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ADDITIONAL NOTES. 
I. Elucidation of the choice of the u- and y-weight: fu} and }y}. 
(Note to sections 2, 4). 
The definitions of {u} and fy} may be elucidated by a simple 
example. Consider first a PrLANCK-resonator. According to the theory 
of quanta its phase q,p must lie either at ¢g=p—O or on one of 
Pianck’s ellipses e =hv, 2hv,... Now two consecutive ellipses are 
known to enclose a ring whose area is 
ffaap=r. SU DA BEDEA EN SOA TOOL) 
The classical theory would admit all the points of the plane and 
ascribe to any portion «of it a “weight” equal to its area sil dq dp. 
It therefore seems natural in the statistical calculations of the 
quantum-theory to ascribe a weight A to each of the ellipses, % 
particular also to the point q=p=O. Since in all statistical 
calculations it is ultimalely only the relative weight that matters, 
the essential thing about this assumption is, that the same weight 
is ascribed to all the ellipses, which moreover is independent of the 
nature of the resonator (say its »). }) 
The choice of h itself as the weight in question has the following 
advantage in connection with (64): if in the y,p plane any portion 
is considered which contains a large number of ellipses, the total 
weight of all the ellipses inside this region coincides with its area 
owing to (64). ’) 
Let us next consider a material point elastically connected to a 
given position of equilibrium, say anisotropically. Its principal 
vibrations may be parallel to the co-ordinates q,, q,, 7s, its frequencies 
being supposed very different 
EEE ee dN Le G2 1. oy Sos meme) 
1) The choice of the weight must be subjected to certain limitations, in order 
that the statistical theory may not get into contradiction with the IInd law of 
thermodynamics. Comp. P. Exrenrest, Phys. Zeitschr. 15 (1914), p. 657; Ann. de 
Phys. 51 (1916), p. 340, § 8 — Versl. Kon. Ak. v. Wetensch. Amsterdam 25 (1) 
(1916), p. 423, § 8 — Proceedings Amsterdam 19 (2 part.) (1917), p. 576, § 8. — 
The above choice is in accordance with the limiting conditions in question. 
2) Hence for sufficiently high temperatures we shall have approximately 
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