ᎪᎡᎢ .

SECTION VIII.

APPLICATION OF THE GENERAL EQUATIONS.

155, 156. In applying the general equation (A) to the case of the cylinder

and of the sphere, we find the same equations as those of Section III.

and of Section II. of this chapter

PAGE

123

SECTION IX.

GENERAL REMARKS.

157-162. Fundamental considerations on the nature of the quantities

x, t, v, K, h, C, D, which enter into all the analytical expressions of the

Theory of Heat. Each of these quantities has an exponent of dimension

which relates to the length, or to the duration, or to the temperature.

These exponents are found by making the units of measure vary

126

CHAPTER III.

Propagation of Heat in an infinite rectangular solid.

SECTION I.

STATEMENT OF THE PROBLEM.

163-166. The constant temperatures of a rectangular plate included be-

tween two parallel infinite sides, maintained at the temperature 0, are

d2v d2v

dx dy

expressed by the equation + =0.

167-170. If we consider the state of the plate at a very great distance from
the transverse edge, the ratio of the temperatures of two points whose
coordinates are x, y and x, y changes according as the value of y
increases; x1 and x, preserving their respective values. The ratio has
a limit to which it approaches more and more, and when y is infinite,
it is expressed by the product of a function of x and of a function of y.
This remark suffices to disclose the general form of v, namely,

It is easy to ascertain how the movement of heat in the plate is

effected

131

134

SECTION II.

ART.

FIRST EXAMPLE OF THE USE OF TRIGONOMETRIC SERIES IN THE

THEORY OF HEAT.

PAGE

From which we conclude

1

4

-

2i -1

(− 1)i+1,

1

4

3

1

cos 3x+008 5x-

1

cos 7x+ etc.

137

SECTION III.

REMARKS ON THESE SERIES.

179-181. To find the value of the series which forms the second member,

the number m of terms is supposed to be limited, and the series becomes

a function of a and m. This function is developed according to powers of

the reciprocal of m, and m is made infinite.

182-184. The same process is applied to several other series

185-188. In the preceding development, which gives the value of the

function of x and m, we determine rigorously the limits within which the

sum of all the terms is included, starting from a given term

189. Very simple process for forming the series

190, 191. Analytical expression of the movement of heat in a rectangular

slab; it is decomposed into simple movements

192–195. Measure of the quantity of heat which crosses an edge or side

parallel or perpendicular to the base. This expression of the flow suffices

to verify the solution

196-199. Consequences of this solution. The rectangular slab must be

considered as forming part of an infinite plane; the solution expresses

the permanent temperatures at all points of this plane

200-204. It is proved that the problem proposed admits of no other solu-

tion different from that which we have just stated.

ART.

SECTION V.

FINITE EXPRESSION OF THE RESULT OF THE SOLUTION.

205, 206. The temperature at a point of the rectangular slab whose co-

ordinates are x and y, is expressed thus

PAGE

SECTION VI.

DEVELOPMENT OF AN ARBITRARY FUNCTION IN TRIGONOMETRIC SERIES.

207-214. The development obtained by determining the values of the un-

known coefficients in the following equations infinite in number:

To solve these equations, we first suppose the number of equations to be

m, and that the number of unknowns a, b, c, d, &c. is m only, omitting

all the subsequent terms. The unknowns are determined for a certain

value of the number m, and the limits to which the values of the coeffi-

cients continually approach are sought; these limits are the quantities

which it is required to determine. Expression of the values of a, b, c, d,

&c. when m is infinite

215, 216. The function (x) developed under the form

a sin x+b sin 2x + c sin 3x+d sin 4x+ &c.,

168

which is first supposed to contain only odd powers of x

217, 218. Different expression of the same development. Application to the

function ex- e-x

219-221. Any function whatever (x) may be developed under the form

a1 sin x+a, sin 2x + sin 3x+ +a, sin ix + &c.

...

2 П

0

The value of the general coefficient a, isda (x) sin ix. Whence we

derive the very simple theorem

ƒ” "da p (a) sin 2a + sin 3x "dap (a) sin 3a +&c.,

0

(x)=Σsin ix ["da p(a) sin ia

1

222, 223. Application of the theorem: from it is derived the remarkable

184

ART.

224, 225. Second theorem on the development of functions in trigono-

metrical series:

PAGE

190

226-230. The preceding theorems are applicable to discontinuous functions,

and solve the problems which are based upon the analysis of Daniel

Bernoulli in the problem of vibrating cords. The value of the series,

T

is

1

2

1

3

sin x versin a+ sin 2x versin 2a + sin 3x versin 3a +&c.,

if we attribute to x a quantity greater than 0 and less than a; and

the value of the series is 0, if x is any quantity included between a and 1⁄4”.

Application to other remarkable examples; curved lines or surfaces which

coincide in a part of their course, and differ in all the other parts.

231–233. Any function whatever, F(x), may be developed in the form

a cos x+a, cos 2x +α cos 3x+&c.,

F(x) = 4+ { cinx+b2 sin 2x+b, sin 3x+&c.

Each of the coefficients is a definite integral. We have in general

We thus form the general theorem, which is one of the chief elements of

our analysis:

2π F(x)=2

193

234. The values of F(x) which correspond to values of x included

between − π and +☛ must be regarded as entirely arbitrary. We may

also choose any limits whatever for x

235. Divers remarks on the use of developments in trigonometric series

204

236, 237. Expression of the permanent temperature in the infinite rectangular

slab, the state of the transverse edge being represented by an arbitrary

function.

209

ART.

CHAPTER IV.

Of the linear and varied Movement of Heat in a ring.

SECTION I.

GENERAL SOLUTION OF THE PROBLEM.

238-241. The variable movement which we are considering is composed of

simple movements. In each of these movements, the temperatures pre-

serve their primitive ratios, and decrease with the time, as the ordinates v

of a line whose equation is v=A.e-mt. Formation of the general ex-

pression.

242-244. Application to some remarkable examples. Different consequences

of the solution

PAGE

213

218

245, 246. The system of temperatures converges rapidly towards a regular
and final state, expressed by the first part of the integral. The sum of
the temperatures of two points diametrically opposed is then the same,
whatever be the position of the diameter. It is equal to the mean tem-
perature. In each simple movement, the circumference is divided by
equidistant nodes. All these partial movements successively disappear,
except the first; and in general the heat distributed throughout the solid
assumes a regular disposition, independent of the initial state

SECTION II,

OF THE COMMUNICATION OF HEAT BETWEEN SEPARATE MASSES.

247-250. Of the communication of heat between two masses. Expression

of the variable temperatures. Remark on the value of the coefficient

which measures the conducibility

251-255. Of the communication of heat between n separate masses, ar-

ranged in a straight line. Expression of the variable temperature of each

mass; it is given as a function of the time elapsed, of the coefficient

which measures the conducibility, and of all the initial temperatures

regarded as arbitrary

256, 257. Remarkable consequences of this solution

221

225

228

236

237

258. Application to the case in which the number of masses is infinite.

259-266. Of the communication of heat between n separate masses arranged

circularly. Differential equations suitable to the problem; integration of

these equations. The variable temperature of each of the masses is ex-

pressed as a function of the coefficient which measures the conducibility,

of the time which has elapsed since the instant when the communication

began, and of all the initial temperatures, which are arbitrary; but in

order to determine these functions completely, it is necessary to effect

the elimination of the coefficients

267-271. Elimination of the coefficients in the equations which contain

these unknown quantities and the given initial temperatures