The fluxion of the ordinate of this curve, answering to the point m, taken with the opposite sign, expresses the velocity with which heat is transferred across the plane. This fluxion of the ordinate is known to be the tangent of the angle formed by the element of the curve with a parallel to the abscissæ.

The result which we have just explained is that of which the most frequent applications have been made in the theory of heat. We cannot discuss the different problems without forming a very exact idea of the value of the flow at every point of a body whose temperatures are variable. It is necessary to insist on this fundamental notion; an example which we are about to refer to will indicate more clearly the use which has been made of it in analysis,

.

100. Suppose the different points of a cubic mass, an edge of which has the length T, to have unequal actual temperatures represented by the equation v cos x cos y cos z. The coordinates x, y, z are measured on three rectangular axes, whose origin is at the centre of the cube, perpendicular to the faces. The points of the external surface of the solid are at the actual temperature 0, and it is supposed also that external causes maintain at all these points the actual temperature 0. On this hypothesis the body will be cooled more and more, the temperatures of all the points situated in the interior of the mass will vary, and, after an infinite time, they will all attain the temperature 0 of the surface. Now, we shall prove in the sequel, that the variable state of this solid is expressed by the equation v = e cos x cos y cos z,

the coefficient g is equal to

3K C.D'

K is the specific conduci

bility of the substance of which the solid is formed, D is the density and C the specific heat; t is the time elapsed.

We here suppose that the truth of this equation is admitted, and we proceed to examine the use which may be made of it to find the quantity of heat which crosses a given plane parallel to one of the three planes at the right angles.

If, through the point m, whose co-ordinates are x, y, z, we draw a plane perpendicular to z, we shall find, after the mode

of the preceding article, that the value of the flow, at this point

dv dz'

and across the plane, is K or Ke cos x.cos y . sin z. The

quantity of heat which, during the instant dt, crosses an infinitely small rectangle, situated on this plane, and whose sides are dx and dy, is

Ke" cos x cos y sin z dx dy dt.

Thus the whole heat which, during the instant dt, crosses the entire area of the same plane, is

If then we take the integral with respect to t, from t = 0 to t=t, we shall find the quantity of heat which has crossed the same plane since the cooling began up to the actual moment.

so that after an infinite time the quantity of heat lost through

to each of the six faces, we conclude that the solid has lost by its

24K

complete cooling a total quantity of heat equal to or 8CD,

since g is equivalent to

3K
CD

The total heat which is dissipated

during the cooling must indeed be independent of the special conducibility K, which can only influence more or less the velocity of cooling.

100. A. We may determine in another manner the quantity of heat which the solid loses during a given time, and this will serve in some degree to verify the preceding calculation. In fact, the mass of the rectangular molecule whose dimensions are dx, dy, dz, is Ddx dy dz, consequently the quantity of heat which must be given to it to bring it from the temperature 0 to that of boiling water is CD dx dy dz, and if it were required to raise this molecule to the temperature v, the expenditure of heat would be v CD dx dy dz.

It follows from this, that in order to find the quantity by which the heat of the solid, after time t, exceeds that which - it contained at the temperature 0, we must take the mul1

tiple integral

v CD dæ dy dz, between the limits a = 122 TT,

We thus find, on substituting for v its value, that is to say

et cos x cos y cos z,

that the excess of actual heat over that which belongs to the temperature 0 is 8CD (1-e); or, after an infinite time, 8CD, as we found before.

We have described, in this introduction, all the elements which it is necessary to know in order to solve different problems relating to the movement of heat in solid bodies, and we have given some applications of these principles, in order to shew the mode of employing them in analysis; the most important use which we have been able to make of them, is to deduce from them the general equations of the propagation of heat, which is the subject of the next chapter.

Note on Art. 76. The researches of J. D. Forbes on the temperatures of a long iron bar heated at one end shew conclusively that the conducting power K is not constant, but diminishes as the temperature increases.-Transactions of the Royal Society of Edinburgh, Vol. XXIII. pp. 133-146 and Vol. xxiv. pp. 73–110.

Note on Art. 98. General expressions for the flow of heat within a mass in which the conductibility varies with the direction of the flow are investigated by Lamé in his Théorie Analytique de la Chaleur, pp. 1-8. [A. F.]

CHAPTER II.

EQUATIONS OF THE MOVEMENT OF HEAT.

SECTION I.

Equation of the varied movement of heat in a ring.

101. WE might form the general equations which represent the movement of heat in solid bodies of any form whatever, and apply them to particular cases. But this method would often involve very complicated calculations which may easily be avoided. There are several problems which it is preferable to treat in a special manner by expressing the conditions which are appropriate to them; we proceed to adopt this course and examine separately the problems which have been enunciated in the first section of the introduction; we will limit ourselves at first to forming the differential equations, and shall give the integrals of them in the following chapters.

102. We have already considered the uniform movement of heat in a prismatic bar of small thickness whose extremity is immersed in a constant source of heat. This first case offered no difficulties, since there was no reference except to the permanent state of the temperatures, and the equation which expresses them is easily integrated. The following problem requires a more profound investigation; its object is to determine the variable state of a solid ring whose different points have received initial temperatures entirely arbitrary.

The solid ring or armlet is generated by the revolution of a rectangular section about an axis perpendicular to the plane of

Fig. 3.

the ring (see figure 3), I is the perimeter of the section whose area is S, the coefficient h measures the external conducibility, K the internal conducibility, C the specific capacity for heat, D the density. The line o.xxx" represents the mean circumference of the armlet, or that line which passes through the centres of figure of all the sections; the distance

of a section from the origin o is measured by the arc whose length is x; R is the radius of the mean circumference. It is supposed that on account of the small dimensions and of the form of the section, we may consider the temperature at the different points of the same section to be equal.

103. Imagine that initial arbitrary temperatures have been given to the different sections of the armlet, and that the solid is then exposed to air maintained at the temperature 0, and displaced with a constant velocity; the system of temperatures will continually vary, heat will be propagated within the ring, and dispersed at the surface: it is required to determine what will be the state of the solid at any given instant.

Letv be the temperature which the section situated at distance a will have acquired after a lapse of time t; v is a certain function of x and t, into which all the initial temperatures also must enter: this is the function which is to be discovered.

104. We will consider the movement of heat in an infinitely small slice, enclosed between a section made at distance x and another section made at distance x + dx. The state of this slice for the duration of one instant is that of an infinite solid terminated by two parallel planes maintained at unequal temperatures; thus the quantity of heat which flows during this instant dt across the first section, and passes in this way from the part of the solid which precedes the slice into the slice itself, is measured according to the principles established in the introduction, by the product of four factors, that is to say, the conducibility K, the area of the