189 
1906-7.] Dr W. Peddie on Vibrating Systems. 
the meaning of the Boltzmann-Maxwell doctrine, as applied to a single 
particle moving, under none but collisional forces, on a smooth surface 
whose boundary is such as to ensure fulfilment of Maxwell’s assumption 
regarding phases, is that a geodetic drawn long enough will pass infinitely 
near every point within the boundary, as often in nearly one direction as in 
another, as often within a given small range in the neighbourhood of one 
point as in that of another. The possibility of passage infinitely near any 
point in infinitely nearly any direction being assumed, the doctrine asserts 
absolute equality in space distribution and direction distribution. To settle 
this point, Lord Kelvin has introduced experimental tests (. Baltimore 
Lectures, App. B) which indicate marked deviation from the Boltzmann- 
Maxwell condition. Lotteries were used to bring in the random element 
which the kinetic theory requires. 
In a recent address to the American Association for the Advancement 
of Science, Professor W. F. Magie says, “ I do not pretend to be able to show 
that these results of Lord Kelvin are of no value as evidence against the 
truth of the theorem, but I would remark that we can at least justify a 
doubt about them by noticing how small a deviation in the experiments 
from perfect impartiality of conditions will suffice to produce a large 
deviation from the expectation of the theory of probabilities ” {Science, 
Feb. 2, 1906). He then shows that great bias may arise through slight 
inequalities in the sizes of cards in a pack, and adds, “ I think we may fairly 
suppose that the discrepancies of 15 per cent, or so, which appeared in his 
experiments, may have been due, not to a failure of the theorem of equi- 
partition, but to trifling departures from impartiality in his methods of 
experimentation.” 
We shall consider specially the case, dealt with by Lord Kelvin in § 38 
of the Appendix, of a plane surface with a circular boundary of infini- 
tesimal semicircular corrugations hollow inwards. He says, “ The Boltz- 
mann-Maxwell doctrine asserts that the time-integrals of the kinetic 
energies of the two components, radial and transversal, according to polar 
co-ordinates, would be equal.” In a set of 143 flights, in which the total 
time-integral of kinetic energy was 121 3, he found an excess of the radial 
over the transverse component amounting to 107, and remarks, “ Out of 
fourteen sets of ten flights I find that the time-integral of the transverse 
component is less than half the whole 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 ultimate time-integral 
of the transverse component is certainly smaller than the time-integral of 
the radial component.” 
