836 
sion Ap (p > 1), since the dimension of / is Jh". In order to illu- 
strate this and the previous results we shall contrast the properties 
of a BontzMann ergode and a conditionally periodic system without 
commensurable relations: 
ergode conditional periodic system 
essential integrals 
Ht 0 Heite ne 
numerical mean 
Op, Ogee, u 
ff aen dn … dan SE ‚ (23) ffe aerden Ten (23') 
density 
e = 0(c,. 4) | oo lt ern) 
essential adiabatic invariants 
V= fap ..- dqn 
H<e 
vip dg = 1, 2, ape 
1 2 2 
0 
density 
vy=e¢(V) | 0 = 0 (P,P, ++. 5%) 
The conditionally periodic system is what BOLTZMANN calls a sub- 
ergode. On the other band the ergode appears as a coherent system 
with a smallest value of 4, viz. k=—=1. These short indications may 
suffice for the present. 
$ 5. Degeneration. 
A conditionally periodic system is called degenerated, if there 
are commensurable relations between the periodicity moduli. It is 
evident, that our system covers a lower set of points with its 
orbital curve everywhere densely, than when there are no such 
relations. Accordingly the; numerical mean will be of a lower 
dimension and more quantities will remain free after the averaging 
process. Thus besides the quantities c the quantities ¢ will play a 
part: the number of essential adiabatic invariants becomes larger 
than the numberof degrees of freedom. The question, whether these 
supernumerary quantities have to be quanticized, we shall not discuss 
here. A good instance for the discussion of the questions which may 
arise here is afforded by the quanta-quantities in Epsrrin’s theory ') 
of the Srark-effect for an infinitely weak external electric field; the 
“parabolic” fquanta-quantities which are found in this case cannot 
1) P. Epstein. Ann. d. Phys. 50. p. 490. 
