581 
may begin with. Here the period ? may still depend in any way 
on d,,d,,.... and on the beginning phase of the motion. 
Then the time integral of the (double) kinetic energy taken over 
the period is an adiabatic invariant : 
P 
a | a2 =0 Rie ch eee eT ae Hal 
0 
é will denote: the difference in the values for two infinitely near, 
adiabatically related motions of the system. [For the proof of (3) 
see appendix I]. If the reciprocal value of the period P is called 
the frequency » and the mean with respect to the time of 7’ 7 
then (3) says: 
27 
Er is an adiabatic invariant . . . . . . (4) 
In the case of a harmonie vibration of one degree of liberty the 
means with respect to the time of kinetic and potential energy are 
known to be equal to each other and therefore also to half the total 
energy, so that: 
& . . . . . 
near adiabatic imvariant:s <40’? «7 4.7) 
YP 
rs i 
§ 5. A geometrical interpretation of the adiabatic — in the (q,p) 
p 
space. Connection with a theorem of P. Hertz. 
In order to find a connection with the quantum hypotheses of 
Puanck, DeBIJE, BonHr and others we shall use a deduction of the 
integral of action to which SOMMERFELD has drawn the attention. ‘) 
ee fe 
a = n 7 = n are 
dT — bat Ni pr an = Va Ldap. pra Yi iqndpn. . (6 
th ala a ee naa 
0 0 
therefore 
hele 
=DE ff tr on. op eee Cred pee eee) 
L 
where the double integrals on the right hand side have the following 
meaning: If the system executes its periodie motion, its phase point 
\) A. SOMMERFELD: Zur Theorie d. Balmerschen Serie, Sitzber. d. Bayr. Akad, 
1916 p. 425 (§ 7), 
