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When we express analytically the temperature of these points, the object of the investigation is not to determine numerically these temperatures, which are not measurable, but to ascertain their ratios. Now these quantities depend certainly on the law according to which the initial heat has been distributed, and the effect of this initial distribution lasts so much the longer as the parts of the prism are more distant from the source. But if the

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and

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terms which form part of the exponent, such as absolute values decreasing without limit, we may employ the approximate integrals.

This condition occurs in problems where it is proposed to determine the highest temperatures of points very distant from the origin. We can demonstrate in fact that in this case the values of the times increase in a greater ratio than the distances, and are proportional to the squares of these distances, when the points we are considering are very remote from the origin. It is only after having established this proposition that we can effect the reduction under the exponent. Problems of this kind are the object of the following section.

SECTION III.

Of the highest temperatures in an infinite solid.

386. We shall consider in the first place the linear movement in an infinite bar, a portion of which has been uniformly heated, and we shall investigate the value of the time which must elapse in order that a given point of the line may attain its highest temperature.

Let us denote by 2g the extent of the part heated, the middle of which corresponds with the origin 0 of the distances x. All the points whose distance from the axis of y is less than g and greater thang, have by hypothesis a common initial temperature f, and all other sections have the initial temperature 0. We suppose that no loss of heat occurs at the external surface of the prism, or, which is the same thing, we assign to the section perpendicular to the axis infinite dimensions. It is required to ascertain what will

F. H.

25

be the time t which corresponds to the maximum of temperature at a given point whose distance is x.

We have seen, in the preceding Articles, that the variable temperature at any point is expressed by the equation

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ducibility, C the capacity for heat, and D the density.

To simplify the investigation, make k=1, and in the result

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measures the velocity with which the heat flows along the axis of

dv dx

the prism. Now this value of is given in the actual problem

without any integral sign. We have in fact

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sign of integration: now it is equal to a fluxion of the first order

hence on equating to zero this value of

dv

dt;

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instantaneous increase of the temperature at any point, we have

the relation sought between x and t. We thus find

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The highest temperatures follow each other according to the law expressed by this equation. If we suppose it to represent the varying motion of a body which describes a straight line, a being the space passed over, and t the time elapsed, the velocity of the moving body will be that of the maximum of temperature.

When the quantity g is infinitely small, that is to say when the initial heat is collected into a single element situated at the origin, the value of t is reduced to and by differentiation or

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0

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CD 2

We have left out of consideration the quantity of heat which escapes at the surface of the prism; we now proceed to take account of that loss, and we shall suppose the initial heat to be contained in a single element of the infinite prismatic bar.

388. In the preceding problem we have determined the variable state of an infinite prism a definite portion of which was affected throughout with an initial temperature f. We suppose that the initial heat was distributed through a finite space from x = 0 to x = = b.

We now suppose that the same quantity of heat bf is contained in an infinitely small element, from x = 0 to x=w. The tempera

fb

The homiru laver will therefore be and from his flows

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To ascertain the highest temperature V, we remark that the

exponent of e1 in equation (a) is ht+

1

22

4kt

Now equation (b)

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2; 4kt

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known value, we have ht +

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ponent of e1 in equation (a), we have

1 h

2'

+x2; substituting this ex

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4kt 2kt

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and replacing Ok by its known value, we find, as the expression

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write g instead of, representing by g the semi-thickness of the

prism whose base is a square. We have to determine V and 0,

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These equations are applicable to the movement of heat in a thin bar, whose length is very great. We suppose the middle of this prism to have been affected by a certain quantity of heat bf which is propagated to the ends, and scattered through the convex surface. V denotes the maximum of temperature for the point whose distance from the primitive source is x; 0 is the time which has elapsed since the beginning of the diffusion up to the instant at which the highest temperature Voccurs. The coeffi

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