x=0 to x; and then that we have changed these curves by multiplying all their ordinates by the corresponding ordinates of a curve whose equation is y = (x). The equations of the reduced curves are y=sin xp(x), y=sin 2x + (x), y = sin 3x p (x), &c. The areas of the latter curves, taken from x=0 to x=π, are the values of the coefficients a, b, c, d, &c., in the equation 1 2 πφ (2) π(x) = a sin x+b sin 2x + c sin 3x + d sin 4x + &c. 221. We can verify the foregoing equation (D), (Art. 220), by determining directly the quantities a,, a,, a,, ... a,, &c., in the equation (x)= a, sin x+a, sin 2x + a, sin 3x+ ... a, sin jx+&c.; for this purpose, we multiply each member of the latter equation by sin ix dx, i being an integer, and take the integral from x = 0 to x=π, whence we have f(x) sin ix dx = af sin x sin ix de + a, fsin 2x sin ix dæ+&c. +a, a, fsin jæ sin iæ dæ + ... &c. Now it can easily be proved, 1st, that all the integrals, which enter into the second member, have a nul value, except only the term a, sin ix sin ixdx; 2nd, that the value of afsin ; whence we derive the value of a,, namely fsin ix ix sin ixdx is The whole problem is reduced to considering the value of the integrals which enter into the second member, and to demonstrating the two preceding propositions. The integral 2 fsin je sin ix dx, taken from x=0 to x =π, in which i and j are integers, is Since the integral must begin when x = 0 the constant C is nothing, and the numbers i and j being integers, the value of the integral will become nothing when x=; it follows that each of the terms, such as afsir a, sin x sin ix dx, a, sin 2x sin ix dx, afsin a, sin 3x sin ixdx, &c., vanishes, and that this will occur as often as the numbers i and j are different. The same is not the case when the numbers i and j are equal, for the term 2 1 sin (ij)x to which the integral re duces, becomes, and its value is 7. Consequently we have 2 fsin ix sin ix dx = π ; thus we obtain, in a very brief manner, the values of a1, a,, a,, ... a,, &c., namely, (x) = sin x(x) sin x dx + sin 2x (x) sin 2x dx + &c. + sin ix ($(x) sin ixdx + &c. 222. The simplest case is that in which the given function has a constant value for all values of the variable x included Jsi between 0 and π; in this case the integral sin ix dx is equal to if the number i is odd, and equal to 0 if the number i is even. It must be remarked that when a function (x) has been developed in a series of sines of multiple arcs, the value of the series a sin x + b sin 2x + c sin 3x + d sin 4x + &c. is the same as that of the function (x) so long as the variable is included between 0 and π; but this equality ceases in general to hold good when the value of x exceeds the number π. Suppose the function whose development is required to be x, we shall have, by the preceding theorem, 1 Tx = sin x x sin a da + sin 2x fæ sin 2x dx + sin 3x fæ sin 3x dx + &c. The integral (* π x sin ixdx is equal to +; the indices 0 and π, ± which are connected with the sign, shew the limits of the inte gral; the sign + must be chosen when i is odd, and the sign We have then the following equation, when i is even. 223. We can develope also in a series of sines of multiple arcs functions different from those in which only odd powers of the variable enter. To instance by an example which leaves no doubt as to the possibility of this development, we select the function cos x, which contains only even powers of x, and which may be developed under the following form: a sin x + b sin 2x + c sin 3x + d sin 4x + e sin 5x + &c., although in this series only odd powers of the variable enter. We have, in fact, by the preceding theorem, The integral cos x sin ix dx is equal to zero when i is an 2i 2-1 when i is an even number. Supposing odd number, and to successively i= 2, 4, 6, 8, etc., we have the always convergent sin 2x+(+) sin 4x+(+) sin 6x+ &c.}. (+) sin 2 3 This result is remarkable in this respect, that it exhibits the development of the cosine in a series of functions, each one of which contains only odd powers. If in the preceding equation ax be made equal to π, we find of 1 T 4 √2 This series is known (Introd. ad analysin. infinit. cap. x.). 224. A similar analysis may be employed for the development any function whatever in a series of cosines of multiple arcs. Let (x) be the function whose development is required, we may write (x) = acos Оx+a, cos x + α, cos 2x + a, cos 3x + &c. If the two members of this equation be multiplied by cos jx, and each of the terms of the second member integrated from x=0 to x=π; it is easily seen that the value of the integral will be nothing, save only for the term which already contains cos jx. This remark gives immediately the coefficient a;; it is sufficient in general to consider the value of the integral fcos jx cos ix dx, taken from x = 0 to π, supposing j and i to be integers. We have 1 1 2 (j + ¿) sin (j + î) x + sin (j-)x+c. i) 2(j-i) This integral, taken from x = 0 to x=π, evidently vanishes whenever j and i are two different numbers. The same is not the case when the two numbers are equal. The last term 0 becomes and its value is 7, when the arc x is equal to π. If then we multiply the two terms of the preceding equation (m) by cos ix, and integrate it from 0 to π, we have [$ (x) cos ix dx = ±πα, an equation which exhibits the value of the coefficient a.. To find the first coefficient a,, it may be remarked that in the integral if j=0 and i=0 each of the terms becomes of each term is from x = 0 to x = are different: it is ; thus the integral cos jx cos ix dx taken is nothing when the two integers j and i when the two numbers j and i are equal but different from zero; it is equal to π when j and i are each equal to zero; thus we obtain the following equation, ¡π4 (x) = { '"]"$(x) dx+cos x ["p(x)cos xdx+cos 2x ["p(x) cos 2x dx πφ 2 1 The process analogous to (4) in Art. 222 fails here; yet we see, Art. 177, that an analogous result exists. [R. L. E.] |