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Similarly it may be shewn, that the density of a given part of the converging wave varies as e, which rapidly increases.

B

33. If a plane-wave be transmitted through the medium, the particles as it successively reaches them are displaced with the same velocity; and there is a certain relation between the density and the velocity of displacement which holds good for all plane-waves.

For, denoting the velocity of displacement of a particle by v, and the density as before,

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and by referring this to a moveable origin (as in Art. 19), we have

v = ± F" (ƒx' + gy' + hx') = ± F" (p);

which is independent of t, and is constant for all particles situated in the same part of the wave, because for such particles p is constant.

Again,

a2 log. £, = − d‚p = aF" (fx + gy + hz − at) = ± av ;

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If v be reckoned positive in the direction of the wave's motion, and negative when in the opposite direction,

p = p'ea.

Hence in plane-waves all points of equal velocity are points of equal density.

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Hence, at points of mean density the particles are stationary; at points of condensation the particles are moving forwards; and at points of rarefaction they are moving backwards.

34. The property which was proved in Art. 23, respecting planewaves, may be extended to curvilinear waves; and we may shew generally,

That no wave-surface, terminated abruptly by sharp edges, can be transmitted through a medium unless its edges rest upon the boundaries of the medium.

For, the only expression for a curvilinear wave which the equation of continuity furnishes is

p=Σ.F{ƒ(x− a) + g ( y − ß) + h (≈—y)—at, f, g, h};

which, as before observed, teaches that we may suppose the wave composed of plane-waves, which we may suppose transmitted through the medium, and then we shall have the true wave-surface by taking that to which they are all tangents.

Now, suppose a wave-surface terminated abruptly to be by some means or other excited in a medium. Upon referring to the above expression for p, we should find that the tangent-planes at the edges of the wave-surface, or rather the component waves, represented by these tangent-planes, and expressed by terms of the form

F. {ƒ'(x − a) +g' (y − ß) + h' (≈ − y) — at, ƒ', g', h'},

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stretch out indefinitely beyond the boundary of the wave-surface into the medium: and when these components are transmitted and afterwards compounded into one wave, the portions of these waves which (as it were) hung over the proper wave-surface must remain. Hence it

appears,

First, That a wave-surface terminated abruptly by sharp edges cannot be excited in a medium: and

Secondly, That if such a wave were excited, it could not be transmitted in that form.

Hence, if a curvilinear wave in traversing a medium meet with a fixed screen in which is an orifice, the part of the wave which passes through the orifice must afterwards abut with its edges upon the back of the screen.

35. In the Undulatory Theory of Light, each point in the front of a wave is considered as the origin of an indefinitely small wave. (See Airy's Tracts, 2d Edit. p. 267. Art. 21). This hypothesis, however, as is well known, is affected by a very troublesome difficulty. "What is to limit the waves diverging from each of these small sources of motion? The disturbance spreads generally in a spherical form, so that the front of each little wave is a sphere: are we to suppose the sphere complete, so that each small undulation is propagated backwards as well as forwards?" (Airy, Art. 22.)

It will have been perceived from what has been done in this paper, that in the transmission of waves by pressure through an elastic medium, the tangent-planes are to be taken, and that these tangent-planes move only in the direction of the wave's motion. Might not the same hypothesis be applied in the Undulatory Theory of Light, in which case the above difficulty would be avoided?

X.

On the Motion of a System of Particles, considered with reference to the Phenomena of Sound and Heat. By PHILIP KELLAND, B. A. Fellow and Tutor of Queens College, Cambridge.

[Read May 16, 1836.]

INTRODUCTION.

IN a former Memoir, it was my endeavour to simplify the equations of motion of a system of particles attracting each other with forces varying according to any law. The discussion of these equations was restricted to their bearing on the phenomena of Light, on which account one of the three was left untouched.

It appeared that the hypothesis of attractive forces led to the result that two of the equations corresponding to the motion in a plane perpendicular to the direction of transmission, indicated vibratory motion, whilst the third assumed a form altogether different, shewing that, as far as it was concerned, the motion was not vibratory.

On the other hand, the hypothesis of repulsive forces would give the motion in the direction of transmission vibratory, whilst the contrary would be the case in a plane perpendicular to this direction.

The discussion of the equations corresponding to motion in the direction of transmission is the object of the present memoir.

It is not improbable that to the action of forces, such as those of which we are treating, a considerable number of the phenomena of nature may be referred; but on account of our imperfect knowledge of the analogies subsisting between phenomena which apparently differ widely from each other in some essential points, we are obliged to restrict ourselves to the most simple, or to those which have been the most carefully examined.

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