at every point with the axis of x. This result is very remarkable, and determines the true sense of the proposition expressed by equation (B). 418. The theorem expressed by equation (II) Art. 234 must be considered under the same point of view. This equation serves to develope an arbitrary function f(x) in a series of sines or cosines of multiple arcs. The function f(x) denotes a function completely arbitrary, that is to say a succession of given values, subject or not to a common law, and answering to all the values of x included between 0 and any magnitude X. The value of this function is expressed by the following equation, The integral, with respect to a, must be taken between the limits a = a, and a = b; each of these limits a and b is any quantity whatever included between 0 and X. The sign Σ affects the integer number i, and indicates that we must give to i every integer value negative or positive, namely, .. — 5, − 4, − 3, − 2, − 1, 0, +1, + 2, + 3, + 4, + 5, ... and must take the sum of the terms arranged under the sign Σ. After these integrations the second member becomes a function of the variable x only, and of the constants a and b. The general proposition consists in this: 1st, that the value of the second member, which would be found on substituting for x a quantity included between a and b, is equal to that which would be obtained on substituting the same quantity for x in the function ƒ(x); 2nd, every other value of x included between 0 and X, but not included between a and b, being substituted in the second member, gives a nul result. Thus there is no function f(x), or part of a function, which cannot be expressed by a trigonometric series. The value of the second member is periodic, and the interval of the period is X, that is to say, the value of the second member does not change when + X is written instead of x. All its values in succession are renewed at intervals X. The trigonometrical series equal to the second member is convergent; the meaning of this statement is, that if we give to the variable x any value whatever, the sum of the terms of the series approaches more and more, and infinitely near to, a definite limit. This limit is 0, if we have substituted for x a quantity included between 0 and X, but not included between a and b; but if the quantity substituted for x is included between a and b, the limit of the series has the same value as f(x). The last function is subject to no condition, and the line whose ordinate it represents may have any form; for example, that of a contour formed of a series of straight lines and curved lines. We see by this that the limits a and b, the whole interval X, and the nature of the function being arbitrary, the proposition has a very extensive signification; and, as it not only expresses an analytical property, but leads also to the solution of several important problems in nature, it was necessary to consider it under different points of view, and to indicate its chief applications. We have given several proofs of this theorem in the course of this work. That which we shall refer to in one of the following Articles (Art. 424) has the advantage of being applicable also to nonperiodic functions. If we suppose the interval X infinite, the terms of the series become differential quantities; the sum indicated by the sign Σ becomes a definite integral, as was seen in Arts. 353 and 355, and equation (4) is transformed into equation (B). Thus the latter equation (B) is contained in the former, and belongs to the case in which the interval X is infinite: the limits a and b are then evidently entirely arbitrary constants. 419. The theorem expressed by equation (B) presents also divers analytical applications, which we could not unfold without quitting the object of this work; but we will enunciate the principle from which these applications are derived. We see that, in the second member of the equation the function f(x) is so transformed, that the symbol of the function f affects no longer the variable x, but an auxiliary F. H. 28 variable a. The variable x is only affected by the symbol cosine. It follows from this, that in order to differentiate the function fƒ (x) with respect to x, as many times as we wish, it is sufficient to differentiate the second member with respect to x under the symbol cosine. We then have, denoting by i any integer number whatever, We take the upper sign when i is even, and the lower sign when i is odd. Following the same rule relative to the choice of sign d2i+1 da2+1ƒ (x) = = 27 [dz ƒ (a) [dp p2i+1 sin (px — p2). f 2π We can also integrate the second member of equation (B) several times in succession, with respect to a; it is sufficient to write in front of the symbol sine or cosine a negative power of p. The same remark applies to finite differences and to summations denoted by the sign Σ, and in general to analytical operations which may be effected upon trigonometrical quantities. The chief characteristic of the theorem in question, is to transfer the general sign of the function to an auxiliary variable, and to place the variable x under the trigonometrical sign. The function f(x) acquires in a manner, by this transformation, all the properties of trigonometrical quantities; differentiations, integrations, and summations of series thus apply to functions in general in the same manner as to exponential trigonometrical functions. For which reason the use of this proposition gives directly the integrals of partial differential equations with constant coefficients. In fact, it is evident that we could satisfy these equations by particular exponential values; and since the theorems of which we are speaking give to the general and arbitrary functions the character of exponential quantities, they lead easily to the expression of the complete integrals. The same transformation gives also, as we have seen in Art. 413, an easy means of summing infinite series, when these series contain successive differentials, or successive integrals of the same function; for the summation of the series is reduced, by what precedes, to that of a series of algebraic terms. 420. We may also employ the theorem in question for the purpose of substituting under the general form of the function a binomial formed of a real part and an imaginary part. This analytical problem occurs at the beginning of the calculus of partial differential equations; and we point it out here since it has a direct relation to our chief object. If in the function f(x) we write μ+v√−1 instead of x, the result consists of two parts +√-14. The problem is to determine each of these functions and in terms of μ and v. We shall readily arrive at the result if we replace ƒ (x) by the expression μ for the problem is then reduced to the substitution of μ+v√−1 instead of x under the symbol cosine, and to the calculation of the real term and the coefficient of -1. We thus have 1 ƒ (x) = ƒ (+ v√ − 1) = 2 fax (2) fdp cos [p (− a) +pv√=1] = 2π 1 [dz ƒ (2) [dp (cos (pμ − p2) (eTM +e ̄TM) 4π f - +√−1 sin (pμ−pa) (erv — e−rv)}; ↓ = 11 [dz ƒ (a) fdp sin (pu – p2) (err — e ̄rr). f Thus all the functions f(x) which can be imagined, even those which are not subject to any law of continuity, are reduced to the form M+N√−1, when we replace the variable x in them by the binomial +-1. 421. To give an example of the use of the last two formulæ, d'v dv 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 when y = 0, we can deduce the required integral from that of the equation value of dy which has long been known; but we find imaginary quantities under the functional signs: the integral is = $ (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 (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, u = 21 [** dx ƒ (1) [** dp cos (px – pa) (e” + eTM), 2π dz F(a) [** dp cos (px — p2) (e» — e ̄TM). Ρ 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 |