Lord Kelvin. Nineteenth-Century Clouds over the Dynamical Theory of Heat and Light. // Phil. Mag. S. 6. Vol. 2. No. 7. July 1901.

iind in (/3) they are in simple proportion to the velocity of its centre of inertia when it leaves F or returns to it.

Fig:. 8.

§ 48. In the cases of §§ 46, 47, we have to consider the time-integral for each flight of the kinetic energy of the component velocity of the particle perpendicular to F, and of the whole velocity of the centre of inertia of the rotator,

which is itself perpendicular to F. If q 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) ^q²T and in case (#) 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 q² = sin²#, and the other component of the energy is cos²#.

In § 47 it is convenient to call the total energy 1; and thus 1—q² is the total rotational energy, which is constant throughout the flight. 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 ^2g² to 2(1 — q²), and in case (/3) they are as £2<?³ and 2g(l — q²)_m

Hence with the following notation—

In § 46 /^ime-integral kinetic energy perpendicular to F, = V

translatory energy=Y, rotatory „ = R.

we

have

Y-U

v+ul

V-R V + R

2(|o²-l) . , s

r = 20^i?)^mCa9e(a)’

₌ W-<7)

_S(f <?*-!) 2(1 -W)

_.W-9)

G8),

(«),

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 summing q, q², and q^z for all the flights, and using the results in § 48, we find the following six results : Phil. Mag. S. 6. Vol. 2. No. 7. July 1901. D