n1, n, n,, &c. being given by the following equation

in which e represents na and the value of μ, is

sin 2na).

2n,a

In the same manner the functions & (y, t), $ (z, t) are found.

337. We may be assured that this value of v solves the problem in all its extent, and that the complete integral of the partial differential equation (a) must necessarily take this form in order to express the variable temperatures of the solid.

In fact, the expression for v satisfies the equation (a) and the conditions relative to the surface. Hence the variations of temperature which result in one instant from the action of the molecules and from the action of the air on the surface, are those which we should find by differentiating the value of v with respect to the time t. It follows that if, at the beginning of any instant, the function v represents the system of temperatures, it will still represent those which hold at the commencement of the following instant, and it may be proved in the same manner that the variable state of the solid is always expressed by the function v, in which the value of t continually increases. Now this function agrees with the initial state: hence it represents all the later states of the solid. Thus it is certain that any solution which gives for v a function different from the preceding must be wrong.

338. If we suppose the time t, which has elapsed, to have become very great, we no longer have to consider any but the first term of the expression for v; for the values n,, n,, n,, &c. are arranged in order beginning with the least. This term is given by the equation

this then is the principal state towards which the system of temperatures continually tends, and with which it coincides without sensible error after a certain value of t. In this state the tempe

rature at every point decreases proportionally to the powers of the fraction e-sk; the successive states are then all similar, or rather they differ only in the magnitudes of the temperatures which all diminish as the terms of a geometrical progression, preserving their ratios. We may easily find, by means of the preceding equation, the law by which the temperatures decrease from one point to another in direction of the diagonals or the edges of the cube, or lastly of a line given in position. We might ascertain also what is the nature of the surfaces which determine the layers of the same temperature. We see that in the final and regular state which we are here considering, points of the same layer preserve always equal temperatures, which would not hold in the initial state and in those which immediately follow it. During the infinite continuance of the ultimate state the mass is divided into an infinity of layers all of whose points have a common temperature.

339. It is easy to determine for a given instant the mean temperature of the mass, that is to say, that which is obtained by taking the sum of the products of the volume of each molecule by its temperature, and dividing this sum by the whole volume. We thus form the expression feddyd, which is that of the dz mean temperature V. The integral must be taken successively with respect to x, y, and z, between the limits a and a: v being equal to the product XYZ, we have

since the three complete

e-knit+ (sinn, a) 1

-knit+ &c.

The quantity na is equal to e, a root of the equation e tan e=

ha K'

We have then, denoting the

different roots of this equation by €, €, €, &c.,

1

3п

2

e, is between 0 andπ, e, is between π and , e, between 2 and

5

2

π, the roots €, €, €, &c. approach more and more nearly to the inferior limits π, 2π, 3π, &c., and end by coinciding with them when the index i is very great. The double arcs 2€,, 2€,, 2€, &c., are included between 0 and π, between 2 and 3π, between 4π and 57; for which reason the sines of these arcs are all positive: the quantities 1+ sin 2e,, &c., are positive and included

sin 2€,, 1+ 2€,

between 1 and 2. It follows from this that all the terms which

enter into the value of V are positive.

340. We propose now to compare the velocity of cooling in the cube, with that which we have found for a spherical mass. We have seen that for either of these bodies, the system of temperatures converges to a permanent state which is sensibly attained after a certain time; the temperatures at the different points of the cube then diminish all together preserving the same ratios, and the temperatures of one of these points decrease as the terms of a geometric progression whose ratio is not the same in the two bodies. It follows from the two solutions that the ratio for the sphere is e and for the cube e3. The quantity n is given by the equation

-kn3

-3

a being the semi-diameter of the sphere, and the quantity e is given

h

by the equation e tan e

Ka, a being the half side of the cube.

This arranged, let us consider two different cases; that in which the radius of the sphere and the half side of the cube are each equal to a, a very small quantity; and that in which the value of a is very great. Suppose then that the two bodies are of

having a very small value, the same is the

case with e, we have therefore =e, hence the fraction

ha

K

Thus the ultimate temperatures which we observe are expressed in

3ht

the form Ae CDa. If now in the equation

suppose the second member to differ very little from unity, we find

We conclude from this that if the radius of the sphere is very small, the final velocities of cooling are the same in that solid and in the circumscribed cube, and that each is in inverse ratio of the radius; that is to say, if the temperature of a cube whose half side is a passes from the value A to the value B in the time t, a sphere whose semi-diameter is a will also pass from the temperature A to the temperature B in the same time. If the quantity a were changed for each body so as to become a', the time required for the passage from A to B would have another value t, and the ratio of the times t and t' would be that of the half sides a and a'. The same would not be the case when the radius a is very great : for e is then equal to 1, and the values of na are the quantities π, 2π, 3π, 4π, &c.

We may then easily find, in this case, the values of the fractions e a2

-kn3

kn2

; they are e 4a and ea2.

From this we may derive two remarkable consequences: 1st, when two cubes are of great dimensions, and a and a' are their halfsides; if the first occupies a time t in passing from the temperature A to the temperature B, and the second the time t' for the same interval; the times t and t will be proportional to the squares a2 and a of the half-sides. We found a similar result for spheres of great dimensions. 2nd, If the length a of the half-side of a cube is considerable, and a sphere has the same magnitude a for radius, and during the time t the temperature of the cube falls from A to 3, a different time will elapse whilst the temperature of the

sphere is falling from A to B, and the times t and t' are in the ratio of 4 to 3.

Thus the cube and the inscribed sphere cool equally quickly when their dimension is small; and in this case the duration of the cooling is for each body proportional to its thickness. If the dimension of the cube and the inscribed sphere is great, the final duration of the cooling is not the same for the two solids. This duration is greater for the cube than for the sphere, in the ratio of 4 to 3, and for each of the two bodies severally the duration of the cooling increases as the square of the diameter.

341. We have supposed the body to be cooling slowly in atmospheric air whose temperature is constant. We might submit the surface to any other condition, and imagine, for example, that all its points preserve, by virtue of some external cause, the fixed temperature 0. The quantities n, p, q, which enter into the value of v under the symbol cosine, must in this case be such that cos nx becomes nothing when x has its complete value a, and that the same is the case with cos py and cos qz. If 2a the side of the cube is represented by π, 2π being the length of the circumference whose radius is 1; we can express a particular value of v by the following equation, which satisfies at the same time the general equation of movement of heat, and the state of the surface,

This function is nothing, whatever be the time t, when x or y or z

temperature cannot have this simple form until after a considerable time has elapsed, unless, the given initial state is itself represented by cos x cos y cos z. This is what we have supposed in Art. 100, Sect. VIII. Chap. I. The foregoing analysis proves the truth of the equation employed in the Article we have just cited.

Up to this point we have discussed the fundamental problems in the theory of heat, and have considered the action of that element in the principal bodies. Problems of such kind and order have been chosen, that each presents a new difficulty of a higher degree. We have designedly omitted a numerous variety of