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4. On the Motion of Rigid Solids in a Liquid circulating Irrotationally through Perforations in them or in any Fixed Solid

Published online by Cambridge University Press:  15 September 2014

William Thomson
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Extract

1. Let ψ, φ … be the values at time t, of generalised co-ordinates fully specifying the positions of any number of solids movable through space occupied by a perfect liquid destitute of rotational motion, and not acted on by any force which could produce it. Some or all of these solids being perforated, let χ, χ′, χ″, &c., be the quantities of liquid which from any era of reckoning, up to the time t, have traversed the several apertures.

Type
Proceedings 1871-72
Information
Proceedings of the Royal Society of Edinburgh , Volume 7 , 1872 , pp. 668 - 682
Copyright
Copyright © Royal Society of Edinburgh 1872

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References

page 668 note * The title and first part (§§ 1 … 13) are new, The remainder (§§ 14, 15) was communicated to the Royal Society at the end of last December.—W. T. September 26, 1872.

page 670 note * Or ∫Fds if F denote the tangential component of the absolute velocity of the fluid at any point of the circuit, and ∫ds line integration once round the circuit.

page 670 note † References distinguished by the initials V. M. are to the part already published of the author's paper on Vortex Motion. (Transactions of the Royal Society af Edinburgh, 1867–8 and 1868–9.)

page 670 note ‡ The general limitation, for impulsive action, that the displacements effected during it are infinitely small, is not necessary in this case. Compare § 5 (11), below.

page 675 note * Proposition I. of article on “The Forces experienced by Solids immersed in a Moving Liquid” (Proceedings R. S. E, February 1870, reprinted in Volume of Electric and Magnetic papers, §§ 733 … 740).

page 675 note † See Proceedings R. S. E., Session 1870–71, or reprint in Philosophical Magazine, Nov. 1871.

page 676 note * Which means that if the globe, after any motion whatever, great or small, comes again to a position in which it has been before, the integral quantity of liquid which this motion has caused ta cross any fixed area is zero.

page 678 note * This follows immediately from the proposition (Thomson and Tait's “Natural Philosophy,” § 496) that any function V, satisfying Laplace's equation throughout a spherical space has for its mean value through this space its value at the centre. For satisfies Laplace's equation.

page 680 note * “On the Forces Experienced by Small Spheres under Magnetic Influence, and some of the Phenomena presented by Diamagnetic Substances” (Cambridge and Dublin Mathematical Journal, May 1847); and “Remarks on the Forces experienced by Inductively Magnetised Ferromagnetic or Diamagnetic Non-crystalline Substances” (Phil. Mag. October 1850). Reprint of Papers on Electrostatics and Magnetism, §§634–668. Macmillan, 1872.

page 680 note † Tait and Steele's “Dynamics of a Particle,” § 149 (15).