1900.] on the Dynamical Theory of Heat and Light. 385 



is genuine, and the discrepance is somewhat approximately of the 

 amount and direction indicated. I am supported in this view by 

 scrutinising the thirty sums for successive sets of twenty nights : 

 thus I find 2 I cos 2 6 to be positive for eighteen out of thirty, and 

 2 I sin 2 to be negative for nineteen out of the thirty. 



§ 38. A very interesting test-case is represented in the accompany- 

 ing diagram, Fig. 6 — a circular boundary of semicircular corrugations. 

 In this case it is obvious from the symmetry that the time-integral 

 of kinetic energy of component motion parallel to any straight line 

 must, in the long run, be equal to that parallel to any other. But the 

 Boltzmann-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. To test this, I 

 have taken the case of an infinite number of the semicircular corruga- 

 tions, so that in the time-integral it is not necessary to include the 

 times between successive impacts of the particle on any one of the 

 semicircles. In this case the geometrical construction would, of 

 course, fail to show the precise point Q at which the free path would 

 cut the diameter AB of the semicircular hollow to which it is ap- 

 proaching ; and I have evaded the difficulty in a manner thoroughly 

 suitable for thermodynamic application such as the kinetic theory 

 of gases. I arranged to draw lots for 1 out of the 199 points divid- 

 ing 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 6 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 6. 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 ; 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 the centre of the large circle, are 

 respectively 6 cos 9, and sin 6 — 6 cos 8. The excess of the latter 



* I had tried numbered billets (small squares of paper") drawn from a bowl, 

 but found this very unsatisfactory. The best mixing 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 con- 

 siderable disturbance may be expected from electrification. 



