248 Prof. G-. A. Schott on Bohr's Hypothesis 



Moreover, denoting vector components in the directions 

 1, 0, and </> by subscripts, and similarly for other directions, 

 we find 



... (6) 



Lastly, as we shall have to deal with periodic motions 

 involving one or more incommensurable periods, we shall 

 calculate the average radiation by means of the equation 



16'rr 2 TR=fj^f^{\J e 2 + \]^\smdd0d<j,dt 1 , . (7) 



where T denotes the period, when the motion is mono- 

 periodic, and an interval of time long compared with the 

 longest period when it is polyperiodic. This equation is 

 easily derived from the first equation [3). 



9. Since the coordinate ^ is the longitude, the quantities 

 C and K, whatever their nature may be, must of necessity be 

 periodic functions of ^ with the period 2tt. For the sake of 

 generality we shall assume that they are also sums of periodic 

 functions of the time, who-e periods are not necessarily com- 

 mensurable. Hence we may expand the components of C 

 and K in series of exponentials of the longitude % and the 

 time t of the form 



{a, c, c;}= s i {Ox(i, *), c 2 (i, *), co - , h)}e^i{jt+k X }, 1 



i (8) 



{K,, K w , K x }= 2 2 {K,(i, k), K.0'. *), K 3 (i, £)}exp «0'' + *x). 1 



— 00 —00 



where k is restricted to integral values, whilst j is not, but 

 may take any values, whether they be commensurable with 

 each other or not, whilst the coefficients Ci, &c, are explicit 

 functions of z and -or, but not of % or t. 



10. For the sake of brevity we shall write 



^= x -4,-7t/2, T=/i 1 + £(<£ + ,r/2). . . (9) 



With these expressions we may write (5) in the form 



jt + k% = r +jz cos 6 4- kyjr —jm sin 6 sin ifr. . (10) 



Substituting from (8) and (10) in (6), omitting the para- 

 meters ;', k from the coefficients C 1? K 2 , &c, as no longer 



