f being an arbitrary constant. This quantity gives rise to a difficulty. For though the proposed problem is sufficiently specific, yet it affords no data for the determination of f, and consequently the curve sought remains as undetermined, and the problem apparently as unsolved, as ever; all that we can gather from the above integral being, that a straight line whose position depends on the value of ƒ will fulfil the proposed conditions.

It may be said, however, that any value given at pleasure to ƒ will determine a line answering the conditions of the question; but it is clear that a line so found can only be considered as a partial solution; inasmuch as the fixing upon a particular value of f tacitly implies the possession of data enabling us to decide upon that value in preference to all others. Now such a decision cannot be received, unless the data which led to it are supplied by the conditions of the proposed problem; and, as we have seen, no such data exist; consequently no particular value of ƒ can be received. Thus it appears, that though the integral

y =ƒx ± √ a2ƒ2 + b2*

∞ to +∞;

furnishes partial solutions of the proposed problem without number, it does not present us with the required curve. The only resource left is, to employ equally all possible values of ƒ from all having an equal claim to have weight in the general interpretation of the above integral. This will present us with an infinite number of straight lines, not drawn at random, but according to a law expressed by

y=ƒx + √ a°ƒ" + b2,

and by the intersections of consecutive lines forming a curve to which they all have an equal relation, being tangents, which is allowed to be the curve required.

*Since ƒ admits the sign

as well as +, we have, by taking all the variations of sign, four straight lines; one in each portion of space comprehended between the coordinate axes. See Arts. 18, 21.

The general inference from this reasoning appears to be this: that when the integral of a differential equation contains constants (introduced by integration), the values of which the proposed conditions of the problem are not sufficient to determine, the singular solution* is the proper integral for that particular problem.

28. Upon these grounds I state the following general principle,

A wave may, at any moment, be resolved into plane component waves, each of which is a tangent to the original wave. These components may

be supposed to be uniformly transmitted with the velocity a, and at any time they may be compounded into a single wave by taking that surface to which they are all simultaneously tangents.

Hence if the form of a wave-surface at any one instant be known, its form at any other time will be determinable from geometrical principles.

The thickness (or, as it is sometimes called, the breadth). of a wave is never altered by transmission.

Ex. 1. In a quiescent medium let us suppose one of its particles to expand, pushing equally from its centre on every side the adjacent particles. The effect of such a disturbance will be a sudden condensation in its neighbourhood, which we may divide into concentric spherical wave-surfaces, for each one of which is constant; though from surface to surface may vary. Now resolve any one of these surfaces into its components by drawing an infinite number of tangent planes to it; each one is transmitted with the velocity a parallel to its edge, and thus at the end of any period they will be equidistant from the centre of original disturbance, and be tangent planes to a spherical surface, which is therefore the form of the wave at any moment.

Ex. 2. In a quiescent medium let all the particles situated in a given

* Or rather the solution determined by the usual method of finding the singular solution, for as is well known such a one may not happen to be a singular solution but a particular integral.

straight line suddenly expand equally. In this case each surface of equal density is a cylinder with hemispherical ends; the given line being the axis, and the extremities of the line being the centres of the hemispheres.

By resolving each wave-surface into component waves, by drawing tangent planes to the cylinder and hemispheres, and supposing these components transmitted, each parallel to itself with the velocity a, it will appear that at any time each wave-surface is of the form of a cylinder with hemispherical ends. The radius of the cylinder increases with the uniform velocity a.

29. It is sufficiently manifest, from what has already been done, that the form of a wave-surface depends only on the form of it at any given moment, or indeed only on its initial form; its magnitude, but not its nature, depends on the time. This is equivalent to saying, that the nature of the equation of a wave-surface depends only on the form of the given disturbance, while the parameters of that equation depend upon the time. This will sometimes enable us to determine the properties of the wave when it is curved, without employing the general integral

4 = ΣF {ƒ (x − a) + g (y − ß) + h (≈ − y) − at, f, g, h}.

Having determined the form of the wave-surface upon the principles of Art. 28, let its equation be x (x, y, z; A, B, C...) = 0; in which A, B, C... are parameters are parameters depending on the time only; also let the state of the fluid be expressed by the equation

Now for any point in a wave-surface o = constant; that is, any values of x, y, z which make x = 0, will make y = constant. therefore can only be made to change in value by writing different values of A, B, C. We may therefore say that is a function of A, B, C... only, which are connected by the equation x = 0. We may therefore consider x, y, z as functions of A, B, C... by virtue of x = 0; and having found the values of dp, dp, dip in terms of the partial

differential coefficients of with regard to A, B, C... we may change the equation

αφ = a2 (d ̧2 + d ̧3p + d22p)

y

into another not containing x, y, z. It is to be observed that A, B, C... are dependent only on t and quantities which are absolutely constant; B, C,... are therefore expressible in terms of A, and consequently there is only one independent parameter A. Hence the process above pointed out will give us an equation between dip, dip, do and A: that is, o will be a function of A and t only.

30. As one of the simplest examples of this process, let us suppose the form of the waves to be spherical. Then x = 0 is

x2 + y2 + x2 = r2,

r being the parameter A. Now being a function of t and r only, and x, y, ≈ being functions of r only, we have

.. di (rp) = a2d2 (rp).

This equation being integrated gives

rp = F(r – at) +f(r + at),

which expresses the state of the fluid when the motion is in spherical waves: the former term F (rat) shews that spherical waves may diverge from a fixed centre; and the latter f(r+at) that they may converge towards a fixed centre. The velocity of either wave is a. The motion of each particle is directed towards or from the fixed centre Art. 3, and its velocity = d,p

31. Since in general the motion of each particle (Art. 3) is perpendicular to the wave-surface in which it is situated, the motions will be directed to or from focal lines, which may be fixed or moveable: and if the motion of the wave were perpendicular to its front we might always deduce the law of variation of the velocity of a particle; but inasmuch as the direction of a wave's motion entirely depends on its form, and is never in the direction of a normal except at those points where the curvature is a maximum or minimum, or when the waves are spherical, no general law of the velocity can be deduced; but we must first find the value of by the method of Art. 29, and then the velocity may be obtained.

32. When the form of the waves is spherical the law of variation of density may be found.

For a given part of this wave (as the front or the middle, ...) r-at is constant, and therefore F" (rat) is constant = Aa suppose,

wherefore as the wave travels through the medium, the density of a given part of it varies as e, which rapidly diminishes.