122 



SCIENCE 



[N. S. Vol. XXXVII. No. 943 



possibilities, except by conceiving the sys- 

 tem to be made up of a definite number of 

 concrete and definite elements — for a con- 

 tinuum can not have countable elements. 

 Hence, an atomistic structure of the system 

 is a fundamental condition for the repre- 

 sentation of its entropy by a probability. 

 All systems, then, which possess an entropy 

 must possess an atomic stracture. Now ex- 

 periment justifies the carrying over of the 

 entropy concept to an enclosure filled with 

 radiant energy, for it is only in this way 

 that the Stefan-Boltzmann law and the 

 Wien displacement law, both of which are 

 found experimentally to be correct, are 

 deduced. Hence we are forced to conclude 

 that an atomistic structure of some sort 

 must be applied to radiant energy. Planck 

 then proceeds to apply it as follows: He 

 imagines an enclosure having perfectly re- 

 flecting walls to be filled with black-body 

 radiation. In this enclosure, and in equi- 

 librium with the black-body radiation, are 

 linear electromagnetic oscillators of a 

 given frequency v. The relation between 

 the energy Uv in each oscillator of fre- 

 quency V, and the energy per unit volume 

 Uy of black-body radiation of frequency v, 

 is given by the ordinary electrodynamie 

 laws as, 



in which c represents the velocity of light. 

 Now let us call in the idea of atoms of 

 energy and assume that each oscillator con- 

 tains at each instant an exact multiple of 

 an element of energy c. From a considera- 

 tion then of the total number of oscillators, 

 and the total number of energy elements 

 in all the oscillators, we can obtain an ex- 

 pression, as in the case of the dice, for the 

 total number of complexions of the system, 

 that is, the total number of possible dis- 

 tributions of the energy elements among 

 the oscillators. This leads to an expression 



for the entropy of the system of the form 

 C7> 



■(f) ■ 



But the second law of thermodynamics, as 

 applied by Wien," had shown that 



--(?)• 



Hence we must place e = /ii', that is, the 

 energy element e is proportional to the nat- 

 ural frequency v of the oscillator, and the 

 proportionality factor h is a universal con- 

 stant, which Planck calls the Wirkungs 

 quantum. He thus arrives at his celebrated 

 formula for the relation between the den- 

 sity of black-body radiation and frequency, 

 namely, 



ghvlckT J 



or the intensity E\ of black-body radiation 

 of wave-length A, and temperature T, is 



•^ X5(ecA/AAT_l)- 



This formula meets the requirements of 

 passing over, at small values of \T, into 

 Wien's equation, namely, 



2c'h _^ 



Ex 



and for large values of \T, into Rayleigh's 

 equation, namely, 



2ckT 



Ex-- 



To this brief sketch of the origin of 

 Planck's equation should be added the 

 statement that Planck^" finds a further 

 proof of the necessity of taking some such 

 step as that which he has taken in the fault- 

 lessness of Jeans 's logic^^ in .showing that 

 the Hamiltonian equations, combined with 

 the theory of probability, lead inevitably to 

 Rayleigh's radiation equation, which is 

 contradicted by experiment. There -is, 



»Wien, Wied. Ann., 52, p. 132, 1894. 



" Planck, Ann. der Phys., 31, p. 758, 1910. 



" Jeans, Phil. Mag., 18, p. 209, 1909. 



