examine more particularly in what the final state consists which is expressed by the equation

If first we seek the point of the ring at which we have the condition

we see that the temperature at this point is at every instant the mean temperature of the ring: the same is the case with the point diametrically opposite; for the abscissa x of the latter point will also satisfy the above equation

Let us denote by X the distance at which the first of these points is situated, and we shall have

If we now take as origin of abscissæ the point which corresponds to the abscissa X, and if we denote by u the new abscissa x-X, we shall have

At the origin, where the abscissa u is 0, and at the opposite point, the temperature is always equal to the mean temperature; these two points divide the circumference of the ring into two parts whose state is similar, but of opposite sign; each point of one of these parts has a temperature which exceeds the mean temperature, and the amount of that excess is proportional to the sine of the distance from the origin. Each point of the

other part has a temperature less than the mean temperature, and the defect is the same as the excess at the opposite point. This symmetrical distribution of heat exists throughout the whole duration of the cooling. At the two ends of the heated half, two flows of heat are established in direction towards the cooled half, and their effect is continually to bring each half of the ring towards the mean temperature.

246. We may now remark that in the general equation which gives the value of v, each of the terms is of the form

We can therefore derive, with respect to each term, consequences analogous to the foregoing. In fact denoting by X the distance for which the coefficient

is nothing, we have the equation b ̧ —— a, tan i

stitution gives, as the value of the coefficient,

a being a constant. It follows from this that taking the point whose abscissa is X as the origin of co-ordinates, and denoting by u the new abscissa x-X, we have, as the expression of the changes of this part of the value of v, the function

If this particular part of the value of v existed alone, so as to make the coefficients of all the other parts nul, the state of the ring would be represented by the function

and the temperature at each point would be proportional to the sine of the multiple i of the distance of this point from the origin. This state is analogous to that which we have already described:

it differs from it in that the number of points which have always the same temperature equal to the mean temperature of the ring is not 2 only, but in general equal to 2i. Each of these points or nodes separates two adjacent portions of the ring which are in a similar state, but opposite in sign. The circumference is thus found to be divided into several equal parts whose state is alternately positive and negative. The flow of heat is the greatest possible in the nodes, and is directed towards that portion which is in the negative state, and it is nothing at the points which are equidistant from two consecutive nodes. The ratios which exist then between the temperatures are preserved during the whole of the cooling, and the temperatures vary together very rapidly in proportion to the successive powers of the fraction

k

If we give successively to i the values 0, 1, 2, 3, &c., we shall ascertain all the regular and elementary states which heat can assume whilst it is propagated in a solid ring. When one of these simple modes is once established, it is maintained of itself, and the ratios which exist between the temperatures do not change; but whatever the primitive ratios may be, and in whatever manner the ring may have been heated, the movement of heat can be decomposed into several simple movements, similar to those which we have just described, and which are accomplished all together without disturbing each other. In each of these states the temperature is proportional to the sine of a certain multiple of the distance from a fixed point. The sum of all these partial temperatures, taken for a single point at the same instant, is the actual temperature of that point. Now some of the parts which compose this sum decrease very much more rapidly than the others. It follows from this that the elementary states of the ring which correspond to different values of i, and whose superposition determines the total movement of heat, disappear in a manner one after the other. They cease soon to have any sensible influence on the value of the temperature, and leave only the first among them to exist, in which i is the least of all. In this manner we form an exact idea of the law according to which heat is distributed in a ring, and is dissipated at its surface. The state of the ring becomes more and more symmetrical; it soon becomes confounded

with that towards which it has a natural tendency, and which consists in this, that the temperatures of the different points become proportional to the sine of the same multiple of the arc which measures the distance from the origin. The initial distribution makes no change in these results.

SECTION II.

Of the communication of heat between separate masses.

247. We have now to direct attention to the conformity of the foregoing analysis with that which must be employed to determine the laws of propagation of heat between separate masses; we shall thus arrive at a second solution of the problem of the movement of heat in a ring. Comparison of the two results will indicate the true foundations of the method which we have followed, in integrating the equations of the propagation of heat in continuous bodies. We shall examine, in the first place, an extremely simple case, which is that of the communication of heat between two equal masses.

Suppose two cubical masses m and n of equal dimensions and of the same material to be unequally heated; let their respective temperatures be a and b, and let them be of infinite conducibility. If we placed these two bodies in contact, the temperature in each would suddenly become equal to the mean temperature (a+b). Suppose the two masses to be separated by a very small interval, that an infinitely thin layer of the first is detached so as to be joined to the second, and that it returns to the first immediately after the contact. Continuing thus to be transferred alternately, and at equal infinitely small intervals, the interchanged layer causes the heat of the hotter body to pass gradually into that which is less heated; the problem is to determine what would be, after a given time, the heat of each body, if they lost at their surface no part of the heat which they contained. We do not suppose the transfer of heat in solid continuous bodies to be effected in a manner similar to that which we have just described: we wish only to determine by analysis the result of such an hypothesis.

Each of the two masses possessing infinite conducibility, the quantity of heat contained in an infinitely thin layer, is sud

F. H.

15

denly added to that of the body with which it is in contact; and a common temperature results which is equal to the quotient of the sum of the quantities of heat divided by the sum of the masses. Let w be the mass of the infinitely small layer which is separated from the hotter body, whose temperature is a; let a and ẞ be the variable temperatures which correspond to the time t, and whose initial values are a and b. When the layer w is separated from the mass m which becomes m — w, it has like this mass the temperature a, and as soon as it touches the second body affected with the temperature ẞ, it assumes at the same time with that body a mB + zw temperature equal to The layer w, retaining the last m + w temperature, returns to the first body whose mass is m-w and temperature a. We find then for the temperature after the second

The variable temperatures a and ẞ become, after the interval

W

dt, a+ (a-B) and B+(x-B); these values are found by

m

m

suppressing the higher powers of w. We thus have

the mass which had the initial temperature ẞ has received in one instant a quantity of heat equal to mdß or (a-B) w, which has been lost in the same time by the first mass. We see by this that the quantity of heat which passes in one instant from the most heated body into that which is less heated, is, all other things being equal, proportional to the actual difference of temperature of the two bodies. The time being divided into equal intervals, the infinitely small quantity o may be replaced by kdt, k being the number of units of mass whose sum contains o as many times as

the unit of time contains dt, so that we have

k 1

-=

W dt'

k

(a-B) dt and dẞ(a — B) — dt.

m

=

m

We thus