1st, The partial differential equation is satisfied; 2nd, the quantity of heat which escapes at the surface accords at the same time with the mutual action of the last layers and with the action of the air dv on the surface; that is to say, the equation +hx=0, which each part of the value of v satisfies when x = X, holds also when we take for v the sum of all these parts; 3rd, the given solution agrees with the initial state when we suppose the time nothing. 292. The roots n,, n,, n,, &c. of the equation are very unequal; whence we conclude that if the value of the time is considerable, each term of the value of v is very small, relatively to that which precedes it. As the time of cooling increases, the latter parts of the value of v cease to have any sensible influence; and those partial and elementary states, which at first compose the general movement, in order that the initial state may be represented by them, disappear almost entirely, one only excepted. In the ultimate state the temperatures of the different layers decrease from the centre to the surface in the same manner as in a circle the ratios of the sine to the arc decrease as the arc increases. This law governs naturally the distribution of heat in a solid sphere. When it begins to exist, it exists through the whole duration of the cooling. Whatever the function F(x) may be which represents the initial state, the law in question tends continually to be established; and when the cooling has lasted some time, we may without sensible error suppose it to exist. 293. We shall apply the general solution to the case in which the sphere, having been for a long time immersed in a fluid, has acquired at all its points the same temperature. In this case the function F(x) is 1, and the determination of the coefficients is reduced to integrating a sin nx dx, from x=0 to xX: the integral is = sin nX - nX cos nX no Hence the value of each coefficient is expressed thus: the order of the coefficient is determined by that of the root n, the equation which gives the values of n being It is easy now to form the general value which is given by the equation Vx = 2Xh nx e-knt sin nx + e-knit sin Denoting by e,, €, e,, &c. the roots of the equation + &c. and supposing them arranged in order beginning with the least; replacing nX, n ̧X, n ̧X, &c. by e1, €, €,, &c., and writing instead of k and h their values K CD h and we have for the expression of K' the variations of temperature during the cooling of a solid sphere, which was once uniformly heated, the equation Note. The problem of the sphere has been very completely discussed by Riemann, Partielle Differentialgleichungen, §§ 61-69. [A. F.] SECTION II. Different remarks on this solution. 294. We will now explain some of the results which may be derived from the foregoing solution. If we suppose the coefficient h, which measures the facility with which heat passes into the air, to have a very small value, or that the radius X of the sphere is very small, the least value of e becomes very small; so that the t CDX substitutions in the general equation we have ve + &c. We may remark that the succeeding terms decrease very rapidly in comparison with the first, since the second root n, is very much greater than 0; so that if either of the quantities h or X has a small value, we may take, as the expression of the variations Sht of temperature, the equation ve Thus the different spherical envelopes of which the solid is composed retain a common temperature during the whole of the cooling. The temperature diminishes as the ordinate of a logarithmic curve, the time being taken for abscissa; the initial temperature 1 is re Sht duced after the time t to e CDX In order that the initial temperature may be reduced to the fraction the value of t 1 m X must be 3h CD log m. Thus in spheres of the same material but of different diameters, the times occupied in losing half or the same defined part of their actual heat, when the exterior conducibility is very small, are proportional to their diameters. The same is the case with solid spheres whose radius is very small; and we should also find the same result on attributing to the interior conducibility K a very great value. The statement holds hX generally when the quantity is very small. We may regard h K K the quantity as very small when the body which is being cooled is formed of a liquid continually agitated, and enclosed in a spherical vessel of small thickness. The hypothesis is in some measure the same as that of perfect conducibility; the temperature decreases then according to the law expressed by the equation ve CDX 295. By the preceding remarks we see that in a solid sphere which has been cooling for a long time, the temperature decreases from the centre to the surface as the quotient of the sine by the arc decreases from the origin where it is 1 to the end of a given arc e, the radius of each layer being represented by the variable length of that arc. If the sphere has a small diameter, or if its interior conducibility is very much greater than the exterior conducibility, the temperatures of the successive layers differ very little from each other, since the whole arc e which represents the radius X of the sphere is of small length. The variation of the temperature v common to all its points is then given by the equation v=e Cox Thus, on comparing the respective times which two small spheres occupy in losing half or any aliquot part of their actual heat, we find those times to be proportional to the diameters. Sht CDX CDX belongs 296. The result expressed by the equation v = e only to masses of similar form and small dimension. It has been known for a long time by physicists, and it offers itself as it were spontaneously. In fact, if any body is sufficiently small for the temperatures at its different points to be regarded as equal, it is easy to ascertain the law of cooling. Let 1 be the initial temperature common to all points; it is evident that the quantity of heat which flows during the instant dt into the medium supposed to be maintained at temperature 0 is hSvdt, denoting by S the external surface of the body. On the other hand, if C is the heat required to raise unit of weight from the temperature 0 to the temperature 1, we shall have CDV for the expression of the quantity of heat which the volume V of the body whose density is D would take from temperature 0 to h Svdt temperature 1. Hence is the quantity by which the CDV temperature v is diminished when the body loses a quantity of heat equal to hSvdt. We ought therefore to have the equation If the form of the body is a sphere whose radius is X, we shall -3ht have the equation v = ex 297. Assuming that we observe during the cooling of the body in question two temperatures v1 and v, corresponding to the times t, and t,, we have ᏂᏚ log v, - log v We can then easily ascertain by experiment the exponent hS CDV If the same observation be made on different bodies, and if we know in advance the ratio of their specific heats C and C', we can find that of their exterior conducibilities h and h'. Reciprocally, if we have reason to regard as equal the values h and h' of the exterior conducibilities of two different bodies, we can ascertain the ratio of their specific heats. We see by this that, by observing the times of cooling for different liquids and other substances enclosed successively in the same vessel whose thickness is small, we can determine exactly the specific heats of those substances. We may further remark that the coefficient K which measures the interior conducibility does not enter into the equation |