satisfied, it is evident (Art. 170) that the quantity of heat which determines the temperature of each molecule can be neither increased nor diminished. The different points of the same solid having received the temperatures expressed by equation (a) or v = p(x, y), suppose that instead of maintaining the edge A at the temperature 1, the fixed temperature 0 be given to it as to the two lines B and C'; the heat contained in the plate BAC will flow across the three edges A, B, C, and by hypothesis it will not be replaced, so that the temperatures will diminish continually, and their final and common value will be zero. This result is evident since the points infinitely distant from the origin A have a temperature infinitely small from the manner in which equation (a) was formed. The same effect would take place in the opposite direction, if the system of temperatures were v(x, y), instead of being v = p(x, y); that is to say, all the initial negative temperatures would vary continually, and would tend more and more towards their final value 0, whilst the three edges A, B, C preserved the temperature 0. 202. Let v = (x, y) be a given equation which expresses the initial temperature of points in the plate BAC, whose base A is maintained at the temperature 1, whilst the edges B and C preserve the temperature 0. Let v = F(x, y) be another given equation which expresses the initial temperature of each point of a solid plate BAC exactly the same as the preceding, but whose three edges B, A, C are maintained at the temperature 0. Suppose that in the first solid the variable state which succeeds to the final state is determined by the equation v = $ (x, y, t), t denoting the time elapsed, and that the equation v = (x, y, t) determines the variable state of the second solid, for which the initial temperatures are F(x, y). Lastly, suppose a third solid like each of the two preceding: let v = f(x, y) + F(x, y) be the equation which represents its initial state, and let 1 be the constant temperature of the base A, 0 and 0 those of the two edges B and C. We proceed to shew that the variable state of the third solid is determined by the equation v = (x, y, t) + Þ(x, y, t). In fact, the temperature of a point m of the third solid varies, because that molecule, whose volume is denoted by M, acquires or loses a certain quantity of heat A. The increase of temperature during the instant dt is the coefficient c denoting the specific capacity with respect to volume. The variation of the temperature of the same point in d D the first solid is dt, and dt in the second, the letters CM CM d and D representing the quantity of heat positive or negative which the molecule acquires by virtue of the action of all the neighbouring molecules. Now it is easy to perceive that A is equal to d+D. For proof it is sufficient to consider the quantity of heat which the point m receives from another point m' belonging to the interior of the plate, or to the edges which bound it. The point m,, whose initial temperature is denoted by f,, transmits, during the instant dt, to the molecule m, a quantity of heat expressed by q,f,-f)dt, the factor q, representing a certain function of the distance between the two molecules. Thus the whole quantity of heat acquired by m is Eq,(f,-ƒ)dt, the sign Σ expressing the sum of all the terms which would be found by considering the other points m2, m,, m, &c. which act on m; that is to say, writing 1,, f, or 3, ƒ, or q1, f, and so on, instead of If In the same manner Eq,(FF)dt will be found to be the expression of the whole quantity of heat acquired by the same point m of the second solid; and the factor q, is the same as in the term Eq,f-f)dt, since the two solids are formed of the same matter, and the position of the points is the same; we have then d = £q1(f1-ƒ)dt and D = Σq1(F, — F)dt. For the same reason it will be found that It follows from this that the molecule m of the third solid acquires, during the instant dt, an increase of temperature equal to the sum of the two increments which the same point would have gained in the two first solids. Hence at the end of the first instant, the original hypothesis will again hold, since any molecule whatever of the third solid has a temperature equal to the sum of those which it has in the two others. Thus the same relation exists at the beginning of each instant, that is to say, the variable state of the third solid can always be represented by the equation v = 0 (x, y, t) + Þ(x, y, t). 203. The preceding proposition is applicable to all problems relative to the uniform or varied movement of heat. It shews that the movement can always be decomposed into several others, each of which is effected separately as if it alone existed. This superposition of simple effects is one of the fundamental elements in the theory of heat. It is expressed in the investigation, by the very nature of the general equations, and derives its origin from the principle of the communication of heat. Let now v(x, y) be the equation (2) which expresses the permanent state of the solid plate BAC, heated at its end A, and whose edges B and C preserve the temperature 1; the initial state of the plate is such, according to hypothesis, that all its points have a nul temperature, except those of the base A, whose temperature is 1. The initial state can then be considered as formed of two others, namely: a first, in which the initial temperatures are -(x, y), the three edges being maintained at the temperature 0, and a second state, in which the initial temperatures are + p(x, y), the two edges B and C preserving the temperature 0, and the base A the temperature 1; the superposition of these two states. produces the initial state which results from the hypothesis. remains then only to examine the movement of heat in each one of the two partial states. Now, in the second, the system of temperatures can undergo no change; and in the first, it has been remarked in Article 201 that the temperatures vary continually, and end with being nul. Hence the final state, properly so called, is that which is represented by v = (x, y) or equation (a). It If this state were formed at first it would be self-existent, and it is this property which has served to determine it for us. If the solid plate be supposed to be in another initial state, the difference between the latter state and the fixed state forms a partial state, which imperceptibly disappears. After a considerable time, the difference has nearly vanished, and the system of fixed temperatures has undergone no change. Thus the variable temperatures converge more and more to a final state, independent of the primitive heating. 204. We perceive by this that the final state is unique; for, if a second state were conceived, the difference between the second and the first would form a partial state, which ought to be self-existent, although the edges A, B, C were maintained at the temperature 0. Now the last effect cannot occur; similarly if we supposed another source of heat independent of that which flows. from the origin A; besides, this hypothesis is not that of the problem we have treated, in which the initial temperatures are nul. It is evident that parts very distant from the origin can only acquire an exceedingly small temperature. Since the final state which must be determined is unique, it follows that the problem proposed admits no other solution than that which results from equation (x). Another form may be given to this result, but the solution can be neither extended nor restricted without rendering it inexact. The method which we have explained in this chapter consists in forming first very simple particular values, which agree with the problem, and in rendering the solution more general, to the intent that v or (x, y) may satisfy three conditions, namely: do d'v = 0, ☀ (x, 0) = 1, † (x, ± 1π) = 0. It is clear that the contrary order might be followed, and the solution obtained would necessarily be the same as the foregoing. We shall not stop over the details, which are easily supplied, when once the solution is known. We shall only give in the following section a remarkable expression for the function (x, y) whose value was developed in a convergent series in equation (a). SECTION V. Finite expression of the result of the solution. 205. The preceding solution might be deduced from the integral of the equation d'v dv + =0,' which contains imaginary quantities, under the sign of the arbitrary functions. We shall confine ourselves here to the remark that the integral v=4(x+y√ − 1) + ↓ (x − y√ − 1), has a manifest relation to the value of v given by the equation In fact, replacing the cosines by their imaginary expressions, we have The first series is a function of x-y-1, and the second series is the same function of x + y√ − 1. Comparing these series with the known development of arc tan z in functions of z its tangent, it is immediately seen that the first is arc tan e ̄ ̄√, and the second is arc tan e√); thus equation (a) takes the finite form πυ = arc tan e ̄(+1) + arc tan e ̄V=T) ̧... 2 In this mode it conforms to the general integral v = $(x + y√ − 1) + (x − y√1) the function (z) is arc tan e, and similarly the function 1 D. F. Gregory derived the solution from the form (B). ..... (4), (2). |