the integral sign in order that the expression fdg

4

qx may be

equal to a given function, the integral being taken from q nothing to q infinite'? But without stopping for different consequences, the examination of which would remove us from our chief object, we shall limit ourselves to the following result, which is obtained by combining the two equations (e) and (e).

If we took the integrals with respect to a from

the result of each integration would be doubled, which is a necessary consequence of the two conditions

and

f(a)=-ƒ(-a) and F(x) = F (− a).

We have therefore the two equations

We have remarked previously that any function

(x) can

always be decomposed into two others, one of which F(x) satisfies the condition F(x) = F(x), and the other f(c) satisfies the condition f(x)=-ƒ(-x). We have thus the two equations

0= ["dal (2) sin qz, and 0 =["dif (a) cos gz,

0:

1 To do this write +x-1 in f(x) and add, therefore

cos qx dq=f (x - 1) + ƒ ( − x √√ − 1),

2 fo cos

which remains the same on writing - x for x,

1

therefore Q== Sdx [ƒ (x√− 1) +ƒ ( − x √ − 1)] cos qx dx.

π

Again we may subtract and use the sine but the difficulty of dealing with imaginary quantities recurs continually. [R. L. E.]

whence we conclude

π [F (x) + ƒ (x)] = π$ (x) = ["dq sin qæ **dıf (a) sin qu

qx

or lastly', () = 1

φ

dap

(4) ["*"dq (sin qæ sin q1+ cos qæ cos q1);

π 81

d. (a) ["dag cos q ( - )............

The integration with respect to q gives a function of x and a, and the second integration makes the variable a disappear.

Thus the function represented by the definite integral fdqcosq (x− a)

has the singular property, that if we multiply it by any function.

(a) and by dx, and integrate it with respect to a between infinite limits, the result is equal to π (x); so that the effect of the integration is to change a into x, and to multiply by the number π.

362. We might deduce equation (E) directly from the theorem 1 Poisson, in his Mémoire sur la Théorie des Ondes, in the Mémoires de l'Académie des Sciences, Tome 1., Paris, 1818, pp. 85—87, first gave a direct proof of the theorem

in which k is supposed to be a small positive quantity which is made equal to 0 after the integrations.

Boole, On the Analysis of Discontinuous Functions, in the Transactions of the Royal Irish Academy, Vol. xxI., Dublin, 1848, pp. 126-130, introduces some analytical representations of discontinuity, and regards Fourier's Theorem as unproved unless equivalent to the above proposition.

Deflers, at the end of a Note sur quelques intégrales définies &c., in the Bulletin des Sciences, Société Philomatique, Paris, 1819, pp. 161-166, indicates a proof of Fourier's Theorem, which Poisson repeats in a modified form in the Journal Polytechnique, Cahier 19, p. 454. The special difficulties of this proof have been noticed by De Morgan, Differential and Integral Calculus, pp. 619, 628.

An excellent discussion of the class of proofs here alluded to is given by Mr J. W. L. Glaisher in an article On sin and coso, Messenger of Mathematics, Ser. 1., Vol. v., pp. 232-244, Cambridge, 1871. [A. F.]

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 nothing.

363. The problem of the propagation of heat in au 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 formule 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

we have e KS as the expression of the final temperature of a point situated at the distance 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

v-e

KS 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)

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

F. H.

23

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

final value 0. At the point A, or x = 0, the quantity ve 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 e*e*. 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 √=dge, when the integral

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

√ = √dqe="