Dynamical Theory of Heat and Light. 29 



§ 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 find if there is a 

 second impact, and, if so, to determine it. Chattering col- 

 lisions 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 these transcendental problems, occurring essen- 

 tially in every case, consists in finding the smallest value of 6 

 which satisfies the equation 



0-i=°^(l-smO); 



where a> 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 2a (the effective dia- 

 meter 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, 

 on the same principle as that which we have just been con- 

 sidering. But I have supposed the distance between the two 

 planes to be very great, practically infinite, in comparison 

 with 2a, 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 Boltzmanr-Maxwell doctrine, which makes the time- 

 integrals of translatory and rotatory energies equal. 



