x=0 to x=π; 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 p (x), y = sin 3x p (x), &c.

0 to x=π,

The areas of the latter curves, taken from x= are the values of the coefficients a, b, c, d, &c., in the equation

1

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 = a, fsin x sin ix de + a, fsin 2x 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 iædæ; 2nd, that the value of fsiniz sin ixda is

; whence we derive the value of a,, namely

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

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=π; it follows that each of the terms, such as

afsin x sin ix da, af sin 2x sin ix dæ, afsin

sin 2x sin ix dx, 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

1

2

sin (i –j) x to which the integral re

duces, becomes, and its value is π. Consequently we have

thus we obtain, in a very brief manner, the values of a,, a, a,,... a,, &c., namely,

≥ πÞ(x) = sin x √4(x) sin x dæ + sin 2x (ø (x) sin 2x dx + &c.

+ sin ix ($(x) sin ixdæ+ &c.

222. The simplest case is that in which the given function has a constant value for all values of the variable x included

between 0 and π; in this case the integral sin ix dx is equal to

2

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 x 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,

fæ sin x dæ + sin 2x fæ sin 2 dæ

1

> T = sin x

The integral ["

0

π

x sin ixda 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,

1

π COS X = sin x

2

cos x sin x dx + sin 2x cos x sin 2x dx

+ sin 3x cos x sin 3x dx + &c.

The integral Jo

odd number, and to

==

cos x sin ix dx is equal to zero when i is an

successively i = 2, 4,

series

or,

2i

1

when i is an even number. Supposing

6, 8, etc., we have the always convergent

2

1

5.7

-={(+3) sin 2x + (+) sin x + (+4) sin 6x + &c.}.

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

This series is known (Introd. ad analysin. infinit. cap. x.).

224. A similar analysis may be employed for the development of any function whatever in a series of cosines of multiple arcs. Let (x) be the function whose development is required, we may write

$(x) = a ̧ cos 0x+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

taken from x = 0 to x=π, supposing j and i to be integers.

have

We

This integral, taken from 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

becomes, and its value is, when the area 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 = žπɑ,

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, and the value

of each term is ; thus the integral cos jx cos ix dx taken

from 0 to π is nothing when the two integers j and i x = are different: it is 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,

1 π

S

¿π+ (x) = {} [ "$ (x) dx + cos x ["p(x)cos xdx+cos 2x ["p(x) cos 2x dx

ἐπφ

0

0

cos 3x dx + &c. (n)1.

1 The process analogous to (A) in Art. 222 fails here; yet we see, Art. 177, that

an analogous result exists. [R. L. E.]