582 
describes a closed ') curve in the 27 dimensional (q,p) space and its 
n projections on the 2 dimensional planes (q,,p‚), a> pa) «++» (Uns Pn) 
describe 2 closed curves. 
{fem dp, is the area enclosed by the 4 projection curve. 
1 
Remarks: A. The numerical value of — does not change if in 
Vv 
the description of the motion we pass from one system of coordi- 
nates g, ... q„ and the corresponding p, ... p„ to another one q',... q'n 
and the corresponding p',...p', Therefore the right hand side of 
(7) can neither change its fdan value. 
B. There are systems, for which if the system of coordinates has 
been chosen rightly, not only the total sum on the right hand 
side of (7) is an adiabatic invariant, but also the single terms 
| [eden (comp. the example in § 7). In this case we obtain thus 
at once more invariants. 
C. For a system of one degree of liberty : 
27 : 
= =| {aq is an adiabatic invarant (8) 
Y 
according to (7) viz.: for a system of one degree of liberty the area 
enclosed by the phase curve in the (g,p) plane is an invariant (in 
this case there are no other invariants independent of this one). 
D. A theorem of P. Hertz (1910) *). Give definite values a,,,... ayo 
to the parameters and consider some motion compatible with these. 
The corresponding path of phase in the (g, p) space lies on a definite 
hypersurface of constant energy ¢&(q ,p,a@,)=&,. This hypersurface 
encloses a definite 27 dimensional volume: 
f° . 
fess fan s+. d= M,. eN 
In the first place an adiabatically reversible influencing a, a 
works on the system. Secondly the hypersurfaces have now 
another position in the (g,p) space than before. We may now 
consider the volume V of that energy surface on which the phase 
path of the system lies after the adiabatic influencing. The theorem 
of P. Hertz teaches now: 
1) This expression needs further interpretation, if e.g. one of the coordinates is 
an angle and this angle increases each period with 27. 
2) P. Hertz: Mechanische Grundlag. d. Thermod. Ann. d. Ph. 33 (1910) p. 
225—274; p. 537-552. [§ 11. Adiabat. Vorgiinge. Comp. 173]. P. Hertz in 
“Repert. d. Physik” (TEUBNER 1916) p. 535 Comp. (7). 
