831 
We will therefore suppose that a changes slowly in the sense of 
the theory of adiabatic invariants. Let Da be the total change of a, i.e. 
De = if dt 
and let De; and Dt, represent the corresponding changes of ¢; and ¢z;_ 
considering further that 
a= const, 
we find 
De, _ 0 (OV mes 6 
meme) TTT 
where the horizontal line indicates the time-mean. If in equation 
(5) we take c’; and ¢’; to mean these time-means, we obtain 
0 n (de De. Ao Dt. 
Ee hes ees ie 1843 beurre (D) 
da ix1 \de, Da dt, Da 
Since @ is independent of ¢, the corresponding term under the 
summation-sign in (7) must be omitted. 
§ 3. Phase-space and adiabatic invariants. 
The stationary density @ need not depend on all the variables 
Cowie) oa Pres bo ee ges og tn 
For example in a conditionally-periodic system without commen- 
surable relations between the periodicity-moduli @ depends on the 
quantities c; only. This follows from the theorem which allows us 
to replace the time-average by an averaging over a @-cell*). For 
an ergodic (or quasi ergodic) system in consequence of the ergode- 
hypothesis 9 depends on the energy c, only. We shall here suppose, 
that @ depends on & quantities (k Sn), which we shall indicate by 
Ein Cay = wann Oke 
These integrals may be called essential integrals. Our supposition 
with regard to o comes to the same as assuming that for our sys- 
tem the time-mean may be replaced by a definite numerical mean. 
To compute this we proceed as follows. 
Suppose the system of equations 
Etn ee er ane is |) 
to be soluble with respect to p‚, ps, --- «> Pk, thus 
1) J. M. Burgers. lc. and my paper l.c. 
