422. We may also remark that we can deduce from equation (B) a very simple expression of the differential coefficient of the ¿th order, d' def (r), 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 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 we must now write the factor p' before the symbol cosine, and i add under this symbol the term +. We shall thus have d' 1 2π +00 π 2 ; ƒ (x) = ___ (** dx ƒ (a) [** dp p1 cos (px−p1+i). The number i, 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 (83) of equation (a), Art. 413, it is here published for the first time. 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 (pë—pr); but suits not only to the trigonometrical function a-x 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 1. (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.] In equation (4), namely, 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, sin r versin r We must then multiply the second member of this equation by daf(x), suppose the number j infinite, and integrate from а=-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. sin r versin The same would be the case with the term sin jr, if it were not multiplied by the factor ; 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 [da ƒ (a) sin jr sin r versin r has no actual values except for certain intervals infinitely small, namely, when the ordinate sin r versin r becomes infinite. This will take place if r or ax 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 dr 2f(x) [* dr sin jr, or 4f(x) for 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 or +π, the construction shews what is the value of the second member of the equation (4), [}ƒ (−π) or {ƒ (π)]. If the limits of integrations are not and +π, but other numbers a and b, each of which is included between -T and +π, we see by the same figure what the values of x are, for which the second member of equation (4) 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 27 limits of integration being - X and + X. sin r a-x 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 versin, 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. 424. We have yet to regard the same propositions under another aspect. If we compare with each other the solutions relative to the varied movement of heat in a ring, a sphere, a rectangular prism, a cylinder, we see that we had to develope an arbitrary function f(x) in a series of terms, such as α ̧$ (μ ̧x) +α ̧$ (μ ̧x) + α ̧¢ (μ‚x) + &c. The function, which in the second member of equation (4) is a cosine or a sine, is replaced here by a function which may be very different from a sine. The numbers μ,, H2, μ, &c. instead of being integers, are given by a transcendental equation, all of whose roots infinite in number are real. ... The problem consisted in finding the values of the coefficients a; they have been arrived at by means of definite integrations which make all the unknowns disappear, except one. We proceed to examine specially the nature of this process, and the exact consequences which flow from it. In order to give to this examination a more definite object, we will take as example one of the most important problems, namely, that of the varied movement of heat in a solid sphere. We have seen, Art. 290, that, in order to satisfy the initial distribution of the heat, we must determine the coefficients a,, a,, "... a, in the equation xF(x) = a ̧ sin (μ‚x) + a ̧ sin (μ‚ ̧x) + a, sin (μ,x) + &c...................(e). The function F(x) is entirely arbitrary; it denotes the value v of the given initial temperature of the spherical shell whose radius is x. The numbers μ1, μ¿ μ are the roots μ, of the transcendental equation ... X is the radius of the whole sphere; h is a known numerical coefficient having any positive value. We have rigorously proved in our earlier researches, that all the values of μ or the roots of the equation (f) are real'. This demonstration is derived from the 1 The Mémoires de l'Académie des Sciences, Tome x, Paris 1831, pp. 119-146, contain Remarques générales sur l'application des principes de l'analyse algébrique |