which escapes from the same slice across the second section, and passes into the contiguous part of the solid, it is only necessary to change x into x + dx in the preceding expression, or, which is the same thing, to add to this expression its differential taken with respect to x; thus the slice receives through one of its faces dv a quantity of heat equal to -KSdt, and loses through the opposite face a quantity of heat expressed by It acquires therefore by reason of its position a quantity of heat equal to the difference of the two preceding quantities, that is d'v On the other hand, the same slice, whose external surface is ld and whose temperature differs infinitely little from v, allows a quantity of heat equivalent to hlv da dt to escape into the air during the instant dt; it follows from this that this infinitely small part of the solid retains in reality a quantity of heat d'v dx represented by KS dx dt - hlv dx dt which makes its temperature vary. The amount of this change must be examined. 105. The coefficient C expresses how much heat is required to raise unit of weight of the substance in question from temperature 0 up to temperature 1; consequently, multiplying the volume Sdx of the infinitely small slice by the density D, to obtain its weight, and by C the specific capacity for heat, we shall have CDS dx as the quantity of heat which would raise the volume of the slice from temperature 0 up to temperature 1. Hence the increase of temperature which results from the addition d'v of a quantity of heat equal to KS dx dt – hlv dx dt will be dx2 found by dividing the last quantity by CDS dx. Denoting therefore, according to custom, the increase of temperature which takes dv place during the instant dt by dt, we shall have the equation We shall explain in the sequel the use which may be made of this equation to determine the complete solution, and what the difficulty of the problem consists in; we limit ourselves here to a remark concerning the permanent state of the armlet. 106. Suppose that, the plane of the ring being horizontal, sources of heat, each of which exerts a constant action, are placed below different points m, n, p, q etc.; heat will be propagated in the solid, and that which is dissipated through the surface being incessantly replaced by that which emanates from the sources, the temperature of every section of the solid will approach more and more to a stationary value which varies from one section to another. In order to express by means of equation (b) the law of the latter temperatures, which would exist of themselves if they were once established, we must suppose that the quantity v does. not vary with respect to t; which annuls the term dv We thus dt 1 This equation is the same as the equation for the steady temperature of a finite bar heated at one end (Art. 76), except that I here denotes the perimeter of a section whose area is S. In the case of the finite bar we can determine two relations between the constants M and N: for, if V be the temperature at the source, where x =0, V=M+N; and if at the end of the bar remote from the source, where x L suppose, we make a section at a distance dx from that end, the flow dv dx' through this section is, in unit of time, - KS and this is equal to the waste of heat through the periphery and free end of the slice, hv (ldx+S) namely; hence ultimately, de vanishing, 107. Suppose a portion of the circumference of the ring, situated between two successive sources of heat, to be divided into equal parts, and denote by v1, v, vg, v1, &c., the temperatures at the points of division whose distances from the origin are X, X, X, X, &c.; the relation between v and a will be given by the preceding equation, after that the two constants have been determined by means of the two values of v corresponding to the sources of heat. Denoting by a the quantity e Denoting by a the quantity e√s, and 19 by the distance x-x, of two consecutive points of division, we shall have the equations: whence we derive the following relation" +"="+a^. 5 We should find a similar result for the three points whose temperatures are v,, v,, v,, and in general for any three consecutive points. It follows from this that if we observed the temperatures V1 V2 V3 V4 Vz &c. of several successive points, all situated between the same two sources m and n and separated by a constant interval λ, we should perceive that any three consecutive temperatures are always such that the sum of the two extremes divided by the mean gives a constant quotient a1+a^. 108. If, in the space included between the next two sources of heat n and p, the temperatures of other different points separated by the same interval A were observed, it would still be found that for any three consecutive points, the sum of the two extreme temperatures, divided by the mean, gives the same quotient a+a. The value of this quotient depends neither on the position nor on the intensity of the sources of heat. 109. Let q be this constant value, we have the equation we see by this that when the circumference is divided into equal parts, the temperatures at the points of division, included between two consecutive sources of heat, are represented by the terms of a recurring series whose scale of relation is composed of two terms 9 and -1. Experiments have fully confirmed this result. We have exposed a metallic ring to the permanent and simultaneous action of different sources of heat, and we have observed the stationary temperatures of several points separated by constant intervals; we always found that the temperatures of any three consecutive points, not separated by a source of heat, were connected by the relation in question. Even if the sources of heat be multiplied, and in whatever manner they be disposed, no change can be effected in the numerical value of the quotient "+"; it depends ra V2 only on the dimensions or on the nature of the ring, and not on the manner in which that solid is heated. 110. When we have found, by observation, the value of the v constant quotient q or ,, the value of a may be derived equation a+aq. One of the roots from it by means of the is 2, and other root is a. This quantity being determined, h we may derive from it the value of the ratio which is K' S (loga). Denoting a by w, we shall have w2-qw+1=0. Thus the ratio of the two conducibilities is found by multiplying S ī by the square of the hyperbolic logarithm of one of the roots of the equation w3 — qw + 1 = 0, and dividing the product by x2. SECTION II. Equation of the varied movement of heat in a solid sphere. 111. A solid homogeneous mass, of the form of a sphere, having been immersed for an infinite time in a medium maintained at a permanent temperature 1, is then exposed to air which is kept at temperature 0, and displaced with constant velocity it is required to determine the successive states of the body during the whole time of the cooling. Denote by the distance of any point whatever from the centre of the sphere, and by v the temperature of the same point, after a time t has elapsed; and suppose, to make the problem more general, that the initial temperature, common to all points situated at the distance x from the centre, is different for different values of x; which is what would have been the case if the immersion had not lasted for an infinite time. Points of the solid, equally distant from the centre, will not cease to have a common temperature; v is thus a function of x and t. When we suppose t=0, it is essential that the value of this function should agree with the initial state which is given, and which is entirely arbitrary. 112. We shall consider the instantaneous movement of heat in an infinitely thin shell, bounded by two spherical surfaces whose radii are x and x+dx: the quantity of heat which, during an infinitely small instant dt, crosses the lesser surface whose radius is a, and so passes from that part of the solid which is nearest to the centre into the spherical shell, is equal to the product of four factors which are the conducibility K, the duration dt, the extent 4πx of surface, and the ratio taken with the negative sign; dv dx dv dx' it is expressed by - 4Kπж2 dt. To determine the quantity of heat which flows during the same instant through the second surface of the same shell, and passes from this shell into the part of the solid which envelops it, x must be changed into x + dx, in the preceding expression: that is to say, to the term 4Kπ2 dt must be added the differen dv tial of this term taken with respect to x. We thus find as the expression of the quantity of heat which leaves the spherical shell across its second surface; and if we subtract this quantity from that which enters through the first surface, we shall have dv 4Kπd (22 du) dt. This difference is evidently the quantity of |