163 



The (Mitropy of a system of X oscillators is also shown to be 



Sn = -kN SPn log Pn«, and since Pn = (1 — ■o)>i.-ri, 

 o 



1 dS k 



— = = — log 



T dV E 



or since E = hv, U 



hv 

 l^ + 1 



hv 



Finally, the vibration intensity of black-body radiation is 



1 1 

 I = . 



P hv 



L kT __ 



IJ 



When T = 0, I = 0, but U = — . This ''Energierest," or energy residue ^ 



2 



is independent of the temperature, and is of importance in comiection with 



specific heats and radio-active transformations. Planck's radiation formula 



readily follows from the above equation. 



Planck finds the following values for k and ^. k = 1.35 -^ 10^"^, and 



h = 6.55 ^ 10"-" in C. C!. S. units. Using these values together with 



e 



— = 1.77 -f- 10«c, e = 4.69 ^ IQ-", c, e = 4 6SXlS-i»,and c = 3 X 10'", the 



mean number of undisturbed vibrations is found to be 1.37 10". Xe , 



14600 

 or 1.37 X 10'. e ^'^ , if X is measured in [j.. In the same units the emission 



14600 

 niunber of an oscillator per second is 2. 18 X 10" e '" , and the mean ac- 



14600 



cumulation time is 4.58 X 10-*. "a- e '"^ . The equations used to obtain the 

 three preceding results are, respectively: 



Number of vibrations 



"^'■'^^ .kT, 



Sx^v 



Emission number = 



8x-v- 

 3c'L 



hv 

 kT 



'Berliner Ber., 5, 13, July, 1911. 

 'Ann. der Physik, 26, p. .30, 1908. 



