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VI. — On Vortex Motion. By Sir W. Thomson. 



(Read 29th April 1867.) 

 (§ § 1-59 recast and augmented 28th August to 12th November 1868.) 



1. The mathematical work of the present paper has been performed to illus- 

 trate the hypothesis, that space is continuously occupied by an incompressible 

 frictionless liquid acted on by no force, and that material phenomena of every 

 kind depend solely on motions created in this liquid. But I take, in the first 

 place, as subject of investigation, a finite mass of incompressible frictionless* fluid 

 completely enclosed in a rigid fixed boundary. 



2. The containing vessel may be either simply or multiply continuous.^ And 

 I shall frequently consider solids surrounded by the liquid, which also may be 

 either simply or multiply continuous. It will not be necessary to exclude the sup- 

 position that any such solid may touch the outer boundary over some finite area, 

 in which case it is not surrounded by the liquid ; but each such solid, whether 

 surrounded by the liquid or not, and whether moveable or fixed, must be con- 

 sidered as a part of the whole boundary of the liquid. 



3. Let the whole fluid be given at rest, and let no force, except pressure from 

 the containing vessel, or from the surfaces of solids immersed in it, ever act on any 

 part of it. Let there be any number of solids, perfectly incompressible, and of the 

 same density as the fluid ; but either perfectly rigid, or more or less flexible, with 

 perfect or imperfect elasticity. Some of these may at times be supposed to lose 

 rigidity, and become perfectly liquid ; and portions of the liquid may be supposed 

 to acquire rigidity, and thus to constitute solids. Let the solids act on one 

 another with any forces, pressures, frictions, or mutual distant actions, subject 

 only to the law of " action and reaction." Let motions originate among them 

 and in the liquid, either by the natural mutual actions of the solids or by the 

 arbitrary application of forces to them during some limited time. It is of no 

 consequence to us whether these forces have reactions on matter outside the con- 

 taining vessel, so that they might be called " natural forces" in the present state 

 of science (which admits action and reaction at a distance); or are applied 

 arbitrarily by supernatural action without reaction. To avoid circumlocution, 



* A frictionless fluid is defined as a mass continuously occupying space, whose contiguous 

 portions press on one another everywhere exactly in the direction perpendicular to the surface 

 separating them. 



■j" Helmholtz — Ueber Integrate der hydrodynamisclien Gleichungen, welche den Wirbelbeicegungen 

 entsjirechen : Crelle (1858); translated by Tait in Phil. Mag. 1867, i. Riemantst — Lehrsatze aus 

 der Analysis situs, fyc. Crelle (1857). See also § 58, below. 



VOL. XXV. PART I. 3 K 



