contained in the prism is so distributed, that it would vanish entirely, if the end A were maintained at the temperature 0.

328. We may add several remarks to the preceding solution.

hl

T

; we

1st, it is easy to see the nature of the equation e tan e= need only suppose (see fig. 15) that we have constructed the curve u = e tan e, the arc e being taken for abscissa, and u for ordinate. The curve consists of asymptotic branches.

312

1

The abscissæ which correspond to the asymptotes are

2

ᅲ,

5

7

2π, 2T, &c.; those which correspond to points of intersection are π, 2, 3π, &c. If now we raise at the origin an ordinate

hl

equal to the known quantity and through its extremity draw a parallel to the axis of abscissæ, the points of intersection will hl The conk

give the roots of the proposed equation e tan e = struction indicates the limits between which each root lies. We shall not stop to indicate the process of calculation which must be employed to determine the values of the roots. Researches of this kind present no difficulty.

329. 2nd. We easily conclude from the general equation (E) that the greater the value of x becomes, the greater that term of the value of v becomes, in which we find the fraction e-√3 +nj2 with respect to each of the following terms. In fact, n,, n,, nz, &c. being increasing positive quantities, the fraction e-*√2m3 is

greater than any of the analogous fractions which enter into the subsequent terms.

Suppose now that we can observe the temperature of a point on the axis of the prism situated at a very great distance, and the temperature of a point on this axis situated at the distance x+1, 1 being the unit of measure; we have then y = 0, z = 0, and the ratio of the second temperature to the first is sensibly equal to the fraction eV2n. This value of the ratio of the temperatures at the two points on the axis becomes more exact as the distance a increases.

It follows from this that if we mark on the axis points each of which is at a distance equal to the unit of measure from the preceding, the ratio of the temperature of a point to that of the point which precedes it, converges continually to the fraction e-√2×12; thus the temperatures of points situated at equal distances end by decreasing in geometrical progression. This law always holds, whatever be the thickness of the bar, provided we consider points situated at a great distance from the source of heat.

It is easy to see, by means of the construction, that if the quantity called I, which is half the thickness of the prism, is very small, n, has a value very much smaller than n,, or n,, &c.; it follows from this that the first fraction e-V is very much greater than any of the analogous fractions. Thus, in the case in which the thickness of the bar is very small, it is unnecessary to be very far distant from the source of heat, in order that the temperatures of points equally distant may decrease in geometrical progression. The law holds through the whole extent of the bar.

330. If the half thickness is a very small quantity, the general value of v is reduced to the first term which contains e-√2. Thus the function v which expresses the temperature of a point whose co-ordinates are x, y, and z, is given in this case by the equation

v=

4 sin nl

2

cos ny cos nz

2nl + sin 2nlc

the arce or nl becomes very small, as we see by the construction.

e, or €1, is

hl

; by inspection of the figure we know the values of the other roots, so that the quantities ¤ ̧, ¤ ¤ ̧, ¤, €, &c. are the following, π, 2π, 3π, 4π, &c. The values of n ̧ ̧ ̧ ̧. ̧,&c.

19

whence we conclude, as was said above, that if I is a very small quantity, the first value n is incomparably greater than all the others, and that we must omit from the general value of v all the terms which follow the first. If now we substitute in the first term the value found for n, remarking that the arcs nl and 2nl are equal to their sines, we have

hl

k

the factor which enters under the symbol cosine being very small, it follows that the temperature varies very little, for different points of the same section, when the half thickness lis very small. This result is so to speak self-evident, but it is useful to remark how it is explained by analysis. The general solution reduces in fact to a single term, by reason of the thinness of the bar, and we have on replacing by unity the cosines of very small

We found the same equation formerly in Article 76; it is obtained here by an entirely different analysis.

331. The foregoing solution indicates the character of the movement of heat in the interior of the solid. It is easy to see that when the prism has acquired at all its points the stationary temperatures which we are considering, a constant flow of heat passes through each section perpendicular to the axis towards the end which was not heated. To determine the quantity of flow which corresponds to an abscissa x, we must consider that the quantity which flows during unit of time, across one element of

the section, is equal to the product of the coefficient k, of the area

dydz, of the element dt, and of the ratio

dv dx

taken with the nega

dv

tive sign. We must therefore take the integral - kfdy fdz dz,

dx

from z = 0 to z=1, the half thickness of the bar, and then from y=0 to y=l. We thus have the fourth part of the whole flow.

The result of this calculation discloses the law according to which the quantity of heat which crosses a section of the bar decreases; and we see that the distant parts receive very little heat from the source, since that which emanates directly from it is directed partly towards the surface to be dissipated into the air. That which crosses any section whatever of the prism forms, if we may so say, a sheet of heat whose density varies from one point of the section to another. It is continually employed to replace the heat which escapes at the surface, through the whole end of the prism situated to the right of the section: it follows therefore that the whole heat which escapes during a certain time from this part of the prism is exactly compensated by that which penetrates it by virtue of the interior conducibility of the solid.

To verify this result, we must calculate the produce of the flow established at the surface. The element of surface is dxdy, and v being its temperature, hvdxdy is the quantity of heat which escapes from this element during the unit of time. Hence the integral h dady v expresses the whole heat which has escaped from a finite portion of the surface. We must now employ the known value of v in y, supposing z = l, then integrate once from y=0 to y=l, and a second time from x = x up to x = ∞. We thus find half the heat which escapes from the upper surface of the prism; and taking four times the result, we have the heat lost through the upper and lower surfaces.

If we now make use of the expression h fdæ fdz v, and give to y in v its value l, and integrate once from z = = 0 to z = l, and a second time from x = 0 to x = x; we have one quarter of the heat which escapes at the lateral surfaces.

The integral h fdæ fdy v, taken between the limits indicated gives

Hence the quantity of heat which the prism loses at its surface, throughout the part situated to the right of the section whose abscissa is x, is composed of terms all analogous to

On the other hand the quantity of heat which during the same time penetrates the section whose abscissa is x is composed of terms analogous to

or k (m2 + n2) sin ml sin nl = hm cos ml sin nl + hn sin ml cos nl :

Hence the equation is satisfied. This compensation which is incessantly established between the heat dissipated and the heat transmitted, is a manifest consequence of the hypothesis; and analysis reproduces here the condition which has already been ex

F. H.

21