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out of the 199 points dividing AB into 200 equal parts.
This was done by taking 100 cards *, 0, 1 ..... 98, 99, to
represent distances from the middle point, and, by the toss of a coin, determining on which side of the middle point it was to be (plus or minus for head or tail, frequently changed to avoid possibility of error by bias). The draw for one of the hundred numbers (0 . . . . 99) was taken after very thorough shuffling of the cards in each case. The point of entry having been found, a large-scale geometrical construction was used to determine the successive points of impact and the inclination 0 of the emergent path to the diameter AB. The inclination of the entering path to the diameter of the semicircular hollow struck at the end of the flight, has the same value 0, If we call the diameter of the large circle unity, the length of each flight is sin 0. Hence, if the velocity is unity and the mass of the particle 2, the time-integral of the whole kinetic energy is sin 0; and it is easy to prove that the time-integrals of the components of the velocity, along and perpendicular to the line from each point of the path to tho centre of the large circle, are respectively 0 cos 0, and sin 0—0 cos 0. The excess of the latter above the former is sin 0 — 20 cos 6. By summation for 143 flights we have found,
2 sin (9 = 121*3 ; 220cos6= 108’3 ;
2 sin 0 — 220 cos 0=13*0.
This is a notable deviation from the Boltzmann-Maxwell doctrine, which makes 2 (sin 0 — 0 cos 0) equal to 20 cos 0. We have found the former to exceed the latter by a difference which amounts to 10*7 of the whole 2 sin 0.
Out of fourteen sets of ten flights, I find that the time-integral of the transverse component is less than half the w hole in twelve sets, and greater in only two. This seems to prove beyond doubt that the deviation from the Boltzmann-Maxwell doctrine is genuine; and that the time-integral of the transverse component is certainly smaller than the time-integral of the radial component.
* I had tried numbered billets (small squares of paper) drawn from a bowl, but found this very unsatisfactory. The best mixing1 we could make in the bowl seemed to be quite insufficient to secure equal chances for all the billets. Full sized cards like ordinary playing-cards, well shuffled, seemed to give a very fairly equal chance to every card. Even with the full-sized cards, electric attraction sometimes intervenes and causes two of them to stick together. In using one’s fingers to mix dry billets of card, or of paper, in a bowl, very considerable disturbance may be expected from electrification.
§ 39. It is interesting to remark that our present result is applicable (see § 38 above) to the motion of a particle, flying about in an enclosed space, of the same shape as the surface of a inarlin-spike (fig. 7). Symmetry shows, that the axes of maximum or minimum kinetic energy must be in the direction of the middle line of the length of the figure and perpendicular to it. Our conclusion is that the time-integral of kinetic energy is maximum for the longitudinal component and minimum for the transverse. In the series of flights, corresponding to the 143 of fig. 6, which we have investigated, the number of flights is of course many times 143 in fig. 7, because of the reflections at the straight sides of the marlin-spike. It will be understood, of course, that we are considering merely motion in one plane through the axis of the marlin-spike.
§ 40. The most difficult and seriously troublesome statistical investigation in respect to the partition of energy which I have hitherto attempted, has been to find the proportions of translational and rotational energies in various cases, in each of which a rotator experiences multitudinous reflections at two fixed parallel planes between which it moves, or at one plane to which it is brought back by a constant force through its centre of inertia, or by a force varying directly as the distance from the plane. Two different rotators were considered, one of them consisting of two equal masses, fixed at* the ends of a rigid massless rod, and each particle reflected on striking either of the planes ; the other consisting of two masses, 1 and 100, fixed at the ends of a rigid massless rod, the smaller mass passing freely across the plane without experiencing any force, while the greater is reflected every time it strikes. The second rotator may be described, in some respects more simply, as a hard massless ball having a mass = 1 fixed anywhere eccentrically within it, and another mass = 100 fixed at its centre. It may be called, for brevity, a biassed ball.