1900.] on the Dynamical Theory of Heat and Light. 387 



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. 



§ 41. In every case of a rotator whose rotation is changed by an 

 impact, a transcendental problem of pure kinematics essentially 

 occurs to find the time and configuration of the first impact ; and 

 another such problem to fiud if there is a second impact, and, if so, 

 to determine it. Chattering collisions of one, two, three, four, five or 

 more impacts, are essentially liable to occur, even to the extreme case 

 of an infinite number of impacts and a collision consisting virtually 

 of a gradually varying finite pressure. Three is the greatest number 

 of impacts we have found in any of our calculations. The first of 

 theso transcendental problems, occurring essentially in every case, 

 consists in finding the smallest value of which satisfies the equation 



6-i = —(1 -sin 5); 

 v 



where w is the angular velocity of the rotator before collision ; a is 

 the length of a certain rotating arm ; i its inclination to the reflecting 

 plane at the instant when its centre of inertia crosses a plane F, 

 parallel to the reflecting plane and distant a from it ; and v is the 

 velocity of the centre of inertia of the rotator. This equation is, in 

 general, very easily solved by calculation (trial and error), but more 

 quickly by an obvious kinematic method, the simplest form of which 

 is a rolling circle carrying an arm of adjustable length. In our 

 earliest work we performed the solution arithmetically, after that 

 kinematically. If the distance between the two parallel planes is 

 moderate in comparison with 2 a (the effective diameter of the 

 rotator), i for the beginning of the collision with one plane has to be 

 calculated from the end of the preceding collision against the other 

 plane by a transcendental equation, en the same principle as that 

 which we have just been considering. But I have supposed the 

 distance between the two planes to be very great, practically in- 

 finite, in comparison with 2 a, and we have therefore found i by 

 lottery for each collision, using 180 cards corresponding to 180° of 

 angle.' In the case of the biassed globe, different equally probable 

 values of i through a range of 360° was required, and we found them 

 by drawing from the pack of 180 cards and tossing a coin for plus or 

 minus. 



§ 42. Summation for 110 flights of the rotator, consisting of two 

 equal masses, gave as the time-integral of the whole energy, 200 ■ 03, 

 and an excess of rotatory above translatory, 42*05. This is just 

 21 per cent, of the whole ; a large deviation from the Boltzmann- 



