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differential, it will be so as long as the motion lasts; this is the mathematical expression of the following physical fact;

If, at any one instant, the motion of the fluid be in wave-surfaces, each surface will travel unbroken through the medium independently of all the rest; that is, as if the others did not exist.

Or, in other words, if the motion at any one instant be in wave-lines (Art. 2), then the motion can never resolve itself into wave-surfaces; and, conversely, if the motion at any one instant be in wave-surfaces, it can never break up into wave-lines.

5. If it happen that a particle be situated in two or more wavesurfaces at once, either the particle must be at rest, or the surfaces must have a contact at that point; for, if in motion, its direction must be perpendicular to all the wave-surfaces.

However complicated the motion of the fluid may be, it will always take place either in wave-lines or wave-surfaces. For the former will be the case when udx + vdy+wdz is not integrable per se or by a multiplier, and the latter when this expression is integrable.

Some of these remarks are illustrated in the following example.

Ex. Suppose the motion of the fluid to be such that udx + vdy+wdz = w2 (ydx - xdy).

In this case the differential equation of the wave surfaces is

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is the general equation of a wave-surface in such a motion of the fluid.

Hence, all the wave-surfaces are planes passing through the axis of, and the motion of the particles, being at right angles to them, will be in circular arcs parallel to the plane of xy.

All the particles in the axis of wave-surfaces intersect each other.

will be at rest, for there the

6. It does not appear possible to carry these investigations much farther in a perfectly general form; it will be necessary therefore to introduce the hypothesis of the expression udx + vdy+wdz being integrable per se. Denote its integral by p, then

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will be the equation of a wave-surface.

The effect of this hypothesis will be, to exclude from our researches many cases of motion in wave-surfaces, and all motion in wave-lines.

FLUID MOTION OF TWO DIMENSIONS.

7. I have preferred commencing my investigations with this simple case because the results more frequently admit of perfect investigation, and are more easily and briefly expressed in words than in the case of three dimensions.

The equation of continuity now to be considered is

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subject to the following condition between the arbitrary constants ƒ and g,

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In this integral the forms of the functions F and F are perfectly arbitrary, to be adapted in any example to express the law of sequence (as to space) of coexistent wave surfaces, according to the nature of the original disturbance. The arbitrariness of these functions shews that the fluid can transmit a disturbance of any kind which does not violate the continuity of the fluid. a, ẞ are arbitrary constants enabling us to fix the origin of co-ordinates in the most convenient position: they may besides contain functions of t, which depend upon the nature of the original agitation. The functions of f, g, t, which enter under

F and F enable us arbitrarily to fix the epoch from which the time is reckoned, and further to accommodate the wave-surfaces to any proposed form.

These observations will be fully illustrated in a subsequent part of this paper.

8. The object to which it will be necessary first to turn our attention in the above integral is the discovery of the meaning of the constants f. g. Whatever forms be given to F, F, whatever origin be taken for co-ordinates, whatever epoch for the time, still ƒ and g are unaffected and as an infinite number of quantities fulfilling the condition ƒ2 + g2 = 0 may be invented, and any one set will satisfy the equation d ̧34+d}2p = 0, which in a general view of the question is the only further condition to which they can be subjected, it follows that all imaginable values of ƒ and g ought equally to appear in the general integral (see Art. 27); one set of values giving only a partial solution of the proposed differential equation. Hence the general integral of the equation of continuity of a moving fluid of two dimensions is

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=F1 {f(x − a1) + g1 (y − ß1), fi, g1, t} + F{ {fi (x — a1) — gì (y — ẞ1), fi, g1, t} +F2 {ƒ1⁄2 (x − α2) +g2 ( y − ß2), fƒ2, 82, t} + F'"' {ƒ1⁄2 (x — a2) — g2 (Y — ß2), ƒ2, 82, t} + &c.............

the quantities of fi, f, f.... gì, gå, gå...... embracing all values from ∞ to +∞o. It is manifest, however, that inasmuch as each set of values can be separately made to satisfy the equation of continuity, each set will represent a possible motion, i. e. a motion of such a nature that the fluid can transmit it. Hence the general integral just exhibited furnishes us with the following physical fact, which I believe has never * before been fully accounted for; —

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-- ...

* It has been remarked that may be represented by F, (x+y√1) + F2 (x + y √ − 1) + +fi (-y-1)+ f (x -3 -1) and thence the superposition of disturbances has been inferred: but before this principle can be inferred, is it not necessary to shew that F, (x+y√√− 1) + F, (x + y √−1)+...

Any number of disturbances separately, though simultaneously, excited in a fluid medium, will be separately, independently and simultaneously transmitted through the fluid, each as perfectly as though the others did not exist. See Art. 4.

9. Having ascertained this to be the meaning of the integral in its general form, it will be sufficient now to consider the transmission of one disturbance only; and if this investigation be carried on upon the general hypothesis of a single disturbance of any kind affecting the fluid, the results will be of a general character also. This point will be gained by keeping our integral under the form

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From this we shall proceed to deduce the following results.

I. Motion cannot be represented by one of these functions alone. For, if possible, let motion be represented by

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Then the (velocity) of the particle whose co-ordinates are (x, y) would

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that is, the medium is at rest. F' is used to denote the differential coefficient of F with regard to the quantity f(x − a) +g (y − ß).

2

...

is a more general expression than one function F(x+y√1)? In the Integral Calculus, we know that C1+ C2+ C... represents only one constant C: are we certain that F1 + F2+ represents more than F? are we sure that F1, F2, F... are so essentially distinct that they cannot be united in one function?

II. Sometimes the disturbance may be such as to render it possible to introduce t entirely into the parts f(x − a) ± g (y - ẞ), so that the integral may be written

Φ = F {ƒ (x − a − T) + g (y − ß − 7), f, g}

+ F, {ƒ(x − a− T) − g (y — ß−7), ƒ, g},

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Let us in this case refer the motion of the fluid to the moveable origin, in the plane of xy, whose co-ordinates are T+a, T+ß; which will be done by writing x' + T + a, y' + + + ẞ for x and y; then the state of the fluid is expressed by

$ = F(ƒx' + gy‚ ƒ, g) + F1 (ƒx' – gy',f,g);

an equation which does not involve t; the state of the medium is therefore perfectly invariable with respect to the moveable origin. The original disturbance, then, of what kind soever it may be, is transmitted through the medium unaltered in all respects, with a velocity equal to that of the origin of co-ordinates, that is, of the point T + a, T+B; hence the velocity of transmission, in the direction of x, = d. T, and in the direction of y, d1T.

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III. It may happen that it will be impossible to introduce t entirely within the parts f(x - a) ± g (y − ß); let it however be done as nearly as possible, so that the equation may be written.

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F {ƒ(x− a− T) + g ( y − ẞ− 7), f, g, t}

+ F1{f(x-a-T) - g (y-ẞ-7), f, g, t}.

After transposing the origin as before, this becomes

$ = F {ƒx' +gy', f, g, t} + F1 {ƒx'-gy', f, g, t}.

Whence it appears that the forms of the equations of the wavesurfaces will remain unchangeable; but t entering into the parameters, shews that the magnitudes of the wave-surfaces will change with the time. In this case therefore the wave-surfaces will be transmitted through

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