - ... given moment of momentum, is a configuration of minimum energy for given moment of momentum, subject to the condition that the shape is constrainedly an ellipsoid. From this proposition, which is easily verified, in the light of § 778 of Thomson and Tait's Natural Philosophy, it follows that, with the ellipsoidal constraint, the equilibrium is stable.