161 



(12)') 



3) 



ö^.s km 



Hence (11) changes into : 

 1 1 



o o 



ƒ 



If &^ changes from O to 1, t' increases by >y, hence qi describes 

 I's full periods^). The expression (13) now becomes: 



If this is introduced into eq. (10), it is found that : öYs =: ; it 

 has thus been shown that Ys is an invariant for the adiabatic disturb- 



%--T— (14) 



da 



ances considered. 



Remarks. 



1. It has been pointed out by Schwarzschild and Epsteik ') that 

 the total energy «^ of the system, when expressed as a function of 

 the lie, depends only on the linear integral combinations Yi^oiiheH^; 

 this is a consequence of the equations (3). From this it follows that 

 it is alwai/s possible to fix the value of the energy by "quantizising" 

 the adiabatic invariants (i.e. by equating the adiab. inv. to integral 

 multiples of Planck's constant). 



2. In the equations : Yg = ^r, . h = adiabatic invariant, an 

 arbitrary primitive system of solutions of eq. (5) may be chosen for 

 the system of coefficients : r\ . . . ;•" {s=:'i . . .n — X). All such systems 

 are connected together by linear integral substitutions, the determinant 

 of which is equal to rb 1. . Hence the same holds for the sets of 

 n — ;. independent Yj^ : if Y[ . . . FU, and Y; . . . Y'„-, be two 

 of these sets, we have : 



1) This follows from the equation: 



dgi = S f I'» dt,n = 2 ƒ "» . a)k,n • dr^ = 2! ƒ '"' . lokm ■ Vs . d^ . 



m 711 k inks 



5) In the diagram of complex values of qi (cf. A. Sommerfeld, Phys. Zeitschr. 

 17, p. 500, 1916) the path of integration goes is limes round the branch points 

 qi= ti^qi = VII of the function pi=^ Fiiqi). 



3) K. Schwarzschild, Sitz. Ber. Berl. Akad. p. 550, 1916. 



P. Epstein, Ann. d. Phys. 51 p. 180, 1916. 



