different consequences, if we were not treating here of entirely new problems, whose results may be of direct use.

86. We know that animated bodies retain a temperature sensibly fixed, which we may regard as independent of the temperature of the medium in which they live. These bodies are, after some fashion, constant sources of heat, just as inflamed substances are in which the combustion has become uniform. We may then, by aid of the preceding remarks, foresee and regulate exactly the rise of temperature in places where a great number of men are collected together. If we there observe the height of the thermometer under given circumstances, we shall determine in advance what that height would be, if the number of men assembled in the same space became very much greater.

In reality, there are several accessory circumstances which modify the results, such as the unequal thickness of the parts of the enclosure, the difference of their aspect, the effects which the outlets produce, the unequal distribution of heat in the air. We cannot therefore rigorously apply the rules given by analysis; nevertheless these rules are valuable in themselves, because they contain the true principles of the matter: they prevent vague reasonings and useless or confused attempts.

87. If the same space were heated by two or more sources of different kinds, or if the first inclosure were itself contained in a second enclosure separated from the first by a mass of air, we might easily determine in like manner the degree of heating and the temperature of the surfaces.

If we suppose that, besides the first source o, there is a second heated surface π, whose constant temperature is B, and external conducibility j, we shall find, all the other denominations being retained, the following equation:

If we suppose only one source o, and if the first enclosure is itself contained in a second, s', h', K', H', e', representing the

elements of the second enclosure which correspond to those of the first which were denoted by s, h, K, H, e; we shall find, p denoting the temperature of the air which surrounds the external surface of the second enclosure, the following equation :

We should obtain a similar result if we had three or a greater number of successive enclosures; and from this we conclude that these solid envelopes, separated by air, assist very much in increasing the degree of heating, however small their thickness may be.

88. To make this remark more evident, we will compare the quantity of heat which escapes from the heated surface, with that which the same body would lose, if the surface which envelopes it were separated from it by an interval filled with air.

If the body A be heated by a constant cause, so that its surface preserves a fixed temperature b, the air being maintained at a less temperature a, the quantity of heat which escapes into the air in the unit of time across a unit of surface will be expressed by h (b-a), h being the measure of the external conducibility. Hence in order that the mass may preserve a fixed temperature b, it is necessary that the source, whatever it may be, should furnish a quantity of heat equal to hS (b− a), S denoting the area of the surface of the solid.

Suppose an extremely thin shell to be detached from the body A and separated from the solid by an interval filled with air; and suppose the surface of the same solid A to be still maintained at the temperature b. We see that the air contained between the shell and the body will be heated and will take a temperature a' greater than a. a permanent state and will transmit to the external air whose fixed temperature is a all the heat which the body loses. It follows that the quantity of heat escaping from the solid will

The shell itself will attain

be hS (b-a), instead of being hS (b-a), for we suppose that the new surface of the solid and the surfaces which bound the shell have likewise the same external conducibility h. It is evident that the expenditure of the source of heat will be less than it was at first. The problem is to determine the exact ratio of these quantities.

89. Let e be the thickness of the shell, m the fixed temperature of its inner surface, n that of its outer surface, and K its internal conducibility. We shall have, as the expression of the quantity of heat which leaves the solid through its surface, hS (b-a').

As that of the quantity which penetrates the inner surface of the shell, hS (am).

As that of the quantity which crosses any section whatever

Lastly, as the expression of the quantity which passes through the outer surface into the air, hS (n − a).

All these quantities must be equal, we have therefore the following equations:

The quantity of heat lost by the solid was hS(b − a), when its surface communicated freely with the air, it is now hS (b-a)

b-a

or hS (na), which is equivalent to hS

he'

3+

K

The first quantity is greater than the second in the ratio of

In order therefore to maintain at temperature b a solid whose surface communicates directly to the air, more than three times as much heat is necessary than would be required to maintain it at temperature b, when its extreme surface is not adherent but separated from the solid by any small interval whatever filled with air.

If we suppose the thickness e to be infinitely small, the ratio of the quantities of heat lost will be 3, which would also be the value if K were infinitely great.

We can easily account for this result, for the heat being unable to escape into the external air, without penetrating several surfaces, the quantity which flows out must diminish as the number of interposed surfaces increases; but we should have been unable to arrive at any exact judgment in this case, if the problem had not been submitted to analysis.

90. We have not considered, in the preceding article, the effect of radiation across the layer of air which separates the two surfaces; nevertheless this circumstance modifies the problem, since there is a portion of heat which passes directly across the intervening air. We shall suppose then, to make the object of the analysis more distinct, that the interval between the surfaces is free from air, and that the heated body is covered by any number whatever of parallel laminæ separated from each other.

If the heat which escapes from the solid through its plane superficies maintained at a temperature b expanded itself freely in vacuo and was received by a parallel surface maintained at a less temperature a, the quantity which would be dispersed in unit of time across unit of surface would be proportional to (b-a), the difference of the two constant temperatures: this quantity

would be represented by H (b-a), H being the value of the relative conducibility which is not the same as h.

The source which maintains the solid in its original state must therefore furnish, in every unit of time, a quantity of heat equal to HS (b-a).

We must now determine the new value of this expenditure in the case where the surface of the body is covered by several successive laminæ separated by intervals free from air, supposing always that the solid is subject to the action of any external cause whatever which maintains its surface at the temperature b.

Imagine the whole system of temperatures to have become fixed; let m be the temperature of the under surface of the first lamina which is consequently opposite to that of the solid, let n be the temperature of the upper surface of the same lamina, e its thickness, and K its specific conducibility; denote also by m1, n, m, n, m, n, m, n, &c. the temperatures of the under and upper surfaces of the different laminæ, and by K, e, the conducibility and thickness of the same lamina; lastly, suppose all these surfaces to be in a state similar to the surface of the solid, so that the value of the coefficient His common to them.

e

The quantity of heat which penetrates the under surface of a lamina corresponding to any suffix i is HS (nm), that which KS crosses this lamina is (m, n), and the quantity which escapes from its upper surface is HS (nm). These three quantities, and all those which refer to the other laminæ are equal; we may therefore form the equation by comparing all these quantities in question with the first of them, which is HS (b− m1); we shall thus have, denoting the number of laminæ by j: