which must elapse before this last effect is set up is exceedingly great when the points of the mass are very distant from the origin. Each of these points which had at first the temperature 0 is imperceptibly heated; its temperature then acquires the greatest value which it can receive; and it ends by diminishing more and more, until there remains no sensible heat in the mass. variable state is in general represented by the equation

The

(a−x)2+(b_y)2+(c−2)2

2it

The integrals must be taken between the limits

a = −a1, a = a, b = − b1, b = b2, c = c1, c = c2.

The limits a,, +α2, —b1, +b,, -c1, +c, are given; they include the whole portion of the solid which was originally heated. The function f(a, b, c) is also given. It expresses the initial temperature of a point whose co-ordinates are a, b, c. The definite integrations make the variables a, b, c disappear, and there remains for v a function of x, y, z, t and constants. To determine the time which corresponds to a maximum of v, at a given point dv m, we must derive from the preceding equation the value of dt we thus form an equation which contains and the co-ordinates of the point m. From this we can then deduce the value of 0. If then we substitute this value of 0 instead of t in equation (E), we find the value of the highest temperature V expressed in x, y, z and constants.

Instead of equation (E) let us write

v = fda [db fdc Pƒ (a, b, c),

denoting by P the multiplier of ƒ (a, b, c), we have

dv

v

db

(a x)2 −

db = − 3 ; + fáa fab fdc (a − x)" + (b − y)" + (c— z)" pƒ(a,b,c)...(•).

dt

2 t

4t2

393. We must now apply the last expression to points of the solid which are very distant from the origin. Any point whatever of the portion which contains the initial heat, having for coordinates the variables a, b, c, and the co-ordinates of the point m

whose temperature we wish to determine being x, y, z, the square of the distance between these two points is (a-x)2 + (b − y)2 + (c − z)2; dv and this quantity enters as a factor into the second term of dt'

Now the point m being very distant from the origin, it is evident that the distance ▲ from any point whatever of the heated portion coincides with the distance D of the same point from the origin; that is to say, as the point m removes farther and farther from the primitive source, which contains the origin of co-ordinates, the final ratio of the distances D and ▲ becomes 1.

of

It follows from this that in equation (e) which gives the value

dv

dt

the factor (a- x)2 + (b − y)2 + (c − 2)2 may be replaced by

x2 + y2 + 22 or r2, denoting by r the distance of the point m from the origin. We have then

If we put for v its value, and replace t by

394. This result belongs only to the points of the solid whose distance from the origin is very great with respect to the greatest dimension of the source. It must always be carefully noticed that it does not follow from this condition that we can omit the variables a, b, c under the exponential symbol. They ought only to be omitted outside this symbol. In fact, the term which enters under the signs of integration, and which multiplies ƒ (a, b, c), is the

х

Now it is not sufficient for the ratio to be always a very

a

great number in order that we may suppress the two first factors. If, for example, we suppose a equal to a decimetre, and x equal to ten metres, and if the substance in which the heat is propagated is iron, we see that after nine or ten hours have elapsed, the factor

2ax

Kt

eCD is still greater than 2; hence by suppressing it we should

dv

reduce the result sought to half its value. Thus the value of dt' as it belongs to points very distant from the origin, and for any time whatever, ought to be expressed by equation (a). But it is not the same if we consider only extremely large values of the time, which increase in proportion to the squares of the distances: in accordance with this condition we must omit, even under the exponential symbol, the terms which contain a, b, or c. Now this condition holds when we wish to determine the highest temperature which a distant point can acquire, as we proceed to prove.

Thus the time which must elapse in order that a very distant point may acquire its highest temperature is proportional to the square of the distance of this point from the origin.

4Kt

If in the expression for v we replace the denominator CD

2

by its value, the exponent of e1 which is

3

3

may be reduced to since the factors which we omit coincide with

2'

unity. Consequently we find

4

i fda fåb fde ƒ (a, b, c).

The integral fda fab fdc f (a, b, c) represents the quantity of

the initial heat: the volume of the sphere whose radius is r is 3, so that denoting by ƒ the temperature which each molecule of this sphere would receive, if we distributed amongst its parts all the initial heat, we shall have v =

6

пе

The results which we have developed in this chapter indicate the law according to which the heat contained in a definite portion of an infinite solid progressively penetrates all the other parts whose initial temperature was nothing. This problem is solved more simply than that of the preceding Chapters, since by attributing to the solid infinite dimensions, we make the conditions relative to the surface disappear, and the chief difficulty consists in the employment of those conditions. The general results of the movement of heat in a boundless solid mass are very remarkable, since the movement is not disturbed by the obstacle of surfaces. It is accomplished freely by means of the natural properties of heat. This investigation is, properly speaking, that of the irradiation of heat within the material solid.

SECTION IV.

Comparison of the integrals.

396. The integral of the equation of the propagation of heat presents itself under different forms, which it is necessary to compare. It is easy, as we have seen in the second section of this Chapter, Articles 372 and 376, to refer the case of three dimensions to that of the linear movement; it is sufficient therefore to integrate the equation

or the equation

dv
dt

=

d'v dx2

.(a).

To deduce from this differential equation the laws of the propagation of heat in a body of definite form, in a ring for example, it was necessary to know the integral, and to obtain it under a certain form suitable to the problem, a condition which could be fulfilled by no other form. This integral was given for the first time in our Memoir sent to the Institute of France on the 21st of December, 1807 (page 124, Art. 84): it consists in the following equation, which expresses the variable system of temperatures of a solid ring:

R is the radius of the mean circumference of the ring; the integral with respect to a must be taken from a = 0 to a= 2πR, or, which gives the same result, from a = πR to α = πR; i is any integer, and the sum Σ must be taken from i∞ to i+∞; v denotes the temperature which would be observed after the lapse of a time t, at each point of a section separated by the arc x from that which is at the origin. We represent by v = F(x) the initial temperature at any point of the ring. We must give to i the successive values

0, +1, +2, +3, &c., and -1, -2, -3, &c.,

We thus obtain all the terms of the value of v. Such is the form under which the integral of equation (a) must be placed, in order to express the variable movement of heat in a ring (Chap. IV., Art. 241). We consider the case in which the form and extent of the generating section of the ring are such, that the points of the same section sustain temperatures sensibly equal. We suppose also that no loss of heat occurs at the surface of the ring.