60 Dr. L. Silberstein on the Quantum 



question.) The energy belonging to any circular orbit is 

 given by 



whence also the corresponding frequency formula. 



The orbits responsible for broad "lines" or bands will be 

 all those and those only for which both sin i and e differ 

 from zero, i. e. n 2 ,n s ^0. 



There is no harm in calling such orbits (diffuse or) broad 

 orbits *, and the remaining ones, sharp orbits. In this 

 nomenclature the passage from a broad to a sharp orbit, or 

 vice versa, will give rise to a line of finite breadth, and a 

 band will be still broader if it is due to the passage from a 

 broad to a broad orbit. If it is assumed, for instance, that, 

 for any fixed e, z, all the longitudes of the perihelion are 

 equally probable (which need by no means be the case), then, 

 the mean value of sin 2 co being i, the mean value of g will 

 be, bv (21-2), 



^=(H-|6 2 )(l-fsin 2 0, .... (23) 



and its two extreme values, corresponding to sina) = and 1, 

 <7±¥sin 2 i ' (23') 



The mean energy belonging to a broad orbit will be 

 obtained by substituting g for g in (21), and the position of 

 the centre of a band will be given by v = (W' — W)/ch, while 

 the breadth of a band due to the passage from, say, a broad 

 orbit (n) to a sharp orbit, will be given by 



Sv=j 7 K -^- c 6 2 sm 2 i, .... (21) 

 4(n-n 3 ) b ' v ' 



and similarly for the passage from a broad to a broad orbit 

 by combining the appropriate extreme values of g and g'. 

 In the case of an equatorial final orbit, for instance, (24) 

 becomes, with <y=.2fca, as in (22), 



8 M**R (u ^ 



To form an idea of the numerical relations take, for 

 example, n — 3, and the least eccentric orbit compatible with 

 it, i. e. % = 1, e 2 = §-. Then 



o>= l^oPR = 8600* 4 * 2 cm." 1 , 



* A "' broad orbit " will thus stand for many orbits, all having the- 

 same a, i, e, but all possible perihelion longitudes o>, from to 2ir f , 

 belonging to the individual atoms of the emitting substance. 



