XX. On the Reduction of the Second Axiom of the Mechanical Theory of Heat to general Mechanical Principles. By R. CLAUSIUS*.

1.

IN

Na memoir recently communicated and published†, I have advanced the following theorem, valid for every stationary motion of any system of material points :-The mean vis viva of the system is equal to its virial. This theorem may be regarded as one of dynamical equilibrium, since it gives a relation which must subsist between the forces and the motions called forth by them in order that the latter may continue with their vis viva, on the average, neither increased by positive work of the forces nor diminished by negative, but, amid passing fluctuations, maintaining a constant mean value.

As the magnitude which I have denoted by the name virial is, with equal coordinates of the material points, proportional to the forces operating upon them, the vis viva of stationary motion is, cæteris paribus, proportional to the forces which it balances. If, then, we regard heat as a stationary motion of the smallest particles of bodies, and absolute temperature as the measure of the vis viva, we shall find no difficulty in recognizing the agreement of the above-mentioned mechanical theorem with the law advanced by me in an earlier memoir‡:-The effective force of heat is proportional to the absolute temperature.

Translated from a separate impression communicated by the Author, having been read before the Niederrheinische Gesellschaft für Natur- und Heilkunde, on November 7, 1870.

122.

† Sitzungsberichte der Niederrheinischen Gesellschaft f. Nat. u. Heilk. June 1870; Pogg. Ann. vol. cxli. p. 124; Phil. Mag. S. 4. vol. xl. p. Pogg. Ann. vol. cxvi. p. 73; Abhandlungen über die mechanischen Wärmetheorie, vol. i. p. 242.

Phil. Mag. S. 4. Vol. 42. No. 279. Sept. 1871.

M

If, however, we wish to make this law the basis of a mathematical development, we must give it a more definite form, because the expression effective force of heat may admit of different interpretations. Hence in that memoir I have, for the purpose of thus applying it, expressed the law more fully, as follows:

The mechanical work which can be done by heat in any alteration of the arrangement of a body, is proportional to the absolute temperature at which the alteration takes place.

In order to express this law by a mathematical equation, let us imagine the body undergoing an infinitely small alteration of its condition, the change proceeding in a reversible manner, in which the quantity of heat contained in the body as well as its constituents may be altered. Work may either be performed (when the internal and external forces operating on the particles are overcome) or expended (when the particles yield to the forces). This infinitesimal work may be denoted by dL; work performed is reckoned as positive, and work expended as negative. Then the following equation will stand as the expression of the above law:

:

in which T denotes the absolute temperature, and A a constant, namely the caloric equivalent of the work, and Z represents a magnitude which is perfectly determined by the present condition of the body, without it being necessary to know in what way the body has come into this condition. This magnitude I have named the disgregation of the body.

If we further assume, as I have done in the above-mentioned memoir, that the absolute temperature of a body is proportional to the quantity of heat present in it, and denote this quantity by H, we can put

T=CH,

in which C will be a constant. The preceding equation is thus transformed into

H
A'

The fraction herein occurring, represents the quantity of

heat present in the body, measured, not according to the usual heat-scale, but mechanically; therefore, in other words, it represents the vis viva of that motion which we name heat. By introducing for this magnitude the simple sign h, the equation becomes

We have now to find for this equation an explanation founded on mechanical principles. For this purpose the above theorem concerning the virial furnishes a clue, inasmuch as it indicates the nature of the considerations which must be employed. But it is not alone sufficient; the investigation requires in addition certain new and peculiar developments, which are to form the subject of the present memoir.

2. To begin with a case as simple as possible in relation to the kind of motion, and thereby facilitate the view of the mode of consideration which here comes into use, we will first suppose a single material point, operated on by a force which may be represented by an ergal—that is, the components of which, referred to three rectangular-coordinate directions, are expressed by the partial differential coefficients of the three coordinates of the point, taken negatively. Under the influence of this force, the point will have a periodical motion in a closed path.

Now let us imagine this motion to undergo an infinitely small alteration, resulting in a new periodical motion. This conversion of the motion can be occasioned in three ways: at any place in the path, through a passing external influence, the velocitydz

dx dy

dt

components'' and may be infinitesimally altered, and then the point may again be left to the operation merely of the original force; or an infinitesimal alteration may occur in the force operating on the point-for example, a change in the value of a constant occurring in the ergal. The third cause of conversion of the motion will not occur in our considerations on heat, but is of interest for a comparison which we shall make further on it is the point being compelled to describe a path somewhat deviating from the one chosen by itself-which is also connected with an alteration of the force, because then to the original force is added the resistance which the new path-curve has to perform.

We will now investigate whether, in all these circumstances, there exists a universally valid relation between the alterations of the different magnitudes occurring in the motion.

3. The alterations undergone by the coordinates of the point, its velocity-components, the components of the force, &c. shall, as differentials of those magnitudes, be denoted as usual by the prefix d; so that, for example, da will signify the variation in x ⚫ during the time dt. On the other hand, the alterations of those magnitudes which result from a different motion taking the place of the original one shall be called variations of the magnitudes, and be denoted by prefixing the letter &; so that, e. g., the difference between a value of x in the original motion and the corresponding value in the altered motion will be signified by dx.

In reference to the latter, however, a special remark must be made, which is of importance for the following. If the altered motion is to be compared with the initial one in such a manner as to show how the values of x in the one differ from the corresponding values of x in the other motion, we must first settle which values of x shall be regarded as corresponding to each other. For this purpose, any two points infinitely near each other in the two paths may first be taken as corresponding points. Starting from these, in order to obtain the remaining corresponding points we take as a measure a magnitude which changes in the course of the motions, and settle that those points in the two paths which belong to equal values of the measuring magnitude are corresponding points. As measuring magnitude, however, one must be chosen which for an entire revolution has equal values in both paths; for through an entire revolution the moving point always arrives again at the chosen initial point in each of the two paths, and these we have already taken as corresponding points.

We will now determine the measuring magnitude in the following manner. Let i be the time of a revolution with the original motion, and t the variable time which the moveable point requires in order to pass from the initial position to another one; then we will put

For the altered motion, let the time of a revolution be denoted by i', and the variable time, reckoned from the point's leaving its initial position, by t'; then we put

t'=i'. p.

If, now, has equal values in both expressions, t and t' are corresponding times. The corresponding times being in this manner determined, the corresponding points of the two paths and, accordingly, the corresponding values of x, y, z, &c. follow of themselves.

The magnitude we will call the phase of the motion. During one revolution the phase increases one unit. With further increase, the phases which differ by a whole number of units may be regarded as equal, in the same sense as angles which differ by multiples of 27.

Subtracting the first of the two preceding equations from the second, there results

t' — t = (i' — i)$.

The difference t'-t is the variation of t, and the difference ¿'—i the variation of i. Denoting these by dt and di, we can

write

whence it follows as a rule that, if we wish to variate equation (3), we must regard the magnitude & as constant. On the contrary, if we wish to differentiate the same equation, we must regard i as constant, because the differentiation refers to the course of a determinate motion, in which the time of a revolution i is a given magnitude. We thus obtain

4. These preliminaries being settled, we can now proceed to the proposed mathematical development. Taking the expression Ex, and differentiating it according to 4, we obtain

dx

dt

do dt

(6)

Now, as in variation the phase 4 is regarded as constant, we can, when a magnitude varies and is to be differentiated according to 4, change the order of these two operations and therefore put

This equation may be transformed in the following manner :

dt

dt

αφ

Putting herein, for the differential coefficient its value from

do'

аф

This equation shall now be multiplied by do and then integrated from 40 to 4=1; that is, for an entire revolution.

The integration on the left-hand side may proceed at once, and we obtain

(' d (da 8.x)dp=(dx dx), − (dx 81);

dt

dt

dt 1