248. If we attributed a greater value to the volume w, which serves, it may be said, to draw heat from one of the bodies for the purpose of carrying it to the other, the transfer would be quicker; in order to express this condition it would be necessary to increase in the same ratio the quantity k which enters into the equations. We might also retain the value of w and suppose the layer to accomplish in a given time a greater number of oscillations, which again would be indicated by a greater value of k. Hence this coefficient represents in some respects the velocity of transmission, or the facility with which heat passes from one of the bodies into the other, that is to say, their reciprocal conducibility.

249. Adding the two preceding equations, we have

dx + dẞ = 0,

and if we subtract one of the equations from the other, we have k

dx − dB + 2 (a− B) — dt = 0, and, making a− B = y,

m

k
dy+2ydt = 0.

m

Integrating and determining the constant by the condition that

_2kt

the initial value is ab, we have y = (a - b) e m. The difference y of the temperatures diminishes as the ordinate of a logarithmic curve, or as the successive powers of the fraction em As the values of a and B, we have

2k

250. In the preceding case, we suppose the infinitely small mass, by means of which the transfer is effected, to be always the same part of the unit of mass, or, which is the same thing, we suppose the coefficient k which measures the reciprocal conducibility to be a constant quantity. To render the investigation in question more general, the constant k must be considered as a function of the two actual temperatures a and B. We should I then have the two equations dx = (1-B) = dt, and

=

m

in which k would be equal to a function of a and ß, which we denote by (a, B). It is easy to ascertain the law which the variable temperatures a and B follow, when they approach extremely near to their final state. Let y be a new unknown equal to the difference between a and the final value which is

1

2

(a + b) or c. Let z be a second unknown equal to the difference c-B. We substitute in place of a and ẞ their values c―y and c-z; and, as the problem is to find the values of y and z, when we suppose them very small, we need retain in the results of the substitutions only the first power of y and z. We therefore find the two equations,

developing the quantities which are under the sign & and omitting the higher powers of y and z. We find dy = (z− y) — ødt,

k

k

m

(z-y)ødt. The quantity being constant, it

m

follows that the preceding equations give for the value of the difference z-y, a result similar to that which we found above for the value of a - B.

From this we conclude that if the coefficient k, which was at first supposed constant, were represented by any function. whatever of the variable temperatures, the final changes which these temperatures would experience, during an infinite time, would still be subject to the same law as if the reciprocal conducibility were constant. The problem is actually to determine the laws of the propagation of heat in an indefinite number of equal masses whose actual temperatures are different.

251. Prismatic masses n in number, each of which is equal to m, are supposed to be arranged in the same straight line, and affected with different temperatures a, b, c, d, &c.; infinitely

thin layers, each of which has a mass w, are supposed to be separated from the different bodies except the last, and are conveyed in the same time from the first to the second, from the second to the third, from the third to the fourth, and so on; immediately after contact, these layers return to the masses from which they were separated; the double movement taking place as many times as there are infinitely small instants dt; it is required to find the law to which the changes of temperature are subject.

Let a, ẞ, y, 8, ... w, be the variable values which correspond to the same time t, and which have succeeded to the initial values a, b, c, d, &c. When the layers w have been separated from the n-1 first masses, and put in contact with the neighbouring masses, it is easy to see that the temperatures become

When the layers w have returned to their former places, we find new temperatures according to the same rule, which consists in dividing the sum of the quantities of heat by the sum of the masses, and we have as the values of a, ß, y, d, &c., after the instant dt,

a-(x-8), B+(x-B-B-7)
y)

m

y+ (3 − y − y − d) — 1, ...

m

w + (y - w) = /.

The coefficient of is the difference of two consecutive dif

W m

ferences taken in the succession a, B, y,... f, w. As to the first

ferences of the second order. It is sufficient to suppose the term a to be preceded by a term equal to a, and the term w to be

252. To integrate these equations, we assume, according to the known method,

a = a,e", В=ae, y = a,c,... w = a2e";

h1, a, a, a,... a2, being constant quantities which must be determined. The substitutions being made, we have the following equations:

for

If we regard a, as a known quantity, we find the expression

a in terms of a1 and h, then that of аз in a2 and h; the same is the case with all the other unknowns, a, a,, &c. The first and last equations may be written under the form

Retaining the two conditions a = a1 and a2 = a+1, the value

of a2 contains the first power of h, the value of a, contains the second power of h, and so on up to a, which contains the nth power of h. This arranged, an+1 becoming equal to a, we have, to determine h, an equation of the nth degree, and a remains undetermined.

It follows from this that we shall find n values for h, and in accordance with the nature of linear equations, the general value of a is composed of n terms, so that the quantities a, ß, y, ... &c. are determined by means of equations such as

The values of h, h', h", &c. are n in number, and are equal to then roots of the algebraical equation of the nth degree in h, which has, as we shall see further on, all its roots real.

The coefficients of the first equation a,, a,, a", a"", &c., are arbitrary; as for the coefficients of the lower lines, they are determined by a number n of systems of equations similar to the preceding equations. The problem is now to form these equations.

We see that these quantities belong to a recurrent series whose scale of relation consists of two terms (q+2) and − 1. We