velopment; and acting on this presumption, investigate the series by an exact and precise process. Now whatever is the angle denoted by 0, we have the following equivalence, sin (n + 1) 0 Let be replaced by x-z; and this being done, let every term be multiplied by ƒ(z) dz; and let the definite integral of each term thus found be taken for the limits b and a, f(z) being finite and continuous throughout this range of integration; then 1 } ['fƒ (z) dz + ['ƒ (z) cos(x−2) dz + [ƒ (z) cos 2 (x −z) dz + ... a | | It is evident that every term in the left-hand member is unaltered when a is increased by 2. Consequently both members denote periodic functions of x, whose period is 2π, whatever is the value of n; and if we trace the value of either member of (80) through this range, that value will be repeated when b and a are replaced by b+2kπ and а+2kл respectively; we may therefore without loss of generality confine our attention within limits of integration, such that b-a is not greater than 2π. When n∞, the number of terms in the first member of (80) is infinite, and the sum of them is denoted by the value which the second member takes when n = ∞. As this is the case of the series which we have to investigate, this latter value must be determined. Its value evidently depends on the relative values of x and the several values of z, and we shall have three cases; (1), when x falls within the range of integration; (2), when a coincides with either limit of the integration; (3), when a falls beyond the range. (1). Let x fall within the range of integration; that is, let a be greater than a and less than b; then, when z = x, the righthand member of (80) takes an indeterminate form, and must be evaluated. Now the element-function vanishes for all other values of z; for since (n+1)(x−≈) varies by 27, when z varies by 0, when n∞; it is evident that over that 2n+1' range of z, which is infinitesimal, f(z) may be considered sin 2 constant, and (sin (n + 1)(~—z)dz = 0. Hence the right-hand member of (80) = 0, except when z = x. To determine the value when z = x, we may consider it only for values of z infinitesimally near to x; that is, if i is an infinitesimal, we may take the limits of the z-integration to be x-i and x+i; and if we replace z-x by §, § is the variable of integration, and is always infinitesimal, its limits of integration being -i and i; so that and as έ is always infinitesimal f(x+8) may be replaced by હું so that ƒ (x)+Eƒ'(x), and sin§ by §; Of the integrals in the right-hand member of this equation, the latter vanishes, because for all values of έ, sin (n + 1) = 0, when 2 n = ∞. And to determine the value of the former, we may enlarge the range to ∞ and +∞, because all the elements of that definite integral vanish when n∞, except those corresponding to small values of §. Hence 2= [f(z)dz + = ['f(z) con (x — z) dz 1 1 = 2π П which gives the periodic series into which f(x) may be developed, when a falls within the limits of the z-integration. As the general term of this series is and this is equal to f(z) f(z) cos n (x-z) dz, it is of the form A, cos nx+B, sin na; and consequently by means of (85) f(x) is developed into a series of the form given in (70), Art. 196. This series is however more general, because the limits of integration in it are b and a, whereas those in (70) Art. 196 are a+27 and a. = = b, say, (2). Let x = one of the limits of the z-integration; the superior limit; then the limits of integration in the righthand member of (81) must be b and b-i, otherwise values of z would be included which are not within the range of integration; and consequently the limits of § in the §-integration are 0 and -i; and the value of the definite integral in the right-hand member of (82) is, the limits being 0 and - ∞ ; and the series is consequently equal to f(b). Similarly if x = a, the inferior limit, the limits in the right-hand member of (81) are a+i and a; so that the limits of έ in the -integration are i and 0; and the value of the definite integral in the right-hand member of π π (82) is, the limits being ∞ and 0; and the series is consequently π equal to f(a). Hence if a is equal to the limit b or a, we have respectively π 1 [ƒ (b) = } ["ƒ (z)dz + Σm=ï [["ƒ (z) cos n (b−z) dz ; π 1 a 1 = ƒ (a) = } ["ƒ (=)dz + Σh=i['ƒ (2) cos n (a—z) dz. (86) (87) (3). Let x fall beyond the range of integration; that is, let x be less than a, or be greater than b; then, as we have shewn in case (1), the element-function vanishes for all values of z, and the right-hand member of (80) always = 0. Thus Thus in recapitulation, b—a being not greater than 27, represents a function of x which is periodic, and of which the period is 2π; and which and these are the values of (89) through a complete period, and are repeated in both directions. This theorem contains the doctrine of periodic series in its most general form. 198.] The preceding results may be geometrically interpreted, and the interpretation is interesting and curious, inasmuch as it exhibits the discontinuity of the function in a striking manner. Take the right-hand member of (85) to be the ordinate of a plane curve, whose abscissa is a; so that if the left-hand member is denoted by y, y = f(x) is the equation to the curve. Then for all values of a between b and a exclusively, the locus is repre sented by the curve y =ƒ (»); when æ = a, y =ƒ(a); and when f(b) 2 x = b, y =); so that the locus then becomes two points which x= 2 are evidently discontinuous points. Also for values of a greater than b, and for values less than a, y = 0, so that the locus becomes the x-axis. This last part of the locus however is subject to a x+47, = but also when z = x+2π, = =x+2kπ; therefore the right-hand member of (85) = f(x), not only when x has values between b and a, but also when a has values between b+2π and a+2, between b+4 and a +4, ...; accordingly the curve, which is the locus between x = b and x = a, is repeated at intervals, such that the distance between two similar points when x=a+2π, =2π. Also as y = f(a), when x = a, so y= 2 f(b) f(a) 2 =α+4π, = ; and y = when xb, b+2π, = b+4π, = ... = ; and consequently we have a repetition of the points of discontinuity. Also the portions of the x-axis which the series expresses are of a finite length, and these finite portions are repeated for as y = 0 for values of a greater than b, so this zero-value continues until x=a+2π, when y abruptly takes the 2 value (a); and then as a increases, y = f(x); and this locus continues until a=b+2, when y abruptly takes the value (b); 2 and again as a increases, y = 0; and this zero-value continues until x = a + 47, when another point of discontinuity occurs. And so on to infinity, in both the positive and negative directions. Thus 2-(b-a) is the length of each portion of the x-axis which the series represents, and these are repeated at intervals, each of which b-a. If b-a2, so that the limits of the z-integration are a and a+2, then these portions of the x-axis disappear, and the locus consists of discontinuous branches of the curve y = f(x); and at those values of x at which the discontinuity takes place, 1 y = 2 2 {f(a)+f(x+a)}. (91) 199.] This last remark leads to a very important result. It shews that f(x) need not be a continuous function, and that it may have points of discontinuity between the limits of the z-integration. When however this is the case, the integrals must be taken separately over the ranges through which the continuity of PRICE, VOL. II. M m |