The expenditure of the source of heat necessary to maintain the surface of the body A at the temperature b is HS (b-a), when this surface sends its rays to a fixed surface maintained at the temperature a. The expenditure is HS (b-m,) when we place between the surface of the body A, and the fixed surface maintained at temperature a, a number j of isolated laminæ; thus the quantity of heat which the source must furnish is very much less in the second hypotheses than in the first, and the ratio of the two quantities is If we suppose the thickness e of the 1 j (1 + K 1 laminæ to be infinitely small, the ratio is The expenditure ງ of the source is then inversely as the number of lamina which cover the surface of the solid. 91. The examination of these results and of those which we obtained when the intervals between successive enclosures were occupied by atmospheric air explain clearly why the separation of surfaces and the intervention of air assist very much in retaining heat. Analysis furnishes in addition analogous consequences when we suppose the source to be external, and that the heat which emanates from it crosses successively different diathermanous envelopes and the air which they enclose. This is what has happened when experimenters have exposed to the rays of the sun thermometers covered by several sheets of glass within which different layers of air have been enclosed. For similar reasons the temperature of the higher regions of the atmosphere is very much less than at the surface of the earth. In general the theorems concerning the heating of air in closed spaces extend to a great variety of problems. It would be useful to revert to them when we wish to foresee and regulate temperature with precision, as in the case of green-houses, dryinghouses, sheep-folds, work-shops, or in many civil establishments, such as hospitals, barracks, places of assembly. In these different applications we must attend to accessory circumstances which modify the results of analysis, such as the unequal thickness of different parts of the enclosure, the introduction of air, &c.; but these details would draw us away from our chief object, which is the exact demonstration of general principles. For the rest, we have considered only, in what has just been said, the permanent state of temperature in closed spaces. We can in addition express analytically the variable state which precedes, or that which begins to take place when the source of heat is withdrawn, and we can also ascertain in this way, how the specific properties of the bodies which we employ, or their dimensions affect the progress and duration of the heating; but these researches require a different analysis, the principles of which will be explained in the following chapters. SECTION VII. On the uniform movement of heat in three dimensions. 92. Up to this time we have considered the uniform movement of heat in one dimension only, but it is easy to apply the same principles to the case in which heat is propagated uniformly in three directions at right angles. Suppose the different points of a solid enclosed by six planes at right angles to have unequal actual temperatures represented by the linear equation v=A+ ax + by + cz, x, y, z, being the rectangular co-ordinates of a molecule whose temperature is v. Suppose further that any external causes whatever acting on the six faces of the prism maintain every one of the molecules situated on the surface, at its actual temperature expressed by the general equation v = A + ax + by + cz (a), we shall prove that the same causes which, by hypothesis, keep the outer layers of the solid in their initial state, are sufficient to preserve also the actual temperatures of every one of the inner molecules, so that their temperatures do not cease to be represented by the linear equation. The examination of this question is an element of the general theory, it will serve to determine the laws of the varied movement of heat in the interior of a solid of any form whatever, for every one of the prismatic molecules of which the body is composed is during an infinitely small time in a state similar to that which the linear equation (a) expresses. We may then, by following the ordinary principles of the differential calculus, easily deduce from the notion of uniform movement the general equations of varied movement. 93. In order to prove that when the extreme layers of the solid preserve their temperatures no change can happen in the interior of the mass, it is sufficient to compare with each other the quantities of heat which, during the same instant, cross two parallel planes. Let b be the perpendicular distance of these two planes which we first suppose parallel to the horizontal plane of x and y. Let m and m' be two infinitely near molecules, one of which is above the first horizontal plane and the other below it: let x, y, z be the co-ordinates of the first molecule, and x', y', z' those of the second. In like manner let M and M' denote two infinitely near molecules, separated by the second horizontal plane and situated, relatively to that plane, in the same manner as m and m' are relatively to the first plane; that is to say, the co-ordinates of M are x, y, z +b, and those of M' are x, y, z' +b. It is evident that the distance mm' of the two molecules m and m' is equal to the distance MM' of the two molecules M and M'; further, let v be the temperature of m, and v' that of m', also let V and V' be the temperatures of M and M', it is easy to see that the two differences v-v' and V- V' are equal; in fact, substituting first the co-ordinates of m and m' in the general equation and then substituting the co-ordinates of M and M', we find also V – V' = a (x − x') + b ( y − y') + c (2-2). Now the quantity of heat which m sends to m' depends on the distance mm', which separates these molecules, and it is proportional to the difference v-v′ of their temperatures. This quantity of heat transferred may be represented by q (v-v') dt; the value of the coefficient q depends in some manner on the distance mm', and on the nature of the substance of which the solid is formed, dt is the duration of the instant. The quantity of heat transferred from M to M', or the action of M on M' is expressed likewise by q (V-V') dt, and the coefficient q is the same as in the expression q (v-v') dt, since the distance MM' is equal to mm' and the two actions are effected in the same solid : furthermore V-V' is equal to v-v', hence the two actions are equal. If we choose two other points n and n', very near to each other, which transfer heat across the first horizontal plane, we shall find in the same manner that their action is equal to that of two homologous points N and N' which communicate heat across the second horizontal plane. We conclude then that the whole quantity of heat which crosses the first plane is equal to that which crosses the second plane during the same instant. We should derive the same result from the comparison of two planes parallel to the plane of x and z, or from the comparison of two other planes parallel to the plane of y and z. Hence any part whatever of the solid enclosed between six planes at right angles, receives through each of its faces as much heat as it loses through the opposite face; hence no portion of the solid can change temperature. 94. From this we see that, across one of the planes in question, a quantity of heat flows which is the same at all instants, and which is also the same for all other parallel sections. In order to determine the value of this constant flow we shall compare it with the quantity of heat which flows uniformly in the most simple case, which has been already discussed. The case is that of an infinite solid enclosed between two infinite planes and maintained in a constant state. We have seen that the temperatures of the different points of the mass are in this case represented by the equation v=A+cz; we proceed to prove. that the uniform flow of heat propagated in the vertical direction in the infinite solid is equal to that which flows in the same direction across the prism enclosed by six planes at right angles. This equality necessarily exists if the coefficient c in the equation v = A +cz, belonging to the first solid, is the same as the coefficient c in the more general equation = A + ax+by+cz which represents the state of the prism. In fact, denoting by Ha plane in this prism perpendicular to z, and by m and u two molecules very near to each other, the first of which m is below the plane H, and the second above this plane, let v be the temperature of m whose co-ordinates are x, y, z, and w the temperature of μ whose co-ordinates are x +α, y + ß, 2+ y. Take a third molecule whose co-ordinates are x-a, y - ẞ, z+y, and whose temperature may be denoted by w'. We see that and v μ are on the same horizontal plane, and that the vertical drawn from the middle point of the line up, which joins these two points, passes through the point m, so that the distances mu and mu are equal. The action of m on u, or the quantity of heat which the first of these molecules sends to the other across the plane H, depends on the difference vw of their temperatures. The action of m on μ depends in the same manner on the difference v-w' of the temperatures of these molecules, since the distance of m from μ is the same as that of m from μ'. Thus, expressing by q (v — w) the action of m on μ during the unit of time, we shall have q (v - w') to express the action of m on μ, q being a common unknown factor, depending on the distance mμ and on the nature of the solid. Hence the sum of the two actions exerted during unit of time is q (v-w+v-w'). If instead of x, y, and z, in the general equation v = A + ax + by +cz, we substitute the co-ordinates of m and then those of μ and μ', we shall find |