We might change the limits of the integrals and write but in each of these two cases it would be necessary to substitute in the first member 7 (x) for (x). 233. The function 4 (x) developed in cosines of multiple arcs, is represented by a line formed of two equal arcs placed sym metrically on each side of the axis of y, in the interval from -to (see fig. 11); this condition is expressed thus, The line which represents the function (x) is, on the contrary, formed in the same interval of two opposed arcs, which is what is expressed by the equation f(x) = − (−x). Any function whatever F(x), represented by a line traced arbitrarily in the interval from π to +π, may always be divided into two functions such as (x) and y (x). In fact, if the line F'F'mFF represents the function F(x), and we raise at the point o the ordinate om, we can draw through the point m to the right of the axis om the arc mff similar to the arc mFF of the given curve, and to the left of the same axis we may trace the arc mƒ'ƒ' similar to the arc mFF; we must then draw through the point m a line d'o'moo which shall divide into two equal parts the difference between each ordinate aF or 'f' and the corresponding ordinate xf or 'F". We must draw also the line ''Oy, whose ordinate measures the half-difference between the ordinate of F'F'mFF and that of f'f'mff. This done the ordinate of the lines FFmFF, and f'f'mff being denoted by F(x) and f(x) respectively, we evidently have ƒ(x) = F (− x); denoting also the ordinate of ''mop by p (x), and that of ''Op by (x), we have F (x) = $ (x) + † (x) and ƒ (x) = $ (x) — y (x) = F (− x), hence 1 $ (x) = ¦ whence we conclude that (x)=(-x) and (x) = − (− x), which the construction makes otherwise evident. Thus the two functions & (x) and y (x), whose sum is equal to F(x) may be developed, one in cosines of multiple arcs, and the other in sines. If to the first function we apply equation (v), and to the second the equation (u), taking the integrals in each case from x=-π to x=π, and adding the two results, we have + sin æ f(x) sin x dx + sin 2x f(x) sin 2x dx + &c. The integrals must be taken from x=-π to x=π. It may now be remarked, that in the integral (a) cosa da we could, -π without changing its value, write (x)+(x) instead of (x): for the function cos x being composed, to right and left of the axis of x, of two similar parts, and the function () being, on the contrary, formed of two opposite parts, the integral (**(x) cos xd.x vanishes. The same would be the case if we wrote cos 2x or cos 3x, and in general cos ix instead of cos x, i being any integer from 0 to infinity. Thus the integral (a) cos ir de is the same as the integral -π [**[¥ (x) + ↓ (x)] cos ix dx, or [**F(x) cos ia dæ. ix It is evident also that the integral (**(a) sin ix de is equal to the integral (**F(x) sin iæ da, since the integral (a) sin iæ dæ vanishes. Thus we obtain the following equation (p), which serves to develope any function whatever in a series formed of sines and cosines of multiple arcs : x [F(x) cos x dx + cos 2x (F(x) cos 2x dx + &c. + cos x + sin x e F(x) sin æ dc + sin 2.x [F(x) sin 2.x d.x + &c. (p) 234. The function F(x), which enters into this equation, is represented by a line F'F'FF, of any form whatever. The arc FFFF, which corresponds to the interval from π to +π, is arbitrary; all the other parts of the line are determinate, and the arc F'F'FF is repeated in each consecutive interval whose length is 2π. We shall make frequent applications of this theorem, and of the preceding equations (u) and (v). If it be supposed that the function F(x) in equation (p) is represented, in the interval from -π to +π, by a line composed of two equal arcs symmetrically placed, all the terms which contain sines vanish, and we find equation (v). If, on the contrary, the line which represents the given function F(x) is formed of two equal arcs opposed in position, all the terms which do not contain sines disappear, and we find equation (u). Submitting the function F(x) to other conditions, we find other results. If in the general equation (p) we write, instead of the variable пх x, the quantity , denoting another variable, and 2r the length r of the interval which includes the arc which represents F(x); the function becomes F(T), which we may denote by ƒ (a). Пх The limits x=-π and x=π become == π, have therefore, after the substitution, *f(x) = } ["ƒ'(x) dx r r to All the integrals must be taken like the first from x =— x=+r. If the same substitution be made in the equations (v) In the first equation (P) the integrals might be taken from from x = 0 to x=2r, and representing by x the whole interval 2r, we should have1 1 It has been shewn by Mr J. O'Kinealy that if the values of the arbitrary function f(x) be imagined to recur for every range of x over successive intervals A, we have the symbolical equation 235. It follows from that which has been proved in this section, concerning the development of functions in trigonometrical series, that if a function f(x) be proposed, whose value in a definite interval from x = 0 to x = X is represented by the ordinate of a curved line arbitrarily drawn; we can always develope this function in a series which contains only sines or only cosines, or the sines and cosines of multiple arcs, or the cosines only of odd multiples. To ascertain the terms of these series we must employ equations (M), (N), (P). The fundamental problems of the theory of heat cannot be completely solved, without reducing to this form the functions. which represent the initial state of the temperatures. These trigonometric series, arranged according to cosines or sines of multiples of arcs, belong to elementary analysis, like the series whose terms contain the successive powers of the variable. The coefficients of the trigonometric series are definite areas, and those of the series of powers are functions given by differentiation, in which, moreover, we assign to the variable a definite value. We could have added several remarks concerning the use and properties of trigonometrical series; but we shall limit ourselves to enunciating briefly those which have the most direct relation to the theory with which we are concerned. The coefficients being determined in Fourier's manner by multiplying both COS Σπα sides by n and integrating from 0 to X. (Philosophical Magazine, August sin λ |