which represents the quantity(). In fact, we have, in a second part of the coefficient of sin x is found by multiplying 1 whose value is "(π). We can determine in this manner the π different parts of the coefficient of sin x, and the components of the coefficients of sin 2x, sin 3x, sin 4x, &c. We may employ for this purpose the equations: 1 2 By means of these reductions equation (A) takes the following form: (π) – 11 4′′ (7) + '% $′′ (π) — — $" (π) + &c. - 1 3a 1* 4 1 r$ (x) = + (x) {sin x − ↓ sin 2x + sin 3x - &c.} &c.} + zi (m) 218. We can apply one or other of these formulæ as often as we have to develope a proposed function in a series of sines of multiple arcs. If, for example, the proposed function is e- eTM*, whose development contains only odd powers of x, we shall have Collecting the coefficients of sin x, sin 2x, sin 3x, sin 4x, &c., We might multiply these applications and derive from them several remarkable series. We have chosen the preceding example because it appears in several problems relative to the propagation of heat. 219. Up to this point we have supposed that the function whose development is required in a series of sines of multiple arcs can be developed in a series arranged according to powers of the variable x, and that only odd powers enter into that series. We can extend the same results to any functions, even to those which are discontinuous and entirely arbitrary. To establish clearly the truth of this proposition, we must follow the analysis which furnishes the foregoing equation (B), and examine what is the nature of the coefficients which multiply sin x, sin 2x, sin 3x, &c. Denoting by the quantity which multiplies 1 n n sin ne in this equation when n is odd, and even, we have Considering s as a function of π, differentiating twice, and 1 d's n2 comparing the results, we find s+=$(π); an equation which the foregoing value of s must satisfy. If n is an integer, and the value of x is equal to π, we have s=±n(x) (x) sin nædæ. The sign + must be chosen when ʼn is odd, and the sign — when that number is even. We must make x equal to the semi-circumference, after the integration indicated; the result may be verified by developing the term [Þ(x) sin næ dæ, by means of integration by parts, remarking that the function () contains only odd powers of the variable x, and taking the integral from x = 0 to x=π. We conclude at once that the term is equal to 8 If we substitute this value of in equation (B), taking the n sign + when the term of this equation is of odd order, and the sign – when ʼn is even, we shall have in general (a) sin nædæ for the coefficient of sin nx; in this manner we arrive at a very remarkable result expressed by the following equation: = sinx (sin xp(x) dx + sin 2x (sin 2x4 (x) dx+ &c. (D), the second member will always give the development required for the function (x), if we integrate from x = 0 to x=π.' 1 Lagrange had already shewn (Miscellanea Taurinensia, Tom. III., 1766, pp. 260-1) that the function y given by the equation y=2(2 Y, sin X, ▲X) sin xπ + 2 (2 Y, sin 2X, AX) sin 2xπ +2( Y, sin 3X, AX) sin 3xπ + +2(2 Y, sin nX, AX) sin nëπ 1 ... 1 receives the values Y1, Y,, Y... Y corresponding to the values X1, X2, X....Xn of x, where X,= n+1 and AX= 1 Lagrange however abstained from the transition from this summation-formula to the integration-formula given by Fourier. Cf. Riemann's Gesammelte Mathematische Werke, Leipzig, 1876, pp. 218-220 of his historical criticism, Ueber die Darstellbarkeit einer Function durch eine Trigonometrische Reihe. [A. F.] 220. We see by this that the coefficients a, b, c, d, e, f, &c., which enter into the equation 1 π(x) = a sin x + b sin 2x + c sin 3x + d sin 4x + &c., and which we found formerly by way of successive eliminations, are the values of definite integrals expressed by the general term sin ix 4 (x) dx, i being the number of the term whose coefficient is required. This remark is important, because it shews how even entirely arbitrary functions may be developed in series of sines of multiple arcs. In fact, if the function (x) be represented by the variable ordinate of any curve whatever whose abscissa extends from x=0 to x=π, and if on the same part of the axis the known trigonometric curve, whose ordinate is y = sin x, be constructed, it is easy to represent the value of any integral term. We must suppose that for each abscissa x, to which corresponds one value of (x), and one value of sina, we multiply the latter value by the first, and at the same point of the axis raise an ordinate equal to the product (x) sin x. By this continuous operation a third curve is formed, whose ordinates are those of the trigonometric curve, reduced in proportion to the ordinates of the arbitary curve which represents (x). This done, the area of the reduced curve taken from x=0 to x=π gives the exact value of the coefficient of sin x; and whatever the given curve may be which corresponds to p(x), whether we can assign to it an analytical equation, or whether it depends on no regular law, it is evident that it always serves to reduce in any manner whatever the trigonometric curve; so that the area of the reduced curve has, in all possible cases, a definite value, which is the value of the coefficient of sin x in the development of the function. The same is the case with the following coefficient b, or p(x) sin 2xdx. In general, to construct the values of the coefficients a, b, c, d, &c., we must imagine that the curves, whose equations are y= sin x, y = sin 2x, y = sin 3x, y sin 4x, &c., = have been traced for the same interval on the axis of x, from |