heat which accumulates in the intervening shell, and whose effect is to vary its temperature. 113. The coefficient C denotes the quantity of heat which is necessary to raise, from temperature 0 to temperature 1, a definite unit of weight; D is the weight of unit of volume, 4πæ3dx is the volume of the intervening layer, differing from it only by a quantity which may be omitted: hence 4πCDx dx is the quantity of heat necessary to raise the intervening shell from temperature O to temperature 1. Hence it is requisite to divide the quantity of heat which accumulates in this shell by 4πCDx'dx, and we shall then find the increase of its temperature v during the time dt. We thus obtain the equation 114. The preceding equation represents the law of the movement of heat in the interior of the solid, but the temperatures of points in the surface are subject also to a special condition which must be expressed. This condition relative to the state of the surface may vary according to the nature of the problems discussed: we may suppose for example, that, after having heated the sphere, and raised all its molecules to the temperature of boiling water, the cooling is effected by giving to all points in the surface the temperature 0, and by retaining them at this temperature by any external cause whatever. In this case we may imagine the sphere, whose variable state it is desired to determine, to be covered by a very thin envelope on which the cooling agency exerts its action. It may be supposed, 1o, that this infinitely thin envelope adheres to the solid, that it is of the same substance as the solid and that it forms a part of it, like the other portions of the mass; 2°, that all the molecules of the envelope are subjected to temperature 0 by a cause always in action which prevents the temperature from ever being above or below zero. To express this condition theoretically, the function v, which contains x and t, must be made to become nul, when we give to x its complete value X equal to the radius of the sphere, whatever else the value of t may be. We should then have, on this hypothesis, if we denote by (x, t) the function of x and t, which expresses the value of v, the two equations Further, it is necessary that the initial state should be represented by the same function (x, t): we shall therefore have as a second condition p (x, 0) = 1. Thus the variable state of a solid sphere on the hypothesis which we have first described will be represented by a function v, which must satisfy the three preceding equations. The first is general, and belongs at every instant to all points of the mass; the second affects only the molecules at the surface, and the third belongs only to the initial state. 115. If the solid is being cooled in air, the second equation is different; it must then be imagined that the very thin envelope is maintained by some external cause, in a state such as to produce the escape from the sphere, at every instant, of a quantity of heat equal to that which the presence of the medium can carry away from it. Now the quantity of heat which, during an infinitely small instant dt, flows within the interior of the solid across the spheri cal surface situate at distance x, is equal to 4Kπж2 4Κπα dv dx dt; and this general expression is applicable to all values of x. Thus, by supposing x = X we shall ascertain the quantity of heat which in the variable state of the sphere would pass across the very thin envelope which bounds it; on the other hand, the external surface of the solid having a variable temperature, which we shall denote by V, would permit the escape into the air of a quantity of heat proportional to that temperature, and to the extent of the surface, which is 4πX3. The value of this quantity is 4hπ X' Vdt. To express, as is supposed, that the action of the envelope supplies the place, at every instant, of that which would result from the presence of the medium, it is sufficient to equate the quantity 4hπ XVdt to the value which the expression - 4KπX2 dv dt the equation dv da receives when we give to x its complete value X; hence we obtain h dv = V1 which must hold when in the functions and v we put instead of x its value X, which we shall denote h K a constant ratio same point. Thus we shall suppose that the external cause of the cooling determines always the state of the very thin envelope, to the value of v, which corresponds to the in such a manner that the value of which results from this dv state, is proportional to the value of v, corresponding to x = X, and that the constant ratio of these two quantities is h K condition being fulfilled by means of some cause always present, dv which prevents the extreme value of from being anything else dx the action of the envelope will take the place of that It is not necessary to suppose the envelope to be extremely thin, and it will be seen in the sequel that it may have an indefinite thickness. Here the thickness is considered to be indefinitely small, so as to fix the attention on the state of the surface only of the solid. 117. Hence it follows that the three equations which are required to determine the function (x, t) or v are the following, dV dv K (d'v + 2 du), Kdv +hV = 0, $ (x, 0) = 1. = dt CD dx The first applies to all possible values of x and t; the second is satisfied when xX, whatever be the value of t; and the third is satisfied when t = 0, whatever be the value of x. It might be supposed that in the initial state all the spherical layers have not the same temperature: which is what would necessarily happen, if the immersion were imagined not to have lasted for an indefinite time. In this case, which is more general than the foregoing, the given function, which expresses the initial temperature of the molecules situated at distance x from the centre of the sphere, will be represented by F(x); the third equation will then be replaced by the following, 4 (x, 0) = F (x). Nothing more remains than a purely analytical problem, whose solution will be given in one of the following chapters. It consists in finding the value of v, by means of the general condition, and the two special conditions to which it is subject. SECTION III. Equations of the varied movement of heat in a solid cylinder. 118. A solid cylinder of infinite length, whose side is perpendicular to its circular base, having been wholly immersed in a liquid whose temperature is uniform, has been gradually heated, in such a manner that all points equally distant from the axis have acquired the same temperature; it is then exposed to a current of colder air; it is required to determine the temperatures of the different layers, after a given time. x denotes the radius of a cylindrical surface, all of whose points are equally distant from the axis; X is the radius of the cylinder; v is the temperature which points of the solid, situated at distance x from the axis, must have after the lapse of a time denoted by t, since the beginning of the cooling. Thus v is a function of x and t, and if in it t be made equal to 0, the function of a which arises from this must necessarily satisfy the initial state, which is arbitrary. Ꮖ 119. Consider the movement of heat in an infinitely thin portion of the cylinder, included between the surface whose radius is x, and that whose radius is x + dx. The quantity of heat which this portion receives during the instant dt, from the part of the solid which it envelops, that is to say, the quantity which during the same time crosses the cylindrical surface whose radius is x, and whose length is supposed to be equal to unity, is expressed by To find the quantity of heat which, crossing the second surface whose radius is x+dx, passes from the infinitely thin shell into the part of the solid which envelops it, we must, in the foregoing expression, change x into x+dx, or, which is the same thing, add to the term Hence the differential of this term, taken with respect to x. the difference of the heat received and the heat lost, or the quantity of heat which accumulating in the infinitely thin shell determines the changes of temperature, is the same differential taken with the opposite sign, or on the other hand, the volume of this intervening shell is 2πxdx, and 2CDπxdx expresses the quantity of heat required to raise it from the temperature 0 to the temperature 1, C being the specific heat, and D the density. Hence the quotient is the increment which the temperature receives during the instant dt. Whence we obtain the equation 120. The quantity of heat which, during the instant dt, crosses the cylindrical surface whose radius is a, being expressed in general by 2Kπж dv da dt, we shall find that quantity which escapes during the same time from the surface of the solid, by making = X in the foregoing value; on the other hand, the |