tained at a constant temperature 0, and carried away by a current with uniform velocity. Within the interior of the solid, heat will pass successively all the parts situate to the right of the source, and not exposed directly to its action; they will be heated more and more, but the temperature of each point will not increase beyond a certain limit. This maximum temperature is not the same for every section; it in general decreases as the distance of the section from the origin increases: we shall denote by v the fixed temperature of a section perpendicular to the axis, and situate at a distance x from the origin A. Before every point of the solid has attained its highest degree of heat, the system of temperatures varies continually, and approaches more and more to a fixed state, which is that which we consider. This final state is kept up of itself when it has once been formed. In order that the system of temperatures may be permanent, it is necessary that the quantity of heat which, during unit of time, crosses a section made at a distance x from the origin, should balance exactly all the heat which, during the same time, escapes through that part of the external surface of the prism which is situated to the right of the same section. The lamina whose thickness is dr, and whose external surface is Slda, allows the escape into the air, during unit of time, of a quantity of heat expressed by 8hlv. dx, h being the measure of the external conducibility of the prism. Hence taking the integral f8hlv.de from x = 0 to x = ∞, we shall find the quantity of heat which escapes from the whole surface of the bar during unit of time; and if we take the same integral from x = 0 to xx, we shall have the quantity of heat lost through the part of the surface included between the source of heat and the section made at the distance x. Denoting the first integral by C, whose value is constant, and the variable value of the second by f8hlv.dx; the difference C-f8hlv. dx will express the whole quantity of heat which escapes into the air across the part of the surface situate to the right of the section. On the other hand, the lamina of the solid, enclosed between two sections infinitely near at distances x and x+dx, must resemble an infinite solid, bounded by two parallel planes, subject to fixed temperatures v and v+ dv, since, by hypothesis, the temperature does not vary throughout the whole extent of the same section. The thickness of the solid is dx, and the area of the section is 4: hence the quantity of heat which flows uniformly, during unit of time, across a section of this solid, is, according to the dv preceding principles, – 4ľk k being the specific internal condx' ducibility: we must therefore have the equation 74. We should obtain the same result by considering the equilibrium of heat in a single lamina infinitely thin, enclosed between two sections at distances x and x+de. In fact, the quantity of heat which, during unit of time, crosses the first section situate at distance x, is – 47k dv dx' To find that which flows during the same time across the successive section situate at distance + dx, we must in the preceding expression change x into xdx, which gives - 4k. +d dv dv Ld.c If we subtract the second expression from the first we shall find how much heat is acquired by the lamina bounded by these, two sections during unit of time; and since the state of the lamina is permanent, it follows that all the heat acquired is dispersed into the air across the external surface 8ldx of the same lamina: now the last quantity of heat is 8hlvdx: we shall obtain therefore the same equation 75. In whatever manner this equation is formed, it is necessary to remark that the quantity of heat which passes into the lamina whose thickness is dx, has a finite value, and that dv its exact expression is 4k dx' The lamina being enclosed between two surfaces the first of which has a temperature v, and the second a lower temperature v', we see that the quantity of heat which it receives through the first surface depends on the difference v-v', and is proportional to it: but this remark is not sufficient to complete the calculation. The quantity in question is not a differential: it has a finite value, since it is equivalent to all the heat which escapes through that part of the external surface of the prism which is situate to the right of the section. To form an exact idea of it, we must compare the lamina whose thickness is dx, with a solid terminated by two parallel planes whose distance is e, and which are maintained at unequal temperatures a and b. The quantity of heat which passes into such a prism across the hottest surface, is in fact proportional to the difference a-b of the extreme temperatures, but it does not depend only on this difference: all other things being equal, it is less when the prism is thicker, and in general a -b it is proportional to This is why the quantity of heat which passes through the first surface into the lamina, whose e We lay stress on this remark because the neglect of it has been the first obstacle to the establishment of the theory. If we did not make a complete analysis of the elements of the problem, we should obtain an equation not homogeneous, and, a fortiori, we should not be able to form the equations which express the movement of heat in more complex cases. It was necessary also to introduce into the calculation the dimensions of the prism, in order that we might not regard, as general, consequences which observation had furnished in a particular case. Thus, it was discovered by experiment that a bar of iron, heated at one extremity, could not acquire, at a distance of six feet from the source, a temperature of one degree (octogesimal'); for to produce this effect, it would be necessary for the heat of the source to surpass considerably the point of fusion of iron; but this result depends on the thickness of the prism employed. If it had been greater, the heat would have been propagated to a greater distance, that is to say, the point of the bar which acquires a fixed temperature of one degree is 1 Reaumur's Scale of Temperature. [A. F.] much more remote from the source when the bar is thicker, all other conditions remaining the same. We can always raise by one degree the temperature of one end of a bar of iron, by heating the solid at the other end; we need only give the radius of the base a sufficient length: which is, we may say, evident, and of which besides a proof will be found in the solution of the problem (Art. 78). 76. The integral of the preceding equation is A and B being two arbitrary constants; now, if we suppose the distance x infinite, the value of the temperature v must be infinitely small; hence the term Be = does not exist in the in represents the permanent tegral: thus the equation v Ae state of the solid; the temperature at the origin is denoted by the constant A, since that is the value of v when x is zero. This law according to which the temperatures decrease is the same as that given by experiment; several physicists have observed the fixed temperatures at different points of a metal bar exposed at its extremity to the constant action of a source of heat, and they have ascertained that the distances. from the origin represent logarithms, and the temperatures the corresponding numbers. 77. The numerical value of the constant quotient of two consecutive temperatures being determined by observation, we easily deduce the value of the ratio h k; for, denoting by v1, v, the temperatures corresponding to the distances x,,,, we have 2h _ log v, - log "1⁄2 √. As for the separate values of h and k, they cannot be determined by experiments of this kind: we must observe also the varying motion of heat. 78. Suppose two bars of the same material and different dimensions to be submitted at their extremities to the same tem perature A; let, be the side of a section in the first bar, and l in the second, we shall have, to express the temperatures of these two solids, the equations v, in the first solid, denoting the temperature of a section made at distance x,, and v,, in the second solid, the temperature of a section made at distance x. When these two bars have arrived at a fixed state, the temperature of a section of the first, at a certain distance from the source, will not be equal to the temperature of a section of the second at the same distance from the focus; in order that the fixed temperatures may be equal, the distances must be different. If we wish to compare with each other the distances x, and x, from the origin up to the points which in the two bars attain the same temperature, we must equate the second members of these equations, and from them we conclude that "= Thus x 2 2 the distances in question are to each other as the square roots of the thicknesses. 79. If two metal bars of equal dimensions, but formed of different substances, are covered with the same coating, which gives them the same external conducibility', and if they are submitted at their extremities to the same temperature, heat will be propagated most easily and to the greatest distance from the origin in that which has the greatest conducibility. To compare with each other the distances x, and x, from the common origin up to the points which acquire the same fixed temperature, we must, after denoting the respective conducibilities of the two substances by k, and k, write the equation Thus the ratio of the two conducibilities is that of the squares of the distances from the common origin to the points which attain the same fixed temperature. 1 Ingenhousz (1789), Sur les métaux comme conducteurs de la chaleur. Journal de Physique, XXXIV., 68, 380. Gren's Journal der Physik, Bd. 1. [A. F.] |