only ones which must be taken into account in an exhaustive treatment of the subject. In a system of bodies free from external action, the sum of the energy and the velocity of the centroid of the whole system are not the only invariable magnitudes; but there are others, too, that always have the same value. According to the theorem of the conservation of areas the moments of the momenta of the system about three axes at right angles to each other have also constant values. By the method here employed it is easily possible to take this proposition of mechanics also into account in the calculation.

If we denote by x, y, z the rectangular coordinates of a particle of mass m whose velocity is made up of the components u, v, w in the directions of the system of coordinate axes, this general proposition of mechanics is expressed by the equations.

c1 = .m(yw-zv)
C2.m(zu-xw)
C3=Σ.m(xv-yu),

in which the magnitudes c are independent of the time and position, and the summations are taken over all the particles of the whole mass. We may either refer the coordinates x, y, z to a fixed system of axes, or choose as origin of coordinates the centroid of the whole gaseous mass, which moves on with unchangeable velocity: we will do the latter in order to gain the advantage of obtaining formulæ which refer only to rotations. about the centroid.

The new formulæ differ from those established earlier in § 12* by containing the coordinates as well as the components of velocity, yet they can be introduced as equations of condition in the same way as those which refer to the energy and the motion of the centroid.

For, to express the fact that the occurrence of other values u + du, v + dv, w + Sw of the velocities has the same degree of probability, provided that they satisfy the laws of mechanics, we have to add to the formulæ before developed the conditions

c1 = Σ.m{y(w + Sw) − z(v + dv)}

c2 = Σ.m{z(u + du) − x(w + Sw)}
c3 = Σ.m{x(v + dv) − y(u + du)};

in these formulæ only the velocities u, v, w, and not the coordinates x, y, z, are varied, since among the changed values

u + Su, v + Sv, w + Sw those velocities are to be understood which replace the original values u, v, w at the same place (and therefore for the same values of x, y, z), but at a different time. From both systems of formulæ it follows that to the equations of condition, to which the variations du, dv, dw are subject, must be further added the three equations

We take these formulæ into account by multiplying them by three provisionally undetermined but constant factors, which we will denote by - 2kg, - 2kn, - 2k, and adding them to the former formulæ, and then the coefficient with which each of the 3N variations Su, Sv, Sw appears to be multiplied in the sum of the equations is to be put equal to 0.

In this way, instead of the three differential equations that stand at the end of § 12", we obtain the more general equations

which may be integrated when u, v, w are taken to vary without limit and x, y, z to be constant.

This integration does not need to be carried out in order to let us see that the result of the calculation will only differ from that in the former case by the magnitudes

having to be subtracted from the variables u, v, w as well as the constants a, ẞ, y. The magnitudes have a similar meaning to those of a, ß, y. While a, ß, y were found equal to the values of the components a, b, c of the velocity of the centroid, §, ŋ, , as it is easy to see, are the values of the angular velocities with which the gaseous mass rotates about the axes of x, y, z; for if this is their interpretation, the sum

a + zn — ys

η,

is the velocity with which a particle at the point x, y, z advances in the direction of the axis of x while rotating about the axis of y from the z-axis to the x-axis with the angular velocity 7, and also about the axis of z from the x-axis to the y-axis with the angular velocity . In exactly similar ways are the two other magnitudes to be interpreted. The formula therefore express motions of the gaseous mass which correspond to those of a nut which moves on a screw-spindle that itself moves along and turns about a second axis.

These kinds of motion of the gaseous mass are, as our examination shows, to be simply subtracted from the molecular motion present in order for us to arrive directly at Maxwell's law of distribution for the state of rest. Hence it follows that the individual motions of the molecules are not disturbed if in addition to a forward translatory motion of a gas there are also rotations about any axis in which the gas as a whole takes part. The actual velocities of a particle are thus made up of three parts: firstly, of the motion which the particle would have in accordance with Maxwell's law if the gas were at rest as a whole; secondly, of the velocity with which the centroid of the whole gas moves; and, thirdly, of the motions which it has in taking part in the general rotation with the mean values §, 7, ( of the angular velocities.

η,

Thus the kinetic energy of the molecular motions breaks up into three parts, and the mean energy of a molecule at the point x, y, z is

+ }m {(a + zn — y§)2 + (b + x5 — z§)2 + (c + y§ — xn)2}.

When the whole amount of energy is known this formula may be used for the determination of the constant k.

In order to obtain Maxwell's law of distribution in its simplest form, we have, according to the foregoing discussion, to subtract from the components u, v, w of the molecular velocity, not always equal values of speed for all the different particles which belong to the system, but for each particle the values of the velocity-components which belong to the whole gas at the point where it actually is. This remark discloses the possibility of widening still further the limits of the region wherein Maxwell's law holds good.

In the simple cases which we have considered the validity of this process would have been easy to see even without mathe

C C

matical proof; for a gaseous body executes a motion of the centroid and rotations about fixed axes exactly as if it were solid. But if in a gaseous mass there are layers which shift together with unequal velocity or rotate unequally quickly, our formulæ which have been used in the proof are no longer valid with absolute strictness, since the energy within any layer need remain constant as little as any other of the observed magnitudes; for the propositions are only strictly true for the gas as a whole. We may, however, look upon them as approximately valid if the interchange of energy or of velocity between the layers occurs only slowly. With this assumption, which is permissible if the differences in the motion of neighbouring layers are small enough, each of the mechanical theorems remains valid with sufficient exactness even for a single layer within an interval of time that is not too long. At the same time, this interval may be long enough to allow the very rapidly resulting arrangement in the distribution of the velocities according to Maxwell's law to occur. In such cases, therefore, the law must also hold good if the gas is divided into unequally moved layers. In each of these layers, then, Maxwell's law holds good for the distribution of the velocities on this condition, that on each occasion the velocity in which a particle shares by the flow or rotation of its layer is to be subtracted from the value which the particle would have in the state of rest and equilibrium of the gas as a whole.

18*. Transformation of Coordinates

In using Maxwell's law it is often of advantage to give it another form by a transformation of coordinates. Naturally it is not always necessary to know how many molecules have a velocity the components of which are u, v, w, that is, a velocity of given magnitude and direction; far oftener the question arises as to the number of particles which possess a given speed, that is, a motion of given magnitude without reference to direction.

This question is answered if we pass over to polar coordinates from the system of rectilinear coordinates to which the components u, v, w are related, and therefore introduce the absolute velocity

w = √(u2 + v2 + w2),

Tait, Trans. Roy. Soc. Edin. xxxiii. 1886, p. 82; Natanson, Wied. Ann. xxxiv. 1888, p. 970.

whose direction is given by the angles s and with reference to a fixed axis, such that

u = w cos s,

vw sin s cos,

ww sin s sin p.

In order not unnecessarily to complicate the calculation, which I wish to carry out without the limiting assumption of a state of rest, I take the position of the coordinate system, which so far has been left arbitrary, such that the axis of u, which is also that of the polar system of coordinates, coincides with the direction of the absolute velocity of translation of the whole system

• = √(a2 + b2 + c2).

Then in the former formulæ o enters instead of a, while b and c vanish altogether. Since, further, the element of volume is now given by the expression w2dw sin s ds do, we have, instead of the first formula of § 16*, the new one

n = · N(π ̄1km)} e − km(w3 — 2wo cos s + o2) w2dw sin s ds do,

and this gives the number of molecules which out of every N move with the velocity in the direction given by s and .

From this we obtain by integration the whole number of all those which move with a speed lying between w and w + dw, viz.

With the special assumption that there is no translatory motion, i.e. that the gas is at rest as a whole or o= 0, this formula, found by Maxwell, becomes

This new formula differs from the former formula in § 16* by an important circumstance; while the latter showed a continuous diminution of the probability as the values of the components increased, this has a maximum which occurs for the value

kmw2 = 1

or

w = (km)1 = W.