421. To give an example of the use of the last two formulæ,

d'v d'v
dx2 dy2

let us consider the equation + =0, which relates to the

uniform movement of heat in a rectangular plate. The general integral of this equation evidently contains two arbitrary functions. Suppose then that we know in terms of x the value of v when y = 0, and that we also know, as another function of x, the

dv

value of when y = 0, we can deduce the required integral from dy that of the equation

which has long been known; but we find imaginary quantities under the functional signs: the integral is

v = $(x+y√− 1) + $ (x − y√− 1) + W.

The second part W of the integral is derived from the first by integrating with respect to y, and changing into y.

It remains then to transform the quantities 4(x+y√−1) and $ (x-y√-1), in order to separate the real parts from the imaginary parts. Following the process of the preceding Article we find for the first part u of the integral,

1

(e2

( — — — [__ da ƒ(a) [" dp cos (p.x – pa) (6′′ + eTM),

u =

dz F(a) [** do cos (px — p2) (e” — e”).

The complete integral of the proposed equation expressed in real terms is therefore vu+ W; and we perceive in fact, 1st, that it satisfies the differential equation; 2nd, that on making y=0 in it, it gives v=f(x); 3rd, that on making y=0 in the

422. We may also remark that we can deduce from equation (B) a very simple expression of the differential coefficient of the

d'

7th order, if (a), or of the integral ["da'f(x).

The expression required is a certain function of x and of the index. It is required to ascertain this function under a form such that the number i may not enter it as an index, but as a quantity, in order to include, in the same formula, every case in which we assign to i any positive or negative value. To obtain it we shall remark that the expression

– sin r,

- cos r, + sin r,+cos r, - sin r, &c.,

if the respective values of i are 1, 2, 3, 4, 5, &c. The same results recur in the same order, when we increase the value of i. In the second member of the equation

f(x) = 1 [da ƒ (a) [dp cos (px — p2),

we must now write the factor p' before the symbol cosine, and

i

π

add under this symbol the term +. We shall thus have

The number, which enters into the second member, may be any positive or negative integer. We shall not press these applications to general analysis; it is sufficient to have shewn the use of our theorems by different examples. The equations of the fourth order, (d), Art. 405, and (e), Art. 411, belong as we have said to dynamical problems. The integrals of these equations were not yet known when we gave them in a Memoir on the Vibrations of

Elastic Surfaces, read at a sitting of the Academy of Sciences', 6th June, 1816 (Art. vI. §§ 10 and 11, and Art. VII. §§ 13 and 14). They consist in the two formulæ & and &, Art. 406, and in the two integrals expressed, one by the first equation of Art. 412, the other by the last equation of the same Article. We then gave several other proofs of the same results. This memoir contained also the integral of equation (c), Art. 409, under the form referred to in that Article. With regard to the integral (88) of equation (a), Art. 413, it is here published for the first time.

α -x

; but suits

423. The propositions expressed by equations (A) and (B′), Arts. 418 and 417, may be considered under a more general point of view. The construction indicated in Arts. 415 and 416 applies sin (pa — px) not only to the trigonometrical function all other functions, and supposes only that when the number p becomes infinite, we find the value of the integral with respect to a, by taking this integral between extremely near limits. Now this condition belongs not only to trigonometrical functions, but is applicable to an infinity of other functions. We thus arrive at the expression of an arbitrary function f(x) under different very remarkable forms; but we make no use of these transformations in the special investigations which occupy us.

With respect to the proposition expressed by equation (A), Art. 418, it is equally easy to make its truth evident by constructions, and this was the theorem for which we employed them at first. It will be sufficient to indicate the course of the proof.

1 The date is inaccurate. The memoir was read on June 8th, 1818, as appears from an abstract of it given in the Bulletin des Sciences par la Société Philomatique, September 1818, pp. 129-136, entitled, Note relative aux vibrations des surfaces élastiques et au mouvement des ondes, par M. Fourier. The reading of the memoir further appears from the Analyse des travaux de l'Académie des Sciences pendant l'année 1818, p. xiv, and its not having been published except in abstract, from a remark of Poisson at pp. 150-1 of his memoir Sur les équations aux différences partielles, printed in the Mémoires de l'Académie des Sciences, Tome ш. (year 1818), Paris, 1820. The title, Mémoire sur les vibrations des surfaces élastiques, par M. Fourier, is given in the Analyse, p. xiv. The object, "to integrate several partial differential equations and to deduce from the integrals the knowledge of the physical phenomena to which these equations refer," is stated in the Bulletin, p. 129. [A. F.]

we can replace the sum of the terms arranged under the sign by its value, which is derived from known theorems. We have seen different examples of this calculation previously, Section III., Chap. III. It gives as the result if we suppose, in order to simplify the expression, 2π=X, and denote a-x by r,

We must then multiply the second member of this equation by daf(x), suppose the number j infinite, and integrate from a=-π to a=+π. The curved line, whose abscissa is a and ordinate cos jr, being conjoined with the line whose abscissa is a and ordinate f(a), that is to say, when the corresponding ordinates are multiplied together, it is evident that the area of the curve produced, taken between any limits, becomes nothing when the number j increases without limit. Thus the first term cos jr gives a nul result.

The same would be the case with the term sin jr, if it were sin r not multiplied by the factor versin r; but on comparing the three curves which have a common abscissa a, and as ordinates

sin jr,

sin r versin r

,f(a), we see clearly that the integral

Jda ƒ (a) sin jr

sin r versin r

has no actual values except for certain intervals infinitely small,

take place if r or a-x is nothing; and in the interval in which a differs infinitely little from x, the value of ƒ (a) coincides with f(x). Hence the integral becomes

2f(x) * dr sin jr, or 4f(x) dr sin jr,

0

which is equal to 2πf(x), Arts. 415 and 356. Whence we conclude the previous equation (A).

When the variable x is exactly equal to π or +, the construction shews what is the value of the second member of the equation (A), [‡ƒ (−π) or {ƒ (π)].

If the limits of integrations are not and +, but other numbers a and b, each of which is included between — and +, we see by the same figure what the values of x are, for which the second member of equation (A) is nothing.

If we imagine that between the limits of integration certain values of ƒ (a) become infinite, the construction indicates in what sense the general proposition must be understood. But we do not here consider cases of this kind, since they do not belong to physical problems.

If instead of restricting the limits and +, we give greater extent to the integral, selecting more distant limits a' and b', we know from the same figure that the second member of equation (4) is formed of several terms and makes the result of integration finite, whatever the function f(x) may be.

We find similar results if we write 2

limits of integration being - X and + X.

sin r versin r

X

instead of r, the

It must now be considered that the results at which we have arrived would also hold for an infinity of different functions of sinjr. It is sufficient for these functions to receive values alternately positive and negative, so that the area may become nothing, when j increases without limit. We may also vary the factor as well as the limits of integration, and we may suppose the interval to become infinite. Expressions of this kind are very general, and susceptible of very different forms. We cannot delay over these developments, but it was necessary to exhibit the employment of geometrical constructions; for they solve without any doubt questions which may arise on the extreme values, and on singular values; they would not have served to discover these theorems, but they prove them and guide all their applications.