transmit heat, and wish to pass to the case of continuous bodies, we must attribute to the coefficient k, which measures the velocity of transmission, a value proportional to the number of infinitely small masses which compose the given body. 3rd. If in the last equation which we obtained to express the value of v or (x, t), we suppose t=0, the equation necessarily represents the initial state, we have therefore in this way the equation (p), which we obtained formerly in Art. 233, namely, 1 #ƒ (x) = = [ƒ (x) dx 2x + sin æ [ƒ(x) sin ædæ + sin 2æf f(x) sin 2x dæ +&c. + cos x [f(x)cos x dx+cos 2x [f(x) cos 2x dx + &c. Thus the theorem which gives, between assigned limits, the development of an arbitrary function in a series of sines or cosines of multiple arcs is deduced from elementary rules of analysis. Here we find the origin of the process which we employed to make all the coefficients except one disappear by successive integrations from the equation $ (x) = a + a1 sin x + a 2 2 These integrations correspond to the elimination of the different unknowns in equations (m), Arts. 267 and 271, and we see clearly by the comparison of the two methods, that equation (B), Art. 279, holds for all values of x included between 0 and 2π, without its being established so as to apply to values of x which exceed those limits. 279. The function (x, t) which satisfies the conditions of the problem, and whose value is determined by equation (E), Art. 277, may be expressed as follows: 2=6(x, t) = fdaƒ(a) + (2 sinxfdaƒ(a) sina + 2 cosa fdaf (a) cos a e-kt + [2sin 2æ fdaf (a) sin 2x+2cos 2x dzƒ (a) cos 22)e-u + (2 sin 3x fdaf (2) sin 32+2cos 3r fdaf (a) cos 32}eTM+ &c. 3a}e32kt or 2π$ (.x,t) = [daƒ(2) {1 + (2 sin æ sin a+ 2 cos x cos x) e ̃kt The sign + (2 sin 2x sin 2x + 2 cos 2x cos 2a) e-22kt + (2 sin 3x sin 3x + 2 cos 3x cos 3) e-3k+ &c.} = [dzƒf(a) [1 + 2Σ cosi (a− x) e ̄ökt]. affects the number i, and indicates that the sum We can also include the must be taken from i=1 to i = ∞. first term under the sign Σ, and we have We must then give to i all integral values from - ∞ to +∞; which is indicated by writing the limits and +∞ next to the sign Σ, one of these values of i being 0. This is the most concise expression of the solution. To develope the second member of the equation, we suppose i=0, and then i=1, 2, 3, &c., and double each result except the first, which corresponds to i=0. When t is nothing, the function (x, t) necessarily represents the initial state in which the temperatures are equal to ƒ (x), we have therefore the identical equation, generally whatever be the form of the function f(x) in the interval from x = = 0 to x = 27; the same is the case with that which is expressed by the equations which give the development of F (x), Art. 235; and we shall see in the sequel that we can prove directly the truth of equation (B) independently of the foregoing considerations. 280. It is easy to see that the problem admits of no solution different from that given by equation (E), Art. 277. The function (x, t) in fact completely satisfied the conditions of the problem, dv d'v and from the nature of the differential equation =k dt dx , no other function can enjoy the same property. To convince ourselves of this we must consider that when the first state of the dv, solid is represented by a given equation v1 =ƒ (x), the fluxion dt is known, since it is equivalent to f(x). Thus denoting by dv dt k ... dx2 v2 or v1 + k dt, the temperature at the commencement of the second instant, we can deduce the value of v, from the initial state and from the differential equation. We could ascertain in the same manner the values v1, v1, v of the temperature at any point whatever of the solid at the beginning of each instant. Now the function (x, t) satisfies the initial state, since we have ¤ (x, 0) = ƒ (x). Further, it satisfies also the differential equation; consequently if it were differentiated, it would give the same dv, dr, dv, dv2 values for &c., as would result from successive dt' dt' dt applications of the differential equation (a). Hence, if in the function (x, t) we give to t successively the values 0, w, 2w, 3w, &c., o denoting an element of time, we shall find the same values v1, v2, v3, &c. as we could have derived from the initial dv dv dt da state by continued application of the equation k = Hence every function (x, t) which satisfies the differential equation and the initial state necessarily coincides with the function (x, t): for such functions each give the same function of x, when in them we suppose t successively equal to 0, w, 2w, 3w ... iw, &c. We see by this that there can be only one solution of the problem, and that if we discover in any manner a function (x, t) which satisfies the differential equation and the initial state, we are certain that it is the same as the former function given by equation (E). 281. The same remark applies to all investigations whose object is the varied movement of heat; it follows evidently from the very form of the general equation. dv = For the same reason the integral of the equation k d'v dt dx can contain only one arbitrary function of x. In fact, when a value of v as a function of x is assigned for a certain value of the time t, it is evident that all the other values of v which correspond to any time whatever are determinate. We may therefore select arbitrarily the function of x, which corresponds to a certain state, and the general function of the two variables x and t then becomes determined. The same is not the case d2v d2v dx2+ dy = with the equation preceding chapter, and which belongs to the constant movement of heat; its integral contains two arbitrary functions of x and y: but we may reduce this investigation to that of the varied movement, by regarding the final and permanent state as derived from the states which precede it, and consequently from the initial state, which is given. 0, which was employed in the The integral which we have given = contains one arbitrary function f(x), and has the same extent as the general integral, which also contains only one arbitrary function of ; or rather, it is this integral itself arranged in a form suitable to the problem. In fact, the equation v, f(x) representing the initial state, and v = (x, t) representing the variable state which succeeds it, we see from the very form of the heated solid that the value of v does not change when x+12π is written instead of x, i being any positive integer. The function satisfies this condition; it represents also the initial state when we suppose t = 0, since we then have an equation which was proved above, Arts. 235 and 279, and is also easily verified. Lastly, the same function satisfies the differ ential equation dv dt = k dx2 Whatever be the value of t, the temperature v is given by a very convergent series, and the different terms represent all the partial movements which combine to form the total movement. As the time increases, the partial states of higher orders alter rapidly, but their influence becomes inappreciable; so that the number of values which ought to be given to the exponent i diminishes continually. After a certain time the system of temperatures is represented sensibly by the terms which are found on giving to i the values 0, 1 and 2, or only 0 and ± 1, or lastly, by the first of those terms, namely, 1 2π - Jda f daƒ (a) ; there is therefore a manifest relation between the form of the solution and the progress of the physical phenomenon which has been submitted to analysis. 282. To arrive at the solution we considered first the simple values of the function v which satisfy the differential equation: we then formed a value which agrees with the initial state, and has consequently all the generality which belongs to the problem. We might follow a different course, and derive the same solution from another expression of the integral; when once the solution is known, the results are easily transformed. If we suppose the diameter of the mean section of the ring to increase infinitely, the function (x, t), as we shall see in the sequel, receives a different form, and coincides with an integral which contains a single arbitrary function under the sign of the definite integral. The latter integral might also be applied to the actual problem; but, if we were limited to this application, we should have but a very imperfect knowledge of the phenomenon; for the values of the temperatures would not be expressed by convergent series, and we could not discriminate between the states which succeed each other as the time increases. The periodic form which the problem supposes must therefore be attributed to the function which represents the initial state; but on modifying that integral in this manner, we should obtain no other result than $ (x, t) = 11 [da ƒ (2) Σc-TMk cos i (a — x). From the last equation we pass easily to the integral in question, as was proved in the memoir which preceded this work. It is not less easy to obtain the equation from the integral itself. These transformations make the agreement of the analytical results more clearly evident; but they add nothing to the theory, |