the total number which after a shorter or longer time reach the mark, we have to integrate with respect to dt from t = 0 to t =∞. Hence

dy dz du dv dw∞ dt ["drNF(u, v, w) ƒ (t, u, v, w)

0

is the required number of particles which pass through the element dy dz in unit time with the velocity whose components

are u, v, u'.

3*. Reaction

This formula, however, only gives the number of particles which cross the surface-element from one-half of the medium to the other in the positive direction of x, that is, with a velocity such that its component u is positive.

If in like manner we obtain the number of the particles which pass in the reverse direction from the second half of the medium to the first, the value of the relative coordinate r, which must be positive since for this motion the component u is negative, must satisfy the condition

0x ut.

Therefore the number of particles crossing in this opposite direction is

dy dz du dv dw foo diso dxNF(u, v, w) ƒ (t, u, v, w)

= – dy dz du dv dwt (dxNF(u, v, w) ƒ (t, u, v, w).

0

In form this formula is distinguished from the other only by its sign, but if the functions N, F, ƒ depend not only on the given variables t, u, v, w, but on the position as well, it may also differ by reason of a difference in the signification of these functions.

4*. Summation Carried Out

The integrations can be immediately effected in the case wherein these functions do not depend on position, but when the state of the motion is the same at every point of the gaseous medium. With this assumption we obtain, by aid of a theorem given in § 1*,

as the number of particles with the velocities u, v, w which, in a unit of time, cross the surface-element dy dz in the interior of the space filled with gas. The upper sign applies to the passage in the positive direction of x, and the lower to that in the opposite direction.

The analogous expressions for the other axial directions are

+NF(u, v, w)v du dv dw dz dx,

+NF(u, v, w)w du dv dw dx dy.

These formulæ have a very simple interpretation. For u dy dz is the volume of an oblique parallelopiped on the base dy dz, of length equal to the velocity, and of altitude equal to the component u; it is thus the volume of the region in which all the particles which cross dy dz in a unit of time with the given velocity must have been at the beginning of the time-unit. Since, from the definition of F, the first of the three expressions denotes the number of particles in this volume which at any moment are moving with the given velocity, it shows, as do also the other two, that as many particles pass through the surface-element as if none had been previously stopped or deviated.

The motion in a gas which is in the same state of equilibrium at all points of the space occupied by it, therefore, goes on just as if the particles never collided, but moved about in all directions without hindrance.

The reason for this (at first sight) surprising result is simply that, when the requisite state of unchangeable equilibrium is attained, for every particle which loses its motion by collision there occurs another which acquires the same motion by another simultaneous collision.

5*. Momentum Carried Over

The possibility thus demonstrated of replacing the hypotheses on which our theory is founded by still simpler assumptions in the case of a gas in equilibrium very considerably facilitates the calculation of the pressure exerted by the gas.

On the theory here assumed, the pressure exerted on a surface within the gaseous medium is measured by the force which one half of the medium exerts on the other from which it is separated by the surface. Since, as has just been proved, 'See Chap. II. § 12.

the collisions do not come into account in the state of equilibrium, we have to deal only with the force which the particles that pass backwards and forwards over the surface exert in virtue of their own motion, and transmit from one half of the medium to the other. What becomes of this motion after the passage from the one half to the other-whether it is destroyed or transformed-is of no moment, since, in the condition of equilibrium, for every particle that loses its velocity another appears which acquires the same motion by collision.

We therefore obtain the momentum carried across the surface dy dz in the unit of time in the positive direction of x from the first half of the medium into the second by multiplying the abovefound number of particles which in unit of time cross the surface with the velocities u, v, w, viz.

NF(u, v, w)u du dv dw dy dz,

into the components of the momentum of each of these particles, which are

mu, mv, mw,

and then integrating the expressions so obtained. The summations are to be carried out with respect to the possible values of the velocity-components u, v, w, and their limits are determined by noting that the passage from the first half of the medium to the second in the positive direction of x can only occur with such velocities as have a positive component u. Consequently, the momenta carried over dy dz in unit of time in the positive direction are

For the particles which pass from the first half to the second, and for the momentum carried with them, the first half is compensated by the passage that occurs in the opposite direction into it from the second half. In this direction the passage can occur only with negative values of u; and the number of particles which pass in this direction with velocity-components u, v, w being

and each carrying with it the momenta mu, mv, mw, the total momenta carried across in the backward direction are

– dy dz Nms_ „duf”_dv[”_dw uwF(u, v, w).

818

6*. Components of Pressure

The second half of the medium loses these latter momenta and gains the former. The resulting increase of its momentum is, therefore, given by the difference of these expressions, so that

dy dz X= dy dz Nm√R du["_dv["_dw u2F(u, v, w),

dy dz Y = dy dz Nm√_du_dv["_dw uxF( u, v, w),

81

∞

dy dz Z1 = dy dz Nm_du_dv_dwuw F(u, v, w)

-∞

are the components of the momenta which, during the unit of time, pass over the surface dy dz from the first half of the medium to the second, or, more briefly, are the components of the force exerted on dy dz by the first half towards the second.

Just as we have here found the force-components which act on a surface perpendicular to the x-axis, and are denoted by the suffix x, we may obtain the analogous magnitudes for the other two axial directions. We thus get, with the corresponding notation, the following values for the forces exerted per unit area, that is, for the pressures,

X1 = Nm√__du_dv[_dw u2F(u, v, w),

81

81

X, Y,= Nm_du_def_dw ucF(x, e, w).

=

dv

81

These formulæ determine the pressure exerted by the half of the medium which is nearer the negative coordinates, and has hitherto been called the first half, on the second, which lies on the positive side. The action of the second half on the first is expressed by the same formula with changed sign.

The first three of these six formulæ give the pressures which act normally on the stressed surface, i.e. the normal pressures, while the last three express the magnitude of the tangential pressures whose directions lie along the surface itself.

7. Interpretation of the Formulæ

Both forms of pressure are expressed by integrals which, from the meaning of the function F, are easily seen to represent probable mean values.

Noting further that

Nm = P,

that is, the mass of gas contained in unit of volume, or its density, we may write the foregoing formulæ thus:-

where the bar denotes the mean value of the magnitude placed under it.1

These formulæ, which were first given in such generality by Maxwell for the pressure in a gas in any state of motion that does not depend on time or position, can be much simplified when there is only the heat-motion of the molecules and not a forward motion of the gas as a whole. Since in this case the motion is symmetrical in all directions of space, all functions depending on uneven powers of the velocity-components u, v, w vanish, and therefore

vw = wu = uv = 0.

These terms also vanish when the direction of motion is along one of the coordinate axes.

[The author uses the notation M(x) to denote the mean value of x, but the ordinary English custom is here followed. Tr.]

Phil. Mag. [4] xxxv. 1868, p. 195.