stated in Article 234, which gives the development of any function F(x) in a series of sines and cosines of multiple arcs. We pass from the last proposition to those which we have just demonstrated, by giving an infinite value to the dimensions. Each term of the series becomes in this case a differential quantity'. Transformations of functions into trigonometrical series are some of the elements of the analytical theory of heat; it is indispensable to make use of them to solve the problems which depend on this theory.

The reduction of arbitrary functions into definite integrals, such as are expressed by equation (E), and the two elementary equations from which it is derived, give rise to different consequences which are omitted here since they have a less direct relation with the physical problem. We shall only remark that the same equations present themselves sometimes in analysis under other forms. We obtain for example this result

which differs from equation (E) in that the limits taken with respect to a are 0 and instead of being ∞ and +∞.

In this case it must be remarked that the two equations (E) and (E) give equal values for the second member when the variable x is positive. If this variable is negative, equation (E') always gives a nul value for the second member. The same is not the case with equation (E), whose second member is equivalent to π (x), whether we give to x a positive or negative value. As to equation (E) it solves the following problem. To find a function of a such that if x is positive, the value of the function may be (x), and if x is negative the value of the function may be always nothing2.

363. The problem of the propagation of heat in an infinite line may besides be solved by giving to the integral of the partial differential equation a different form which we shall indicate in

1 Riemann, Part. Diff. Gleich. § 32, gives the proof, and deduces the formulæ corresponding to the cases F(x) = ± F ( − x).

These remarks are essential to clearness of view. The equations from which (E) and its cognate form may be derived will be found in Todhunter's Integral Calculus, Cambridge, 1862, § 316, Equations (3) and (4). [A. F.]

the following article. We shall first examine the case in which the source of heat is constant.

Suppose that, the initial heat being distributed in any manner throughout the infinite bar, we maintain the section A at a constant temperature, whilst part of the heat communicated is dispersed through the external surface. It is required to determine the state of the prism after a given time, which is the object of the second problem that we have proposed to ourselves. Denoting by 1 the constant temperature of the end A, by 0 that of the medium,

HL
KS

we have e as the expression of the final temperature of a point situated at the distance x from this extremity, or simply

HL

e, assuming for simplicity the quantity to be equal to unity.

KS

Denoting by v the variable temperature of the same point after the time t has elapsed, we have, to determine v, the equation

KS

v-e

is that of the difference between the actual and the final temperatures; this difference u', which tends more and more to vanish, and whose final value is nothing, is equivalent at first to

F(x) -e

denoting by F(x) the initial temperature of a point situated at the distance x. Let f(x) be the excess of the initial temperature over

the final temperature, we must find for u a function which satisfies

du d'u
dt dx

the equation t =h -hu, and whose initial value is f(x), and

-A

HL

KS

final value 0. At the point A, or x = 0, the quantity v-e has, by hypothesis, a constant value equal to 0. We see by this that u represents an excess of heat which is at first accumulated in the prism, and which then escapes, either by being propagated to infinity, or by being scattered into the medium. Thus to represent the effect which results from the uniform heating of the end A of a line infinitely prolonged, we must imagine, 1st, that the line is also prolonged to the left of the point A, and that each point situated to the right is now affected with the initial excess of temperature; 2nd, that the other half of the line to the left of the point A is in a contrary state; so that a point situated at the distance from the point A has the initial temperature —ƒ(x): the heat then begins to move freely through the interior of the bar, and to be scattered at the surface.

The point A preserves the temperature 0, and all the other points arrive insensibly at the same state. In this manner we are able to refer the case in which the external source incessantly communicates new heat, to that in which the primitive heat is propagated through the interior of the solid. We might therefore solve the proposed problem in the same manner as that of the diffusion of heat, Articles 347 and 353; but in order to multiply methods of solution in a matter thus new, we shall employ the integral under a different form from that which we have considered up to this point.

to eet. This function of x and t may also be put under the form of a definite integral, which is very easily deduced from the known

value of [dgeTM. We have in fact √T=fdge, when the integral

is taken from q= − ∞o to q = + ∞. We have therefore also

b being any constant whatever and the limits of the integral the same as before. From the equation

hence the preceding value of u or e* et is equivalent to

a and n being any two constants; and we should find in the same way that this function is equivalent to

We can therefore in general take as the value of u the sum of an infinite number of such values, and we shall have

The constants a,, a,, a,, &c., and n1, n,, n,, &c. being undetermined, the series represents any function whatever of x + 2q/kt; we have

therefore u = [dqe−42 4(x+2q√/kt). The integral should be taken

from

q=

=− ∞ to q=+∞, and the value of u will necessarily satisfy du d'u the equation = k dt da

This integral which contains one arbi

trary function was not known when we had undertaken our researches on the theory of heat, which were transmitted to the Institute of France in the month of December, 1807: it has been

given by M. Laplace', in-a work which forms part of volume VIII of the Mémoires de l'École Polytechnique; we apply it simply to the determination of the linear movement of heat, From it we conclude

-x

HL
As

when t = 0 the value of u is F(x) - e√ or f(x);

hence

1

HL

KS

ƒ (x) = f ** dqe−4a 4(x) and 4 (x) = ——ƒ (x).

Thus the arbitrary function which enters into the integral, is determined by means of the given function f(x), and we have the following equation, which contains the solution of the problem,

it is easy to represent this solution by a construction.

365. Let us apply the previous solution to the case in which all points of the line AB having the initial temperature 0, the end A is heated so as to be maintained continually at the temperature 1. It follows from this that F(x) has a nul value when

-x

BL

differs from 0. Thus f(x) is equal to -e KS whenever x differs from 0, and to 0 when x is nothing. On the other hand it is necessary that on making a negative, the value of f(x) should change sign, so that we have the condition ƒ(− x) = −ƒ (x). We thus know the nature of the discontinuous function f(x); it becomes

-x

HL

HL

e Ks when x exceeds 0, and + e when x is less than 0. We must now write instead of x the quantity +2q√kt. To find

u or

from

[*° dqe-a — — f (x+2q√kt), we must first take the integral

x+2q√kt = 0 to x+2q√kt = ∞o,

1 Journal de l'École Polytechnique, Tome VIII. pp. 235-244, Paris, 1809. Laplace shews also that the complete integral of the equation contains only one arbitrary function, but in this respect he had been anticipated by Poisson. [A. F.]