of Stationary States of Motion. 253 



The new coefficients a 1? &c, are all real constants ; hence 

 each term of the integrand of (17) is essentially positive 

 throughout the whole range of integration, the onty excep- 

 tion being when^' vanishes, in which case each term vanishes 

 identically. Therefore the principal term (17) in the radiation 

 R can only vanish if each of the four functions inside the 

 curly brackets vanishes identically for every value of & 

 between and ir, and for every pair of values of j and k 

 excepting only j = 0. 



13. There are two possible cases : — 



(1) p = 0. 



The principal term in R vanishes identically whatever 

 values the coefficients a ly &c, may have, and the radiation 

 becomes small of order a at least. This occurs for a uni- 

 form spherical electron rotating, or even oscillating about 

 a diameter, a case already considered by Herglotz and 

 Sommerfeld *, provided only that the period of oscillation 

 be properly adjusted, and the electron have a surface- 

 charge; also under similar conditions for a pair of spherical 

 electrons oscillating about a common diameter (Oseen, loc. cit. 

 p. 646) ; and lastly, as is well known, for an axially sym- 

 metrical system rotating uniformly about its axis. In all 

 these examples, however, the centre of the electron remains 

 at rest, and consequently not one of them has any bearing 

 on Bohr's theory. 



14. (2) a l = 0, &c, for all pairs of values of j, Jc except 

 j=0. 



We can express these conditions more conveniently by 

 multiplying (8) by dizdz, integrating over the area of the 

 meridian plane swept through by the electron and using (13) 

 and (16). The only terms left in the result are those for 

 which j=0, and these are independent of the time t, so that 

 we obtain 



!-. (18> 



= 2 {a 10 + di w , a 20 + t6 S o> «so + '03oi exp ik% 



— oo 

 00 



= S {l 10 + wi 10 , l 20 + tm 20 , ho + Lm Bo} exp ik.% . 



-oo J 



The zero suffix indicates that in each coefficient j=0, whilst 

 k takes all integral values between + go . 



* Soinmerfeld, Gott. Nach. p. 431 (1904). 



