510 Dr. N. Bolir on the Efect of 



the kinetic theory of gases. For higher values of n, how- 

 ever, 2a is great compared with the values of ordinary atomic 

 dimensions. As I pointed out in ray former paper, this 

 result may he connected with the non-appearance in vacuum- 

 tubes of hydrogen lines corresponding to high numbers in 

 Balmer's formula and observed in the spectra of stars. 

 Further, it will appear from the considerations of the next 

 section that the large diameter of the orbits offers an expla- 

 nation of the surprisingly great magnitude of the Stark 

 effect. 



From (10) it appears that the condition (8) holds, not only 

 for large values of n but for all values of n. In addition, for 

 a stationary orbit W is equal to the mean value of the total 

 kinetic energy T of the particles ; from (10) we therefore 

 get 



T n =in7ia> n (11) 



In using the expressions (6) we have assumed that the 

 motion of the particles in the stationary states of the system 

 can be determined by help of the ordinary mechanics. On 

 this assumption it can be shown generally that the conditions 

 (8) and (11) are equivalent. Consider a particle moving in 

 a closed orbit in a stationary field. Let co be the frequency 

 of revolution, T the mean value of the kinetic energy during 

 a revolution, and — W the mean value of the sum of the 

 kinetic energy and the potential energy of the particle 

 relative to the stationary field. Applying Hamilton's prin- 

 ciple, we get for a small variation of the orbit 



8W=-2fi>8^) (12) 



If the new orbit is also one of dynamical equilibrium, we get 

 8A= — SW, where A is the total energy of the system, and 

 it will be seen that the equivalence of (8) and (11) follows 

 immediately from (12). 



In these deductions we have made no assumptions about 

 the degree of eccentricity of the orbits. If the orbits are 

 circular (11) is equivalent to the simple condition that the 

 angular momentum of the system in the stationary states is 



equal to an entire multiple of — *. 



In Planck's vibrators the particles are held by quasi- 

 elastic forces, and the mean value of the kinetic energy is 



* Comp. J. W. Nicholson, Month. Not. Roy. Astr. Soc. Ixxii. p. 679 

 (1912). 



