$ (x, y) = f (x, y, 0), † (x, y) =ƒ, (x, y, 0).

Lastly, let the proposed differential equation be

the coefficients a, b, c, d are known numbers, and the order of the equation is indefinite.

The most general value of v can only contain one arbitrary function of x; for it is evident, from the very form of the equation, that if we knew, as a function of x, the value of v which corresponds to t = 0, all the other values of v, which correspond to successive values of t, would be determined. To express v, we should have therefore the equation vetDp (x).

We denote by Do the expression

=

that is to say, in order to form the value of v, we must develop according to powers of t, the quantity

d

and then write

d.x

et{aa2+ba1+ca®+da®+&c.) ̧

instead of a, considering the powers of a as orders

of differentiation. In fact, this value of v being differentiated with respect to t only, we have

It would be useless to multiply applications of the same process. For very simple equations we can dispense with abridged expressions; but in general they supply the place of very complex investigations. We have chosen, as examples, the preceding equations, because they all relate to physical phenomena whose analytical expression is analogous to that of the movement of heat. The two first, (a) and (b), belong to the theory of heat; and the three

following (c), (d), (e), to dynamical problems; the last (ƒ) expresses what the movement of heat would be in solid bodies, if the instantaneous transmission were not limited to an extremely small distance. We have an example of this kind of problem in the movement of luminous heat which penetrates diaphanous media.

404. We can obtain by different means the integrals of these equations we shall indicate in the first place that which results from the use of the theorem enunciated in Art. 361, which we now proceed to recal,

we see that it represents a function of x; for the two definite integrations with respect to a and p make these variables disappear, and a function of a remains. The nature of the function will evidently depend on that which we shall have chosen for $ (a). We may ask what the function 4 (z), ought to be, in order that after two definite integrations we may obtain a given function f(x). In general the investigation of the integrals suitable for the expression of different physical phenomena, is reducible to problems similar to the preceding. The object of these problems is to determine the arbitrary functions under the signs of the definite integration, so that the result of this integration may be a given function. It is easy to see, for example, that the general integral of the equation

would be known if, in the preceding expression (a), we could determine (a), so that the result of the equation might be a given function f(x). In fact, we form directly a particular value of v, expressed thus,

v=e-mt

cos px,

and we find this condition,

m = ap2 + bp* + cp" + &c.

giving to the constant a any value. We have similarly

v = [ d1¢ (4) e −t(ap2+bp1+cp2+ &e) cos (px − px).

It is evident that this value of v satisfies the differential equation (f); it is merely the sum of particular values.

Further, supposing t = 0, we ought to find for v an arbitrary function of x. Denoting this function by ƒ (x), we have

ƒ (x) = [d14 (1) fdp cos (px − p2).

f

Now it follows from the form of the equation (f), that the most general value of v can contain only one arbitrary function of x. In fact, this equation shews clearly that if we know as a function of the value of v for a given value of the time t, all the other values of which correspond to other values of the time, are necessarily determined. It follows rigorously that if we know, as a function of t and x, a value of v which satisfies the differential equation; and if further, on making t = 0, this function of x and t becomes an entirely arbitrary function of x, the function of x and t in question is the general integral of equation (f). The whole problem is therefore reduced to determining, in the equation above, the function (a), so that the result of two integrations may be a given function f(x). It is only necessary, in order that the solution may be general, that we should be able to take for f(x) an entirely arbitrary and even discontinuous function. It is merely required therefore to know the relation which must always exist between the given function f(x) and the unknown function (2). Now this very simple relation is expressed by the theorem of which we are speaking. It consists in the fact that when the integrals are taken between infinite limits, the function (2) is 1 f(a); that is to say, that we have the equation

2π

From this we conclude as the general integral of the proposed equation (ƒ),

v =

1

2 = [ "dx f ( x ) [ " " dp e- t (ap2 + bp2 +cp2+&c) cos (px—p2) .......(c). (x) e-t(ap2+bp1+cp'+&c)

2π

-∞

which expresses the transverse vibratory movement of an elastic plate', we must consider that, from the form of this equation, the most general value of v can contain only two arbitrary functions of x for, denoting this value of v by f(x, t), and the function

d

dtf(x, t) by f'(x, t), it is evident that if we knew f(x, 0) and

dv dt

f'(x, 0), that is to say, the values of v and at the first instant, all the other values of v would be determined.

This follows also from the very nature of the phenomenon. In fact, consider a rectilinear elastic lamina in its state of rest: x is the distance of any point of this plate from the origin of coordinates; the form of the lamina is very slightly changed, by drawing it from its position of equilibrium, in which it coincided with the axis of x on the horizontal plane; it is then abandoned to its own forces excited by the change of form. The displacement is supposed to be arbitrary, but very small, and such that the initial form given to the lamina is that of a curve drawn on a vertical plane which passes through the axis of x. The system will successively change its form, and will continue to move in the vertical plane on one side or other of the line of equilibrium. The most general condition of this motion is expressed by the equation

Any point m, situated in the position of equilibrium at a distance x from the origin 0, and on the horizontal plane, has, at

1 An investigation of the general equation for the lateral vibration of a thin elastic rod, of which (d) is a particular case corresponding to no permanent internal tension, the angular motions of a section of the rod being also neglected, will be found in Denkin's Acoustics, Chap. 1x. §§ 169-177. [A. F.]

the end of the time t, been removed from its place through the perpendicular height v. This variable flight v is a function of x and t. The initial value of v is arbitrary; it is expressed by any function (x). Now, the equation (d) deduced from the fundamental principles of dynamics shews that the second fluxion and the fluxion of the fourth

of v, taken with respect to t, or

d'v

order taken with respect to x, or are two functions of x and t,

dx

which differ only in sign. We do not enter here into the special question relative to the discontinuity of these functions; we have in view only the analytical expression of the integral.

We may suppose also, that after having arbitrarily displaced the different points of the lamina, we impress upon them very small initial velocities, in the vertical plane in which the vibrations ought to be accomplished. The initial velocity given to any point m has an arbitrary value. It is expressed by any function (x) of the distance x.

It is evident that if we have given the initial form of the system or (x) and the initial impulses or (x), all the subsequent states of the system are determinate. Thus the function v or f(x, t), which represents, after any time t, the corresponding form of the lamina, contains two arbitrary functions (x) and (x).

To determine the function sought f(x, t), consider that in the equation

denoting by q and a any quantities which contain neither x nor t. We therefore also have