the functions extends; and the values at the limits must be given by means of (86) and (87). Thus, if f (x) becomes discontinuous when x = a, and the value of f(x) changes abruptly from A, to A2, then, when x = a, y = -742; similarly, if there is a point of discontinuity when x=B, and the value of f(x) changes abruptly from B, to By, then at that point y = B: B; and so on. 200.] We now return to the general theorems contained in (90), and consider certain forms which they assume for particular values of the limits, for particular forms of the variables, and in some special cases. As the period of the function is 27, if the difference between the limits of the z-integration is 27, the values of the functions for the variables contained between these limits recur without intervals; and if we take either 0 and 27, or — 7 and 7, to be limits of integration, there is no loss of generality. In these cases we have the following theorems. 2 Sof(z)dz+ =1*/*** f (2) cos n (x – 2) dz = f(0) = -f(x) = f(27), when ! (92) 0, 0 < x < 27, X = 27. = $(-), = + f(x), = f(n), when > (93) x = -1, < X < T, X = . If the limits of the z-integration are a and 0; then "r(e)dz+ 2 =1*f(z) cos n (0–2)dz = 0, = f(0) = f(x), = f(t), when ? -=<x<0, x = 0, 0<x<+, x = 1; and these assign the values of the integrals through a complete period. Now if in (94) x is replaced by -X; then -1<x<0, x = 0, 0< x <T, X = 16. As the limits of integration are the same in (94) and (95), we may add these equations, and we have ["f(z)dz +2 = cos nx "f(z) cos nz dz ] == f(-x) = f(0) = f(x) = f(t), when -1<x<0, x = 0, 0<x<TT, X = t; thus the definite integral in the left-hand member of (96) is equal to a f (x) for all values contained between 0 and a inclusively; and to a f(-x) for all values between – 7 and 0 inclusively. Again, subtracting (95) from (94), 2 E1=1 sin næsof(z) sin nz dz =-=f(-x), = 0, = f(x), = 0, when (97) -<x<0, 2 = 0, 0<x< T, X = ; so that the definite integral in the left-hand member of (97) = - f(-x) for values of x between – and 0 exclusively, = 0 at the limits of the z-integration, and = f (x) for values of x between 0 and 7 exclusively. If f (x) is such a function of x that f(-x) = f (w); then the function has the same sign for positive and negative values of x, and it will consist of the series of cosines given in (96). And if f(x) is such a function of x that f(-x) = -f(x), then f (x) will have opposite signs for positive and negative values of x, and will consist of the series of sines given in (97). 201.] Another form, which is frequently required, may be given to these theorems by means of a change of form of the variables x and z. Let x and z be respectively replaced by *** and **, so that the period of the function becomes 2c instead of 27. Also let the functions f (*) and f (*2) be replaced respectively by f (x) and f(z). Then (92) takes the following form, = 0, = TX, = 7?, according as x = 0, 0< x < 27, X = 27. Consequently, for all values of x between 27 and 0 exclusively, sin x sin 2 x sin 3x. 2 100 X = 11-23 + + +... i (102) and thus x is expressed in a series of sines of multiples of x. The geometrical interpretation of this result deserves notice. If the right-hand member of (102), which is a function of x, represents the ordinate of a locus, say, y, the equation to that locus is y = x. That is, the locus is a straight line passing through the origin and making an angle of 45° with the x-axis. That line however is limited by x = 0 and x = 27, and (102) expresses no part of that line beyond these limits. If x = 21, y = 1; and as the point expressed by these equations is not on the line, it is a point of discontinuity. As this geometrical interpretation of (102) recurs at every interval of 2t, so the equation represents a series of broken lines and of points of discontinuity exactly similar to those described above.. If in (102) x is replaced by 1—, then for all values of x between — 4 and 7 exclusively, sin x sin 2 x sin 3 x ) x = 24 11- 2 + 3^-... (103) This series may also be deduced from (93). Again, applying (96) we have the following result, If « is replaced by x +, then by (104) for all values of * between – and, inclusive, 4 S sinx sin 3x sin 5x sin 7 x , x = = {12- 32 + 52 - 73 + ...} (106) Also applying (97), we have 2 EET sin næ *zsin nz dz = -2- Er Fnza sin nx COSNT sin x sin 2 x sin 3 x H1T-2"+-...{ = TX, = 0, = 7x, = 0, according as — <x<0, x=0,0<x<Ti, x = t; and consequently for all values of x between – 7 and 1 exclusively sin x sin 2 x sin 3x = % T - 2"+ 2"-... (107) which is the same as (103). From these series others may be deduced by the substitution of particular values for the variables, provided that such values are within the limits of application. Thus if x = 0 in (104), 11,11 m. (108) Also these series may be the subjects of differentiation. Thus from (106) by differentiation we have for all values of x between - and inclusively, sin 2 and from (107), for all values between — and to exclusively, Also these series may be the element-functions of definite integration according to the method of the preceding section. Thus if we take the two members of (107) to be the element-functions of a definite integral for the limits x and 0, then, when x is between - 7 and , X“_ cos x cos 2x cos 3.2 |