257. If we divide the semi-circumference into n equal parts, and, having drawn the sines, take the difference between two consecutive sines, the n differences are proportional to the co efficients of e 2kt versin u or to the second terms of the values of For this reason the later values of a, B, y...w are such that the differences between the final temperatures and the a, B, y,...w. 1 mean initial temperature (a+b+c+ &c.) are always propor n tional to the differences of consecutive sines. In whatever manner the masses have first been heated, the distribution of heat is effected finally according to a constant law. If we measured the temperatures in the last stage, when they differ little from the mean temperature, we should observe that the difference between the temperature of any mass whatever and the mean temperature decreases continually according to the successive powers of the same fraction; and comparing amongst themselves the temperatures of the different masses taken at the same instant, we should see that the differences between the actual temperatures and the mean temperature are proportional to the differences of consecutive sines, the semi-circumference having been divided into n equal parts. 258. If we suppose the masses which communicate heat to each other to be infinite in number, we find for the arc u an infinitely small value; hence the differences of consecutive sines, taken on the circle, are proportional to the cosines of the corresponding sin mu - sin (m − 1) u arcs; for is equal to cos mu, when the sin u arc u is infinitely small. In this case, the quantities whose temperatures taken at the same instant differ from the mean temperature to which they all must tend, are proportional to the cosines which correspond to different points of the circumference divided into an infinite number of equal parts. If the masses which transmit heat are situated at equal distances from each other on the perimeter of the semi-circumference, the cosine of the arc at the end of which any one mass is placed is the measure of the quantity by which the temperature of that mass differs yet from the mean temperature. Thus the body placed in the middle of all the others is that which arrives most quickly at that mean temperature; those which are situated on one side of the middle, all have an excessive temperature, which surpasses the mean temperature the more, according as they are more distant from the middle; the bodies which are placed on the other side, all have a temperature lower than the mean temperature, and they differ from it as much as those on the opposite side, but in contrary sense. Lastly, these differences, whether positive or negative, all decrease at the same time, proportionally to the successive powers of the same fraction; so that they do not cease to be represented at the same instant by the values of the cosines of the same semi-circumference. Such in general, singular cases excepted, is the law to which the ultimate temperatures are subject. The initial state of the system does not change these results. We proceed now to deal with a third problem of the same kind as the preceding, the solution of which will furnish us with many useful remarks. 259. Suppose n equal prismatic masses to be placed at equal distances on the circumference of a circle. All these bodies, enjoying perfect conducibility, have known actual temperatures, different for each of them; they do not permit any part of the heat which they contain to escape at their surface; an infinitely thin layer is separated from the first mass to be united to the second, which is situated towards the right; at the same time a parallel layer is separated from the second mass, carried from left to right, and joined to the third; the same is the case with all the other masses, from each of which an infinitely thin layer is separated at the same instant, and joined to the following mass. Lastly, the same layers return immediately afterwards, and are united to the bodies from which they had been detached. Heat is supposed to be propagated between the masses by means of these alternate movements, which are accomplished twice during each instant of equal duration; the problem is to find according to what law the temperatures vary: that is to say, the initial values of the temperatures being given, it is required to ascertain after any given time the new temperature of each of the masses. We shall denote by a,, a, a,,...a....a the initial temperatures whose values are arbitrary, and by a, a, a,...,...", the values of the same temperatures after the time t has elapsed. Each of the quantities a is evidently a function of the time t and of all the initial values a, a, a,...a: it is required to determine the functions 2. 260. We shall represent the infinitely small mass of the layer which is carried from one body to the other by w. We may remark, in the first place, that when the layers have been separated from the masses of which they have formed part, and placed respectively in contact with the masses situated towards the right, the quantities of heat contained in the different bodies become (m-w) α, +wx, (m − w) α2 + wx, (m—w) α, + wx,,..., (m — w) a +w11; dividing each of these quantities of heat by the mass m, we have for the new values of the temperatures + (− &n); m that is to say, to find the new state of the temperature after the first contact, we must add to the value which it had formerly the product of by the excess of the temperature of the body m from which the layer has been separated over that of the body to which it has been joined. By the same rule it is found that the temperatures, after the second contact, are The time being divided into equal instants, denote by dt the duration of the instant, and suppose to be contained in k units of mass as many times as dt is contained in the units of time, we thus have w=kdt. Calling dz,, da, dë ̧...dz..........d2„ the infinitely small increments which the temperatures a,,,,....... receive during the instant dt, we have the following differential equations: m k - dt (x, − 2a, + a ̧), = m = m k = = ՊՆ k m 261. To solve these equations, we suppose in the first place, according to the known method, a1 = b1e", a=b,e", The quantities b1, b2, b1, ... b, a=be", a = be1. are undetermined constants, as also is the exponent h. It is easy to see that the values of a1, a....a satisfy the differential equations if they are subject to the following conditions: It follows from this that we may take, instead of b1, b, b,... b...b, the n consecutive sines which are obtained by dividing the whole circumference 2π into n equal parts. In fact, denoting the arc 2 n by u, the quantities sin Ou, sin lu, sin 2u, sin 3u, ..., sin (n - 1) u, whose number is n, belong, as it is said, to a recurring series whose scale of relation has two terms, 2 cos u and 1: so that we always have the condition Take then, instead of b1, b, b„,... b, the quantities sin Ou, sin lu, sin 2u, sin (n - 1) u, ... 262. The last equations furnish only a very particular solution of the problem proposed; for if we suppose t = 0 we have, as the initial values of a,, a, a, ... a, the quantities sin Ou, sin lu, sin 2u, sin (n-1) u, ... n which in general differ from the given values a1, α, α, ... a1: but the foregoing solution deserves to be noticed because it expresses, as we shall see presently, a circumstance which belongs to all possible cases, and represents the ultimate variations of the F. H. 16 |