It is easy to express the sum of this series. To obtain the result, develope as follows the function cos (a sin x) in cosines of multiple arcs. We have by known transformations

Developing the second member according to powers of w, we find the term which does not contain w in the development of 2 cos (a sin x) to be

The coefficients of w', w3, w, &c. are nothing, the same is the case with the coefficients of the terms which contain w1, w3, w3, &c.; the coefficient of wis the same as that of w2; the coefficient of wis

the coefficient of wis the same as that of w. It is easy to express the law according to which the coefficients succeed; but without stating it, let us write 2 cos 2x instead of (w2+w72), or 2 cos 4x instead of (w* + w ̄1), and so on: hence the quantity 2 cos (a sin x) is easily developed in a series of the form

A+B cos 2x + C'cos 4x + D cos 6x + &c.,

and the first coefficient A is equal to

if we now compare the general equation which we gave formerly

with the equation

2 cos (a sin x) = A + B cos 2x + C cos 4x + &c.,

we shall find the values of the coefficients A, B, C expressed by definite integrals. It is sufficient here to find that of the first coefficient A. We have then

the integral should be taken from x=0 to x=. Hence the

manner by comparison of two equations the values of the successive coefficients B, C, &c.; we have indicated these results because they are useful in other researches which depend on the same theory. It follows from this that the particular value of u which satisfies the equation

the integral being taken from r=0 to rπ. Denoting by q this

value of u, and making u =qS, we find S=a+bfda

as the complete integral of the equation gu +

d'u

+

and we have

1 du

dx x dr

= 0,

u

=[a+bf;

dr

= \ a + b) x {cos (x√g sin r) drcos (x √g sin r) dr.

a and b are arbitrary constants. If we suppose b=0, we have,

as formerly,

u = fcos (x Jg sin r) dr.

With respect to this expression we add the following remarks.

and integrating from u=0 to u=π, denoting by S1, S., S., &c.

it remains to determine S,, S, S, &c. The term sin" u, n being an even number, may be developed thus

sin” u = A„ + B„ cos 2u + C, cos 4u + &c.

Multiplying by du and integrating between the limits u = 0 and u =π, we have simply sin" u du = A„, the other terms vanish. From the known formula for the development of the integral powers of sines, we have

Substituting these values of S2, S4, S, &c., we find

Suppose then that we have a function (z) which may be developed thus

Now, it is easy to see that the values of S, S1, S, &c. are nothing. With respect to S, S, S, &c. their values are the quantities which we previously denoted by A, 4, 4, &c. For this reason, substituting these values in the equation (e) we have generally, whatever the function & may be,

in the case in question, the function (z) represents cos z, and we have $ = 1, 4′′ = − 1, 4′′ = 1, 4" = − 1, and so on.

312. To ascertain completely the nature of the function ƒ (0), and of the equation which gives the values of g, it would be necessary to consider the form of the line whose equation is

which forms with the axis of abscissæ areas alternately positive and negative which cancel each other; the preceding remarks, also, on the expression of the values of series by means of definite integrals, might be made more general. When a function of the variable x is developed according to powers of x, it is easy to deduce the function which would represent the same series, if the powers x, x2, x3, &c. were replaced by cos x, cos 2.x, cos 3x, &c. By making use of this reduction and of the process employed in the second paragraph of Article 235, we obtain the definite integrals which are equivalent to given series; but we could not enter upon this investigation, without departing too far from our main object.

It is sufficient to have indicated the methods which have enabled us to express the values of series by definite integrals.

65 (0)
f(0)

We will add only the development of the quantity 6continued fraction,

313. The undetermined y or f(0) satisfies the equation

in a

314. We shall now state the results at which we have

this point arrived.

up to

If the variable radius of the cylindrical layer be denoted by x, and the temperature of the layer by v, a function of x and the time t; the required function v must satisfy the partial differential equation

1 dv);

dt

dx2

for v we may assume the following value

v = uemt;

u is a function of x, which satisfies the equation