We have, therefore, if we denote the mean temperature by z, an equation in which the coefficients of the exponentials are all positive. 302. Let us consider the case in which, all other conditions remaining the same, the value X of the radius of the sphere Taking up the construction described hX becomes infinitely great'. in Art. 285, we see that since the quantity becomes infinite, K the straight line drawn through the origin cutting the different branches of the curve coincides with the axis of x. We find then for the different values of e the quantities T, 2π, 3π, etc. K €12 Since the term in the value of z which contains e CD X3 becomes, as the time increases, very much greater than the following terms, the value of z after a certain time is expressed Kn without sensible error by the first term only. The index CD being equal to we see that the final cooling is very slow Кп2 in spheres of great diameter, and that the index of e which measures the velocity of cooling is inversely as the square of the diameter. 303. From the foregoing remarks we can form an exact idea of the variations to which the temperatures are subject during the cooling of a solid sphere. The initial values of the temperatures change successively as the heat is dissipated through the surface. If the temperatures of the different layers are at first equal, or if they diminish from the surface to the centre, they do not maintain their first ratios, and in all cases the system tends more and more towards a lasting state, which after no long delay is sensibly attained. In this final state the temperatures decrease 1 Riemann has shewn, Part. Diff. gleich. § 69, that in the case of a very large sphere, uniformly heated initially, the surface temperature varies ultimately as the square root of the time inversely. [A. F.] from the centre to the surface. If we represent the whole radius of the sphere by a certain arce less than a quarter of the circumference, and, after dividing this arc into equal parts, take for each point the quotient of the sine by the arc, this system of ratios will represent that which is of itself set up among the temperatures of layers of equal thickness. From the time when these ultimate ratios occur they continue to exist throughout the whole of the cooling. Each of the temperatures then diminishes as the ordinate of a logarithmic curve, the time being taken for abscissa. We can ascertain that this law is established by observing several successive values z, z′, z′′, z′′, etc., which denote the mean temperature for the times t, t+, t+20, t + 30, etc.; the series of these values converges always towards a geometrical Z z z" progression, and when the successive quotients Z' 2 2 etc. no longer change, we conclude that the relations in question are established between the temperatures. When the diameter of the sphere is small, these quotients become sensibly equal as soon as the body begins to cool. The duration of the cooling for a given interval, that is to say the time required for the mean tem 2 m perature 2 to be reduced to a definite part of itself increases as the diameter of the sphere is enlarged. 304. If two spheres of the same material and different dimensions have arrived at the final state in which whilst the temperatures are lowered their ratios are preserved, and if we wish to compare the durations of the same degree of cooling in both, that is to say, the time ℗ which the mean temperature of the first occupies in being reduced to, and the time ℗ in m consider three different cases. If the diameter of each sphere is small, the durations and are in the same ratio as the diameters. If the diameter of each sphere is very great, the durations and O' are in the ratio of the squares of the diameters; and if the diameters of the spheres are included between these two limits, the ratios of the times will be greater than that of the diameters, and less than that of their squares. SECT. II.] EQUATION OF CONDITION HAS ONLY REAL ROOTS. 289 The exact value of the ratio has been already determined'. The problem of the movement of heat in a sphere includes that of the terrestrial temperatures. In order to treat of this problem at greater length, we have made it the object of a separate chapter2. 305. The use which has been made above of the equation tan € € =λ is founded on a geometrical construction which is very well adapted to explain the nature of these equations. The construction indeed shows clearly that all the roots are real; at the same time it ascertains their limits, and indicates methods for determining the numerical value of each root. The analytical investigation of equations of this kind would give the same results. First, we might ascertain that the equation e-λ tane = 0, in which is a known number less than unity, has no imaginary root of the form m+n√-1. It is sufficient to substitute this quantity for e; and we see after the transformations that the first member cannot vanish when we give to m and n real values, unless n is nothing. It may be proved moreover that there can be no imaginary root of any form whatever in the equation In fact, 1st, the imaginary roots of the factor belong to the equation € - λ tan e = 0, since these roots are all of the form m+n-1; 2nd, the equation sin e necessarily all its roots real when λ is less than unity. To prove this proposition we must consider sine as the product of the infinite number of factors 1 12 It is : 0','X2 : e2X", as may be inferred from the exponent of the first term in the expression for z, Art. 301. [A. F.] 2 The chapter referred to is not in this work. It forms part of the Suite du mémorie sur la théorie du mouvement de la chaleur dans les corps solides. See note, page 10. The first memoir, entitled Théorie du mouvement de la chaleur dans les corps solides, is that which formed the basis of the Théorie analytique du mouvement de la chaleur published in 1822, but was considerably altered and enlarged in that work now translated. [A. F.] F. H. 19 · (1-5) (1 − −) (1-6) (1 - 4) &c., and consider cos e as derived from sine by differentiation. Suppose that instead of forming sine from the product of an infinite number of factors, we employ only the m first, and denote the product by 4, (e). To find the corresponding value of cose, we take Now, giving to the number m its successive values 1, 2, 3, 4, &c. from 1 to infinity, we ascertain by the ordinary principles of Algebra, the nature of the functions of e which correspond to these different values of m. We see that, whatever m the number of factors may be, the equations in e which proceed from them have the distinctive character of equations all of whose roots are real. Hence we conclude rigorously that the equation in which is less than unity, cannot have an imaginary root'. The same proposition could also be deduced by a different analysis which we shall employ in one of the following chapters. Moreover the solution we have given is not founded on the property which the equation possesses of having all its roots real. It would not therefore have been necessary to prove this proposition by the principles of algebraical analysis. It is sufficient for the accuracy of the solution that the integral can be made to coincide with any initial state whatever; for it follows rigorously that it must then also represent all the subsequent states. 1 The proof given by Riemann, Part. Diff. Gleich. § 67, is more simple. The method of proof is in part claimed by Poisson, Bulletin de la Société Philomatique, Paris, 1826, p. 147. [A. F.]. CHAPTER VI. OF THE MOVEMENT OF HEAT IN A SOLID CYLINDER. 306. THE movement of heat in a solid cylinder of infinite length, is represented by the equations which we have stated in Articles 118, 119, and 120. To integrate these equations we give to v the simple particular value expressed by the equation v=ue-mt; m being any number, and K CD u a function of x. We denote by k the coefficient which h enters the first equation, and by h the coefficient K which enters the second equation. Substituting the value assigned to v, we find the following condition Next we choose for u a function of x which satisfies this differential equation. It is easy to see that the function may be expressed by the following series in the sequel the differential equation from which this series |