1123 



the integration extending over the region enclosed by the energy- 

 surface H =: c^. We thus have 



dV dV- 



^ -f ^ c', = . . . . . . . . (47) 



da dc, ' 



hence V is an adiabatic invariant. It can also easily be shown 

 that this quantity has a meaning which is independent of the system 

 of co-ordinates used; it therefore also satisfies the condition mentioned 

 in the introduction under (3). The same is true for the quantity 

 called V in § 3, b. 



It remains to be seen under what conditions the quantities v^ 

 defined by equation (32) also satisfy this requirement. It may be 

 expected that this enquiry will teach us how to quanticize systems 

 which are "degeneiated" in different ways. It also seems very pro- 

 bable, that this question will be decided on the lines indicated by 

 Planck ') and Schwahzschild '). For instance, as regards the movement 

 of a top on which tio exteinal forces are acting, of the three adia- 

 batic invariants: the moment of momentum, its projection on the 

 axes of the figure and its projection on S-axes of a fixed system of 

 coordinates of arbitrary orientation (all three multiplied by 2jr) only 

 the first two may be quanticized. The "elementary region" thus 

 will be not h^ but A' {2)i^ -\- 1), where ii^ is the quantum-number 

 corresponding to the moment of momentum. On this ground excep- 

 tion may be taken to Epstein's calculation of the specific heat of 

 hydrogen'). To all these problems — problems relating to the 

 adaptation of the quantum-hypothesis to different cases — 1 hope 

 to return soon. 



The method above developed is independent of this question, it 

 is the solution of a purely mechanical problem. It seems advisable 

 to try and apply it to systems which cannot be integrated by a 

 separation of the variables in Hamilton-Jacobi's partial differential 

 equation, e.g. to the PoiNsoT-motion. About this question also I hope 

 to be able to make a communication shortly. 



Petrograd, September 1918. Physical Laboratory of 



the Uiiiversity. 



1) M. Planck. 1. c. 



2) K. Schwarzschild. Silzungsber. Berlin 1916, p. 550. 



») P. S. Epstein. Ber. d. D. Phys. Ges. 1916 p. 398. Compare especially 

 (10) on p. 401. Objections may also be made to the quanticizing proposed on 

 p. 407, seeing that the quantum-quantities in that case are not adiabatic invariants, 



