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, +cos 6+cos 20+...+cosno = (79) 2 sin Let o be replaced by x—2; and this being done, let every term be multiplied by f(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 It is evident that every term in the left-hand member is unaltered when x is increased by 27. Consequently both members denote periodic functions of X, whose period is 27, 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+2ks and a +2kt respectively; we may therefore without loss of generality confine our attention within limits of integration, such that b-a is not greater than 2. When n = oo, 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 = 00. 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 x coincides with either limit of the integration; (3), when x falls beyond the range. (1). Let x fall within the range of integration; that is, let x 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 + -)(-2) varies by 27, when z varies by 47 7, which = 0, when n = 00; it is evident that over that 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 2 - x by , & is the variable of integration, and is always infinitesimal, its limits of integration being - i and +i; so that sin (n + 5) (2 — 2) dz (81) Sin 1056) Of the integrals in the right-hand member of this equation, the latter vanishes, because for all values of ļ, = 0, when n = 00. And to determine the value of the former, we may en. large the range to –0 and +00, because all the elements of that definite integral vanish when n = oo, except those corresponding to small values of & Hence which gives the periodic series into which f(x) may be developed, when x falls within the limits of the z-integration. As the general term of this series is - cos n (x —%) dz, and this is equal to 1 Ja T Ja it is of the form A, COS NX + B, sin nx ; and consequently by means of (85) f () 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. (2). Let x = one of the limits of the z-integration; = b, say, 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 E-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 — 00; 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 & in the A-integration are i and 0; and the value of the definite integral in the right-hand member of (82) is 5, the limits being co and 0; and the series is consequently equal . Hence if x is equal to the limit b or a, we have (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 2, and the right-hand member of (80) always = 0. Thus lof(z)dz +2=1"(z) cos n (x –z)dz = 0. (88) Thus in recapitulation, b–a being not greater than 27, 'f(z)dz +2=18°f(z) cos n (2–2)dz (89) represents a function of x which is periodic, and of which the period is 27; and which = f(a), = -f(x), = f(K), = 0, 1 (90) according as x = a, a<x<b, x = b, b<x<27 +a;J 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 x; 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 x between b and a exclusively, the locus is represented by the curve y=f(x); when a =a, y ="); and when x = b, y = 10; so that the locus then becomes two points which 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 certain restriction ; for since sin“ * = 0, not only when z = x, but also when z = x+271, = x + 47, = ... = x+2 km; therefore the right-hand member of (85) = f(x), not only when x has values between b and a, but also when x has values between b+ 211 and a+2, between 6+47 and a +47, ...; 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 = 25. Also as y="@, when x = a, so y=+49 when a=a+21, = a +47, = ...; and y = °? when x=b, = 6+21, = 6+ 47, when x=0, = 0+% = ...; 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 6, so this zero-value continues until x=a+27, when y abruptly takes the value ; and then as x increases, y = f(x); and this locus continues until x=b+27, when y abruptly takes the value ; and again as x 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 21-(6-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-a= 27, 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 curvey = f(x); and at those values of x at which the discontinuity takes place, y = a {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 |