TRANSACTIONS OF SECTION A. 409 
The electrical moment Z (¢) of a resonator vibrating parallel to the Z-axis is deter- 
mined by the equation® 
16m'v09Z + 4e've2L—I2Z=200Er 2 wl, (2) 
where ¥ is the frequency of the resonator, o its logarithmic decrement, c the velocity 
- of light and Ez the Z component of the electric intensity of the heat radiation at 
the resonator. 
Writing E, as a Fourier series for the time 7 between two collisions, 
0,00 
E;= = Cn (cos On ) Be EE 8 0: 3) 
n = 
and putting ~ = v we obtain the complete integral of (2) in the form 
Z (t) = Zo(t) + 2n(t) 
= Pe —o%+ cos (farvot — 8) + 
3c* 9, Cy sin Ya ee 
lon : PPT) cos (Savy — On — Yn) . - (4) 
V9 (Vo? — v? 
where cot yn = aie ide 
The rate at which energy is absorbed by the resonator is 
Fz. —pe — ov% { g cos (2ryvf—9,) + 2m sin (2mvot — 45) \ PE,z+ E.Zn (5) 
Out of all the resonators which suffered the last collision at the time t=o let us 
select a group in which, at the time ¢, Pand Ez have the same value for every resonator 
in the group. 
The part of the mean value of E: Z or this group, which depends on the first 
term of (5), vanishes if we assume that all values of @ are equally probable. 
But all the resonators can be arranged in groups in a similar manner; so that the 
mean value of the rate of absorption is independent of the free vibrations. 
The mean rate at which energy is absorbed by a very large number of resonators, 
N, is therefore 
és 3c N oS? CR sin 
N Es oc . n n k 
pee l€m x v : , * (6) 
Reasoning in the same way as Planck,‘ we arrive at the equation 
2 
iy =e" Nip iuttrei “cnwbire ised! ok t 206-107) 
Vo 
where Ky is the intensity of the monochromatic plane polarised radiation of frequency v. 
The mean emitted energy can be calculated in a similar manner. A resonator 
whose energy is me emits per second, on the average, the energy 
Qm = 2me vp ; : : ‘ ee (8) 
Out of N resonators 
Nn =N (emma “OD S) Jamasatind d:(8) 
vibrate with the energy me at the temperature 7. 
The total energy emitted per second by the N resonators is therefore 
0, co 
= Qm Nm = 2avoN 
m 
€ 
(10) 
= 
ext —1 
8 Planck, ‘ Theorie d. Warmestrahlung,’ p. 113. 
* «Theorie d. Warmestrahlung,’ p. 122 (1906). 
