ART.

to the cosine of the angle which its direction makes with the normal to
the surface. Divers remarks, and considerations on the object and extent
of thermological problems, and on the relations of general analysis with
the study of nature.

PAGE

22

26

22-24.

SECTION II.

GENERAL NOTIONS AND PRELIMINARY DEFINITIONS.

Permanent temperature, thermometer. The temperature denoted

by 0 is that of melting ice. The temperature of water boiling in a

given vessel under a given pressure is denoted by 1.

25. The unit which serves to measure quantities of heat, is the heat

required to liquify a certain mass of ice

26. Specific capacity for heat.

27-29. Temperatures measured by increments of volume or by the addi-

tional quantities of heat. Those cases only are here considered, in which

the increments of volume are proportional to the increments of the

quantity of heat. This condition does not in general exist in liquids;

it is sensibly true for solid bodies whose temperatures differ very much

from those which cause the change of state

30. Notion of external conducibility

31. We may at first regard the quantity of heat lost as proportional to the

temperature. This proposition is not sensibly true except for certain

limits of temperature

32-35. The heat lost into the medium consists of several parts. The effect
is compound and variable. Luminous heat

ib.

36. Measure of the external conducibility

31

37. Notion of the conducibility proper. This property also may be observed

in liquids..

38, 39. Equilibrium of temperatures. The effect is independent of contact.

40-49. First notions of radiant heat, and of the equilibrium which is

established in spaces void of air; of the cause of the reflection of rays

of heat, or of their retention in bodies; of the mode of communication

between the internal molecules; of the law which regulates the inten-

sity of the rays emitted. The law is not disturbed by the reflection of

heat.

50, 51. First notion of the effects of reflected heat

52-56. Remarks on the statical or dynamical properties of heat. It is the

principle of elasticity. The elastic force of aeriform fluids exactly indi-

cates their temperatures

SECTION III.

PRINCIPLE OF THE COMMUNICATION OF HEAT.

57-59. When two molecules of the same solid are extremely near and at

unequal temperatures, the most heated molecule communicates to that

which is less heated a quantity of heat exactly expressed by the product

of the duration of the instant, of the extremely small difference of the

temperatures, and of a certain function of the distance of the molecules.

41

ᎪᎡᎢ.

60. When a heated body is placed in an aeriform medium at a lower tem-
perature, it loses at each instant a quantity of heat which may be
regarded in the first researches as proportional to the excess of the
temperature of the surface over the temperature of the medium

61-64. The propositions enunciated in the two preceding articles are founded

on divers observations. The primary object of the theory is to discover

all the exact consequences of these propositions. We can then measure

the variations of the coefficients, by comparing the results of calculation

with very exact experiments

PAGE

43

. ib.

SECTION IV.

OF THE UNIFORM AND LINEAR MOVEMENT OF HEAT.

65. The permanent temperatures of an infinite solid included between two
parallel planes maintained at fixed temperatures, are expressed by the
equation (va) e = (b− a) z; a and b are the temperatures of the two
extreme planes, e their distance, and v the temperature of the section,
whose distance from the lower plane is z

.

66, 67. Notion and measure of the flow of heat

68, 69. Measure of the conducibility proper.

70. Remarks on the case in which the direct action of the heat extends to

a sensible distance

71. State of the same solid when the upper plane is exposed to the air

72. General conditions of the linear movement of heat

53

55

SECTION V.

LAW OF THE PERMANENT TEMPERATURES IN A PRISM OF SMALL THICKNESS.

73-80. Equation of the linear movement of heat in the prism. Different
consequences of this equation

SECTION VI.

THE HEATING OF CLOSED SPACES.

81–84. The final state of the solid boundary which encloses the space
heated by a surface b, maintained at the temperature a, is expressed by
the following equation :

56

The value of P is

air, n the temperature of the external air, g, h, H measure respectively
the penetrability of the heated surface σ, that of the inner surface of the
boundary s, and that of the external surfaces; e is the thickness of the
boundary, and K its conducibility proper.

+ + m is the temperature of the internal

8 h K H

85, 86. Remarkable consequences of the preceding equation.

87-91. Measure of the quantity of heat requisite to retain at a constant

temperature a body whose surface is protected from the external air by

ART.

PAGE

several successive envelopes. Remarkable effects of the separation of the

surfaces. These results applicable to many different problems

SECTION VII.

OF THE UNIFORM MOVEMENT OF HEAT IN THREE DIMENSIONS.

92, 93. The permanent temperatures of a solid enclosed between six rec-
tangular planes are expressed by the equation

v=A+ax+by+cz.

x, y, z are the coordinates of any point, whose temperature is v; ▲, a,

b, c are constant numbers. If the extreme planes are maintained by any

causes at fixed temperatures which satisfy the preceding equation, the

final system of all the internal temperatures will be expressed by the

same equation.

SECTION VIII.

MEASURE OF THE MOVEMENT OF HEAT AT A GIVEN POINT OF A GIVEN SOLID.

96-99. The variable system of temperatures of a solid is supposed to be

expressed by the equation v=F (x, y, z, t), where v denotes the variable

temperature which would be observed after the time t had elapsed, at the

point whose coordinates are x, y, z. Formation of the analytical expres-

sion of the flow of heat in a given direction within the solid

100. Application of the preceding theorem to the case in which the function

F is e- cos x cos y cos z

78

82

CHAPTER II.

Equation of the Movement of Heat.

SECTION I.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A RING.

101-105. The variable movement of heat in a ring is expressed by the

equation

The arc a measures the distance of a section from the origin 0; v is

the temperature which that section acquires after the lapse of the time t;

K, C, D, h are the specific coefficients; S is the area of the section, by

the revolution of which the ring is generated; I is the perimeter of

the section

85

ART.

equal distances are

Observation of the

106-110. The temperatures at points situated at

represented by the terms of a recurring series.

temperatures v1, v2, v3 of three consecutive points gives the measure

The distance between two consecutive points is λ, and log w is the decimal

logarithm of one of the two values of w

PAGE

86

SECTION II.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A SOLID SPHERE.

111-113.

x denoting the radius of any shell, the movement of heat in the

sphere is expressed by the equation

114-117.

Conditions relative to the state of the surface and to the initial

state of the solid

90

SECTION III.

EQUATION OF THE VARIED MOVEMENT OF HEAT IN A SOLID CYLINDER,

118-120. The temperatures of the solid are determined by three equations;

the first relates to the internal temperatures, the second expresses the

continuous state of the surface, the third expresses the initial state of

the solid

SECTION IV.

EQUATIONS OF THE VARIED MOVEMENT OF HEAT IN A SOLID PRISM

OF INFINITE LENGTH.

121-123. The system of fixed temperatures satisfies the equation

v is the temperature at a point whose coordinates are x, y, z

124, 125. Equation relative to the state of the surface and to that of the
first section

SECTION V.

EQUATIONS OF THE VARIED MOVEMENT OF HEAT IN A SOLID CUBE.

126-181. The system of variable temperatures is determined by three

equations; one expresses the internal state, the second relates to the

state of the surface, and the third expresses the initial state

ART.

SECTION VI.

GENERAL EQUATION OF THE PROPAGATION OF HEAT IN THE INTERIOR

OF SOLIDS.

132-139. Elementary proof of properties of the uniform movement of heat

in a solid enclosed between six orthogonal planes, the constant tem-

peratures being expressed by the linear equation,

v=A-ax-by- cz.

The temperatures cannot change, since each point of the solid receives

as much heat as it gives off. The quantity of heat which during the

unit of time crosses a plane at right angles to the axis of z is the same,

through whatever point of that axis the plane passes. The value of this

common flow is that which would exist, if the coefficients a and b

were nul

140, 141. Analytical expression of the flow in the interior of any solid. The

dv

equation of the temperatures being r=f(x, y, z, t) the function - Kw dz

expresses the quantity of heat which during the instant dt crosses an

infinitely small area & perpendicular to the axis of 2, at the point whose

coordinates are x, y, z, and whose temperature is v after the time t

has elapsed

PAGE

104

• 109

142-145. It is easy to derive from the foregoing theorem the general
equation of the movement of heat, namely

dv K (dv d2v dr\

=

dt CD dx dy3TM dz2

112

SECTION VII.

GENERAL EQUATION RELATIVE TO THE SURFACE.

146-154. It is proved that the variable temperatures at points on the

surface of a body, which is cooling in air, satisfy the equation

being the differential equation of the surface which bounds the solid,
and q being equal to (m2+n2+p2). To discover this equation we
consider a molecule of the envelop which bounds the solid, and we express
the fact that the temperature of this element does not change by a finite
magnitude during an infinitely small instant. This condition holds and
continues to exist after that the regular action of the medium has been
exerted during a very small instant. Any form may be given to the
element of the envelop. The case in which the molecule is formed by
rectangular sections presents remarkable properties. In the most simple
case, which is that in which the base is parallel to the tangent plane,
the truth of the equation is evident.

. 115