considerable, the value of a, is represented without sensible error by the equation,

Denoting by a and b the coefficients of sin (j- 1)

2π

n

2π

cos (j-1)

by w, we have

n

and of

The quantities a and b are constant, that is to say, independent of the time and of the letter j which indicates the order of the mass whose variable temperature is a. These quantities are the same for all the masses. The difference of the variable temperature a; from the final temperature - Ea, decreases therefore for

n

each of the masses, in proportion to the successive powers of the fraction w. Each of the bodies tends more and more to acquire

1

the final temperature Ea, and the difference between that

n

final limit and the variable temperature of the same body ends always by decreasing according to the successive powers of a fraction. This fraction is the same, whatever be the body whose changes of temperature are considered; the coefficient of wor

2π

(a sin u; + b cos u;), denoting by u; the arc (j − 1) may be put

α

=

n

under the form A sin (u; + B), taking A and B so as to have A cos B, and b= = A sin B. If we wish to determine the coefficient of wt with regard to the successive bodies whose temperature is oj, aj, aj, &c., we must add to u, the arc

276. We see, by these equations, that the later differences between the actual temperatures and the final temperatures are represented by the preceding equations, preserving only the first term of the second member of each equation. These later differences vary then according to the following law: if we consider only one body, the variable difference in question, that is to say, the excess of the actual temperature of the body over the final and common temperature, diminishes according to the successive powers of a fraction, as the time increases by equal parts; and, if we compare at the same instant the temperatures of all the bodies, the difference in question varies proportionally to the successive sines of the circumference divided into equal parts. The temperature of the same body, taken at different successive equal instants, is represented by the ordinates of a logarithmic curve, whose axis is divided into equal parts, and the temperature of each of these bodies, taken at the same instant for all, is represented by the ordinates of a circle whose circumference is divided into equal parts. It is easy to see, as we have remarked before, that if the initial temperatures are such, that the differences of these temperatures from the mean or final temperature are proportional to the successive sines of multiple arcs, these differences will all diminish at the same time without ceasing to be proportional to the same sines. This law, which governs also the initial temperatures, will not be disturbed by the reciprocal action of the bodies, and will be maintained until they have all acquired a common temperature. The difference will diminish for each body according to the successive powers of the same fraction. Such is the simplest law to which the communication of heat between a succession of equal masses can be submitted. When this law has once been established between the initial temperatures, it is maintained of itself; and when it does not govern the initial temperatures, that is to say, when the differences of these temperatures from the mean temperature are not proportional to successive sines of multiple arcs, the law in question tends always to be set

up, and the system of variable temperatures ends soon by coinciding sensibly with that which depends on the ordinates of a circle and those of a logarithmic curve.

Since the later differences between the excess of the temperature of a body over the mean temperature are proportional to the sine of the arc at the end of which the body is placed, it follows that if we regard two bodies situated at the ends of the same diameter, the temperature of the first will surpass the mean and constant temperature as much as that constant temperature surpasses the temperature of the second body. For this reason, if we take at each instant the sum of the temperatures of two masses whose situation is opposite, we find a constant sum, and this sum has the same value for any two masses situated at the ends of the same diameter.

277. The formula which represent the variable temperatures of separate masses are easily applied to the propagation of heat in continuous bodies. To give a remarkable example, we will determine the movement of heat in a ring, by means of the general equation which has been already set down.

Let it be supposed that n the number of masses increases successively, and that at the same time the length of each mass decreases in the same ratio, so that the length of the system has a constant value equal to 2. Thus if n the number of masses be successively 2, 4, 8, 16, to infinity, each of the masses will

π π П

2' 4' 8

be T, facility with which heat is transmitted increases in the same ratio as the number of masses m; thus the quantity which k represents when there are only two masses becomes double when there are four, quadruple when there are eight, and so on. Denoting this quantity by g, we see that the number k must be successively replaced by g, 2g, 4g, &c. If we pass now to the hypothesis of a continuous body, we must write instead of m, the value of each infinitely small mass, the element da; instead of n,

&c. It must also be assumed that the

the number of masses, we must write

;

instead of k write

dx

As to the initial temperatures a,, a, a,...a, they depend on the value of the arc x, and regarding these temperatures as the successive states of the same variable, the general value a, represents an arbitrary function of x. The index i must then be

x

dx

replaced by With respect to the quantities a, a, a, ..., these are variable temperatures depending on two quantities x and t. Denoting the variable by v, we have v = (x, t). The index j, which marks the place occupied by one of the bodies, should be replaced by. Thus, to apply the previous analysis to the case of an infinite number of layers, forming a continuous body in the form of a ring, we must substitute for the quantities n, m, k, a, i, a;, j, their corresponding quantities, namely, x Let these substitutions be

2π

dx'

x

dx, Tx, f(x), Zx, 4(x, t), dx

dx'

dx'

of versin da, and i and j instead of i-1

dx2 be written instead

x=0 to x = 2; the quantity sin (j − 1)

2π

becomes sinjdx or

n

2π

2

sin

n

is

x; the value of cos (j-1) is cos a; that of Ea, sin (i −1) dx

2π

n

[ƒ (a) sin ædæ, the integral being taken from a = 0 to 2=2π ;

and the value of Ea cos (i-1) is = f (x)

integral being taken between the same limits. Thus we obtain the equation

+ = (sin x [ƒ (4) sin æda + cos a fƒ (4) cos xde )e-st

1

+= (sin 2r [f(x) sin 2x dx + cos 2rff(x) cos 2x de)e-29t

π

+ &c.

(E)

and representing the quantity gπ by k, we have

πV=

1

= {} dx

2

[f(x) da + (sin x [f(x) sin æda+cosxff(x) cos xde) e

+ + (sin 2r f(x) sin 2cdx + cos 2rff (x) cos 2x dx) e-ťúi

+ &c.

278. This solution is the same as that which was given in the preceding section, Art. 241; it gives rise to several remarks. 1st. It is not necessary to resort to the analysis of partial differential equations in order to obtain the general equation which expresses the movement of heat in a ring. The problem may be solved for a definite number of bodies, and that number may then be supposed infinite. This method has a clearness peculiar to itself, and guides our first researches. It is easy afterwards to pass to a more concise method by a process indicated naturally. We see that the discrimination of the particular values, which, satisfying the partial differential equation, compose the general value, is derived from the known rule for the integration of linear differential equations whose coefficients are constant. The discrimination is moreover founded, as we have seen above, on the physical conditions of the problem. 2nd. To pass from the case of separate masses to that of a continuous body, we supposed the coefficient k to be increased in proportion to n, the number of masses. This continual change of the number k follows from what we have formerly proved, namely, that the quantity of heat which flows between two layers of the same prism is proportional to the value dv of x denoting the abscissa which corresponds to the section, dx'

and the temperature. If, indeed, we did not suppose the coefficient k to increase in proportion to the number of masses, but were to retain a constant value for that coefficient, we should find, on making n infinite, a result contrary to that which is observed in continuous bodies. The diffusion of heat would be infinitely slow, and in whatever manner the mass was heated, the temperature at a point would suffer no sensible change during a finite time, which is contrary to fact. Whenever we resort to the consideration of an infinite number of separate masses which