assistance in understanding the theorem, and I trust that they may prove useful to others.

Consider a particle moving in the straight line OX (fig. 1) under any law of force. Let M be the position of the particle at any instant, and let the velocity of the particle at this instant be represented by the ordinate MP drawn at right angles to OX. Then, since the momentum is proportional to the velocity, the coordinates OM, MP represent the coordinate and momentum of the particle at the given instant, and we may call P the representative point.

Now let four such particles of equal mass be projected simultaneously, having the initial coordinates and x+dx and the initial velocities v and v+Sv. The representative points will form a small rectangle P Q R S of area Sr. Sv.

Let P'Q'R'S' be the corresponding representative points. at any subsequent instant t.

Then the determinantal relation asserts that the area of the small parallelogram P'Q'R'S' is equal to that of the rectangle PQR S.

[Instead of taking four particles we might suppose the points P, Q, R, S to refer to the same particle projected with different initial conditions and allowed to move for a fixed time-interval t.]

This property may be verified from first principles in the following simple cases :

CASE I. Let the motion be uniformly accelerated. Then from the equations

v=v+ft,

x' = x+vt + {ft2,

and the corresponding equations obtained by substituting x+dx for x and v+dv for v, it is easy to see (fig. 1) that the parallelogram P'Q'R'S' hus its base P'Q' parallel to OX and equal to PQ or da, and its altitude equal to PQ or dv Therefore

area P'Q'R'S'=area PQRS.

The parallelogram will, however, have undergone a shear, the points R, S having advanced beyond P, Q by an amount (M'K'=) Sv. t. Hence

cot RPQ=t

Sx'

and is proportional to t. As the time increases, the diagonal P'S' becomes more and more elongated, but the area of the parallelogram remains the same.

CASE II. Let the motion be simple harmonic, the acceleration varying as the distance from a fixed point O (fig. 2). Then, by properly choosing the scale of representation of Fig. 2.

N

velocity, the representative points of different particles will all describe concentric circles about O with uniform angular velocity. Hence the figure PQRS will be brought into the position P'Q'R'S' by rotating about O through a certain angle, and the areas of the two figures will of course be equal.

Case I. might be deduced as the limit of Case II. by (i.) reducing the scale of representation of velocity so that

the circles become projected into ellipses; (ii.) supposing the centre O to go off to infinity, so that these ellipses gradually become elongated into parabolas.

The case of a repulsive force varying directly as the distance would be a little more complicated; and it therefore seems hardly worth while to give a proof for it, though the legitimacy of the corresponding inference for this case might be inferred by means of "imaginary projection." The theorem might possibly then be extended to the case of any variable law of force by dividing up the times, and therefore the corresponding spaces decribed, into elements so small that the force might be supposed to vary uniformly with the distance along OX in any single element. This is not put forward as a satisfactory proof, but then the object of this note was to show, not how to prove the theorem, but how to convince oneself of its truth after proving it by highly analytical methods. This I found hard in the case of systems like that of Case I., where a slight variation in the initial conditions (viz. the difference of velocity dv of the points M'K') causes two such systems to separate indefinitely. I could not see how this was compatible with the multiple differential (dr. Sv) remaining constant till I had worked out the above explanation. It is a pity that systems with more than one degree of freedom could not be treated by this graphic method, but a similar objection applies equally to the graphic proof of the formula for uniformly accelerated motion of our text-books and to many other valuable illustrations of the principles of dynamics.

DISCUSSION.

Dr. STONEY thought the arguments were based on actions depending on the distances of the molecules and the supposition that they were rigid. In his opinion events occur in nature which are not represented by this simple theory, and great reservation should be shown in accepting dynamical problems which leave out of account actions occurring between matter and the ether. In nature nothing was large and nothing was small except relatively. Even molecules might possess infinite detail of structure. Their interaction with the ether must be considered in any complete theory.

XLI. On the Mechanical Analogue of Thermal Equilibrium between Bodies in Contact. By G. H. BRYAN and LUDWIG BOLTZMANN.

[Abstract.]

THE great disadvantage of the Kinetic Theory has been that it has not hitherto furnished a very satisfactory proof of the fundamental property of temperature, namely, that two bodies in contact are at equal temperature when neither gains or loses heat from the other.

In a medium composed of two or more different kinds of gas molecules mixed together, the conditions of equilibrium of kinetic energy involve Avogadro's well-known law†, namely, that the mean kinetic energies of translation of the different kinds of molecules shall be equal. From this we naturally infer that the quantity which represents temperature in the kinetic theory is proportional to the mean vis viva of translation. But our experimental knowledge of the properties of temperature is necessarily derived from observations of bodies which do not mix, and it is impossible to form a purely physical conception of the temperatures of two mixed gases apart from each other. We can only experiment on the temperature of the whole mixture.

It is therefore of great interest to devise a mechanical representation of the phenomena presented by two bodies between which transference of heat takes place by conduction, but whose molecules do not become indefinitely interdiffused. The system now to be described affords such a repre

sentation.

Let X and Y be two infinite parallel planes (or other surfaces) at a small distance apart, dividing space into three

*This paper is taken from a memoir by Prof. Ludwig Boltzmann and G. H. Bryan, communicated by Prof. Boltzmann to the Imperial Academy of Vienna, Dec. 13, 1894. Read at the meeting of the Physical Society, April 26, 1895.

+ Wien. Sitzungsber. vol. lxiii. 1871, vol. lxvi. 1872, vol. xciv. 1886, vol. xcv.

1887.

VOL. XIII.

2 K

regions, S1, S2, and σ, of which the region S1 lies to the left of X, the region S2 lies to the right of Y, and the region lies between the surfaces X, Y, and may be supposed to be very thin by supposing X, Y to be very near together.

Let there be two different kinds of molecules A, B, the former represented in the figure by light, and the latter by dark points, and let these molecules be acted on by different laws of force specified as follows :

:

(i) The plane Y shall be supposed to repel the A molecules with a force which vanishes at all points on the left-hand side of X, but which becomes infinitely great as we approach the plane Y itself so as to prevent any of the A molecules from crossing beyond Y and passing into the region S2. This force shall not act on the B molecules.

(ii) In like manner, the plane X shall be supposed to repel the B molecules with a force which vanishes at all points on the right-hand side of Y, but which becomes infinitely great as we approach the plane X itself, so as to prevent any of the B molecules from crossing beyond X and passing into the region S1. This force shall not act on any of the A

molecules.

Then in the region S, we have a gas composed only of A molecules, and in the region S, we have a gas composed entirely of B molecules. Between these regions there is the region containing a mixture of both kinds of molecules, and these molecules will collide with one another, and energy will thus be transferred from the molecules of one gas to those of the other.