in number, namely, the unit of length, the unit of time, that of temperature, that of weight, and finally the unit which serves to measure quantities of heat. For the last unit, we might have chosen the quantity of heat which raises a given volume of a certain substance from the temperature 0 to the temperature 1. The choice of this unit would have been preferable in many respects to that of the quantity of heat required to convert a mass of ice of a given weight, into an equal mass of water at 0, without raising its temperature. We have adopted the last unit only because it had been in a manner fixed beforehand in several works on physics; besides, this supposition would introduce no change into the results of analysis.

158. The specific elements which in every body determine the measurable effects of heat are three in number, namely, the conducibility proper to the body, the conducibility relative to the atmospheric air, and the capacity for heat. The numbers which express these quantities are, like the specific gravity, so many natural characters proper to different substances.

We have already remarked, Art. 36, that the conducibility of the surface would be measured in a more exact manner, if we had sufficient observations on the effects of radiant heat in spaces deprived of air.

It may be seen, as has been mentioned in the first section of Chapter 1, Art. 11, that only three specific coefficients, K, h, C, enter into the investigation; they must be determined by observation; and we shall point out in the sequel the experiments adapted to make them known with precision.

159. The number C which enters into the analysis, is always multiplied by the density D, that is to say, by the number of units of weight which are equivalent to the weight of unit of volume; thus the product CD may be replaced by the coefficient c. In this case we must understand by the specific capacity for heat, the quantity required to raise from temperature 0 to temperature 1 unit of volume of a given substance, and not unit of weight of that substance.

With the view of not departing from the common definition, we have referred the capacity for heat to the weight and not to

the volume; but it would be preferable to employ the coefficient c which we have just defined; magnitudes measured by the unit of weight would not then enter into the analytical expressions: we should have to consider only, 1st, the linear dimension x, the temperature v, and the time t; 2nd, the coefficients c, h, and K. The three first quantities are undetermined, and the three others are, for each substance, constant elements which experiment determines. As to the unit of surface and the unit of volume, they are not absolute, but depend on the unit of length.

160. It must now be remarked that every undetermined magnitude or constant has one dimension proper to itself, and that the terms of one and the same equation could not be compared, if they had not the same exponent of dimension. We have introduced this consideration into the theory of heat, in order to make our definitions more exact, and to serve to verify the analysis; it is derived from primary notions on quantities; for which reason, in geometry and mechanics, it is the equivalent of the fundamental lemmas which the Greeks have left us without proof.

161. In the analytical theory of heat, every equation (E) expresses a necessary relation between the existing magnitudes x, t, v, c, h, K. This relation depends in no respect on the choice of the unit of length, which from its very nature is contingent, that is to say, if we took a different unit to measure the linear dimensions, the equation (E) would still be the same. Suppose then the unit of length to be changed, and its second value to be equal to the first divided by m. Any quantity whatever x which in the equation (E) represents a certain line ab, and which, consequently, denotes a certain number of times the unit of length, becomes mx, corresponding to the same length ab; the value t of the time, and the value v of the temperature will not be changed; the same is not the case with the specific elements h, K, c: the first, h, becomes; for it expresses the quantity of heat which escapes, during the unit of time, from the unit of surface at the temperature 1. If we examine attentively the nature of the coefficient K, as we have defined it in Articles 68 and 135,

h

K

we perceive that it becomes ;

m

for the flow of heat varies

directly as the area of the surface, and inversely as the distance between two infinite planes (Art. 72). As to the coefficient c which represents the product CD, it also depends on the unit of

с

length and becomes hence equation (E) must undergo no

m

change when we write mx instead of x, and at the same time Kh C instead of K, h, c; the number m disappears after

m

m2,

m

37

these substitutions: thus the dimension of x with respect to the unit of length is 1, that of K is -1, that of h is 2, and that of c is 3. If we attribute to each quantity its own exponent of dimension, the equation will be homogeneous, since every term will have the same total exponent. Numbers such as S, which represent surfaces or solids, are of two dimensions in the first case, and of three dimensions in the second. Angles, sines, and other trigonometrical functions, logarithms or exponents of powers, are, according to the principles of analysis, absolute numbers which do not change with the unit of length; their dimensions must therefore be taken equal to 0, which is the dimension of all abstract numbers.

n'

If the unit of time, which was at first 1, becomes, the number t will become nt, and the numbers x and v will not change. The Kh coefficients K, h, c will become

n n

C. Thus the dimensions

of x, t, v with respect to the unit of time are 0, 1, 0, and those of K, h, c are 1, -1, 0.

-

If the unit of temperature be changed, so that the temperature 1 becomes that which corresponds to an effect other than the boiling of water; and if that effect requires a less temperature, which is to that of boiling water in the ratio of 1 to the number p; v will become vp, x and t will keep their values, and the coeffiKh с

cients K, h, c will become.

p' p' p

The following table indicates the dimensions of the three undetermined quantities and the three constants, with respect to each kind of unit.

F. H.

9

162. If we retained the coefficients C and D, whose product has been represented by c, we should have to consider the unit of weight, and we should find that the exponent of dimension, with respect to the unit of length, is 3 for the density D, and 0 for C.

On applying the preceding rule to the different equations and their transformations, it will be found that they are homogeneous with respect to each kind of unit, and that the dimension of every angular or exponential quantity is nothing. If this were not the case, some error must have been committed in the analysis, or abridged expressions must have been introduced.

If, for example, we take equation (b) of Art. 105,

we find that, with respect to the unit of length, the dimension of each of the three terms is 0; it is 1 for the unit of temperature, and 1 for the unit of time.

2h

In the equation v=Ae* Kt of Art. 76, the linear dimension of each term is 0, and it is evident that the dimension of the

exponent x/2h

ΚΙ

is always nothing, whatever be the units of

length, time, or temperature.

CHAPTER III.

PROPAGATION OF HEAT IN AN INFINITE RECTANGULAR SOLID.

SECTION I.

Statement of the problem.

163. PROBLEMS relative to the uniform propagation, or to the varied movement of heat in the interior of solids, are reduced, by the foregoing methods, to problems of pure analysis, and the progress of this part of physics will depend in consequence upon the advance which may be made in the art of analysis. The differential equations which we have proved contain the chief results of the theory; they express, in the most general and most concise manner, the necessary relations of numerical analysis to a very extensive class of phenomena; and they connect for ever with mathematical science one of the most important branches of natural philosophy.

It remains now to discover the proper treatment of these equations in order to derive their complete solutions and an easy application of them. The following problem offers the first example of analysis which leads to such solutions; it appeared to us better adapted than any other to indicate the elements of the method which we have followed.

164. Suppose a homogeneous solid mass to be contained between two planes B and C vertical, parallel, and infinite, and to be divided into two parts by a plane A perpendicular to the other two (fig. 7); we proceed to consider the temperatures of the mass BAC bounded by the three infinite planes A, B, C. The other part BAC of the infinite solid is supposed to be a constant source of heat, that is to say, all its points are maintained at the temperature 1, which cannot alter. The two