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R. corpore ovali; latere sinistro scabriusculo, etuberculato, olivaceofusco, ferruginascente, annellis punctorum albidorum ocellatim picto: pinnæ dorsalis analisque radiis inclusis, indivisis: dentibus minutis, uniseriatis: maxillæ superiore ambituque oculorum antice tuberculato-cornutis.

D. 9195; A. 6971; C. 15-17.

P. sinistra 10 v. 11; dextra 9 v. 10.

V. sin. 6; dext. 5 v. 6.

R. maderensis, nob. in Proceedings of the Zool. Soc. 1833. I. p. 143.

TAB. VI. f. 1. magn3, nat3.

f. 2. Ejusdem pars anterior lateris dextri.

Hab. rarior in statione navium prope urbem Funchalensem. Ad Insulam Portûs Sancti frequentior dicitur.

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IX.

On Fluid Motion, so far as it is expressed by the Equation of
Continuity. By S. EARNSHAW, M.A. of St John's College.

[Read March 21, 1836.]

THE difficulty of this subject is so universally admitted, that I hope it will be received as a sufficient excuse for bespeaking the reader's indulgence should any thing occur, in the course of this paper, which he may judge not sufficiently borne out by the arguments on which it is sought to be established.

Though the subject of this communication is by no means new, yet what is brought forward in it will be found to possess some novelty both as to the results obtained, and the manner of treating the subject. Hitherto, nothing more could be done, beyond investigating the differential equations of fluid motion, than to endeavour to generalize the results obtained from a particular integral of the equation of continuity d2+d ̧2+d, p=0. It is manifest, however, that the results of such generalization from a particular case, how skilfully soever deduced, must at least be clogged with some degree of uncertainty, and be therefore in some measure unsatisfactory. But in consequence of the discovery of the general integral of the equation

2

d22 + d ̧2 + d2' p = 0,

not only is the difficulty of the subject shifted farther from the threshold of our researches, being reduced to that of interpreting this integral, but we are able to proceed with a much greater degree of generality.

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GENERAL PROPERTY.

1. If in a fluid medium we describe a surface whose differential equation is

udx + vdy + wdz

= 0,

the motion of each particle through which this surface passes is in the direction of the normal at the point where the particle is situated.

u, v, w are the velocities, estimated parallel to the co-ordinate axes, of the particle whose co-ordinates are x, y, z. Let there be another particle in the surface very near to this; and let its co-ordinates be x+dx, y+dy, z+ds; and let ds be their distance from each other; by a, ẞ, y denote the inclinations of ds to the axes. Let also V be the velocity of the former particle, and by a', B', y denote the inclinations of the direction in which takes place to the three axes.

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=

u dx v dy w dz
V'ds Vds V'ds

+

+

Vds. (cos a' cos a + cos ẞ' cos ß + cos y cos y):

and since, from the nature of the case, neither nor ds is equal to

zero,

.. cos a' cosa + cos B' cos ẞ + cos y cos y = 0.

But the left hand member expresses the cosine of the inclination of V to ds, which being equal to zero, V and ds must be at right angles to each other; that is, the motion of the particle whose coordinates are x, y, z, takes place in a direction perpendicular to the surface whose equation is

udx + vdy+wdx = 0.

We may simultaneously draw surfaces of this nature through all parts of the fluid in motion, and shall thus obtain the direction of the motion of every particle. It is manifest that these surfaces are very analogous to the level-surfaces, which occur in investigations concerning the equilibrium of heterogeneous fluids.

If the expression udx + vdy+wdz be integrable either immediately or by a multiplier, the integral of the equation

2.

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which will furnish the surfaces alluded to in last article. But if the above expression should neither be integrable at once nor by a multiplier, the integral of the above equation will be of the form

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and will denote, not a series of surfaces, but a series of curve lines.

3. It appears that all the particles through which the surface passes whose equation is

udx + vdy+wdz = 0,

are connected by the common property proved in Art. 1; and as we have no other idea of a wave-surface than that it is the locus of particles in a similar state of disturbance, we may be permitted to take the above equation as the expression of that similarity which constitutes a wave-surface; or in other words, we may assume the equation

udx + vdy+wdz = 0,

as the mathematical definition of a wave-surface, or of a wave-line, as the case may be.

By the assistance of this definition we may enunciate the proposition of Art. 1 in these terms;

The motion of every particle of the fluid is perpendicular to the wave-surface in which it is situated.

4. It is proved by Pontécoulant in his "Théorie Analytique du Système du Monde," Tom. I. p. 163, and by most other writers on Hydrodynamics, that if udx + vdy+wdz be at any one instant a complete

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