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which is itself perpendicular to F. If q denotes the velocity perpendicular to F of the particle, or of the centre of inertia of the rotator, at the instants of crossing F at the beginning and end of the flight, and if 2 denotes the mass of the particle or of the rotator so that the kinetic energy is the same as the square of the velocity, the time-integral is in case (a) ^q2T and in case (#) the time of the flight being denoted
in each case by T. In both (a) and (/3), § 46, if we call 1 the velocity of the particle, which is always the same, we have q2 = sin2#, and the other component of the energy is cos2#.
In § 47 it is convenient to call the total energy 1; and thus 1—q2 is the total rotational energy, which is constant throughout the flight. Hence, remembering that the times of flight are all the same in case (a) and are proportional to the value of q in case (/3); in case (a), whether of § 46 or § 47, the time-integrals of the kinetic energies to be compared are as ^2g2 to 2(1 — q2), and in case (/3) they are as £2<?3 and 2g(l — q2)m
Hence with the following notation—
In § 46 /^ime-integral kinetic energy perpendicular to F, = V
translatory energy=Y, rotatory „ = R.
V-R V + R
2(|o2-l) . , s
r = 20^i?)mCa9e(a)’
_S(f <?*-!) 2(1 -W)
§ 49. By the processes described above, q was calculated for the single particle and corrugated floor (§ 46), and for the rotator of two equal masses each impinging on a fixed plane (§§ 41, 42), and for the biassed ball (central and eccentric masses 100 and 1 respectively, §§41, 43). Taking these values of summing q, q2, and qz for all the flights, and using the results in § 48, we find the following six results : Phil. Mag. S. 6. Vol. 2. No. 7. July 1901. D
Single particle bounding from corrugated floor (semicircular hollows), 143 flights :—
V —U f = +*197 for isochronous sinusoidal flights.
V + U L = +‘136 for gravitational parabolic „
Rotator of two equal masses, 110 flights :—
V—R f =r —'179 for isochronous sinusoidal flights.
V + R \ = “*-’150 for gravitational parabolic „
Biassed ball, 400 flights :—
V — R f = + '025 for isochronous sinusoidal flights.
V + R \ = —’014 for gravitational parabolic „
The smallness o£ the deviation of the last two results from what the Boltzmann-Maxwell doctrine makes them, is very remarkable when we compare it with the 15 per cent, which we have found (§ 43 above) for the biassed ball bounding free from force, to and fro between two parallel planes.
§ 50. The last case of partition of energy which we have worked out statistically, relates to an impactual problem belonging partly to Class I., § 28, and partly to Class II. It was designed as a nearer approach to practical application in thermodynamics than any of those hitherto described. It is, in fact, a one-dimensional illustration of the kinetic theory of gases. Suppose a row of a vast number of atoms, of equal masses, to be allowed freedom to move only in a straight line between, fixed bounding planes L and K. Let P the atom next K be caged between it and a parallel plane 0, at a distance from it very small in comparison with the average of the free paths of the other particles ; and let Q, the atom next to P, be perfectly free to cross the cage-front C, without experiencing force from it. Thus, while Q gets freely into the cage to strike P, P cannot follow it out beyond the cage-front. The atoms being all equal, every simple impact would produce merely an interchange of velocities between the colliding atoms, and no new velocity could be introduced, if the atoms were perfectly hard (§ 1 i> above), because this implies that no three can be in collision at the same time. [ do not, however, limit the present investigation to perfectly hard atoms. But, to simplify our calculations, we shall suppose P and Q to be infinitely hard. All the other atoms we shall suppose to have the property defined in § 21 above. They may pass through one another in a simple collision, and go asunder each with its previous velocity unaltered, if the differential velocity be sufficiently great;