Dynamical Tlieory of Heat and Light. 33 



which is itself perpendicular to F. If a denotes the velocity 

 perpendicular to F of the particle, or of the centre of inertia 

 of the rotator, at the instants of crossing F at the beginning 

 and end of the flight, and if 2 denotes the mass of the particle 

 or of the rotator so that the kinetic energy is the same as the 

 square of the velocity, the time-integral is in case (a) ^</ 2 T 

 and in case (#)^</ 2 T, the time of the flight being denoted 

 in each case by T. In both (a) and (/3), § 46, if we call 1 

 the velocity of the particle, which is always the same, we have 

 2 2 = sin 2 0, and the other component of the energy is cos 2 #. 

 In § 47 it is convenient to call the total energy 1 ; and thus 

 1— q 2 is the total rotational energy, which is constant 

 throughout the tlight. Hence, remembering that the times 

 of flight are all the same in case (a) and are proportional to 

 the value of q in case ((3) ; in case (a), whether of § 46 or 

 § 47, the time-integrals of the kinetic energies to be compared 

 are as ^%q 2 to 2(1 — q 2 ), and in case (/3) they are as J-2<? 3 

 and 2^(1 — q 2 ). 



Hence with the following notation — 



t c ... f Time-integral of kinetic energy perpendicular to F, = V. 

 [_ r >? » parallel to r,= U. 



InS47 r » translatory energy = V, 



in54 '\ „ rotatory „ = R. 



we have 



v^r = s(w?) mca9e(a)j 



v+ul = %s 



(fi), 



V-R 



, S(fg 2 -l) 



${q-*q*) 



08). 



§ 49. By the processes described above, q was calculated 

 for the single particle and corrugated floor (§ 46), and for 

 the rotator of two equal masses each impinging on a fixed 

 plane (§§ 41, 42), and for the biassed ball (central and 

 eccentric masses 100 and 1 respectively, §§ 41, 43). Taking 

 these values of q, summing q y ,f\ and g z for all the flights, and 

 using the results in § 48, we rind the following six results : 



Phil. Man. S. 6. Vol. 2. No. 7. July 1901. D 



