260 Bohr's Hypothesis of Stationary States of Motion. 

 terms. We find for a plane nearly circular orbit 



dcmjS 2e 2 /3 z e 2 cos <f> c 2 m/3 2 _ e 2 sin </> __ e 2 p 



. . . (25) 



where p denotes the perpendicular from the positive nucleus 

 on the tangent, and cf> the angle between that tangent and 

 the radius vector r. For the sake of brevity we shall write 

 for the angular momentum of the electron 



K = cm/3p (26) 



Since p = rdr/dp, and f = 6'/3cos </>, we find from (25) and 

 (26) 



e 2 cos <f> _ H dp dH__ 2e 2 ^p 



r 2 ~p*di> dt ~ 3/9 2 (l-/3 2 ) 2 ' 



We have r 2 Q — c$p ; hence changing the variable from t to 

 we find by means of (25) and (26) 



, H ,«*H_ 2«y , H 3 - 2 *T P W 



6 dd ~ c 3 (l-/8 2 )V ° c 3 J (1-/3 2 )V 



. . . (27) 



Thus the angular momentum of the electron, H, diminishes 

 continually, instead of remaining constantly equal to h\2ir y 

 as it should do on the hypothesis adopted by Nicholson and 

 Bohr. With the usual values of e and c we have 



26 6 /c 3 = 8-8.10- 88 , 



whilst 1i\2it = 1-05. 10- 27 ; 



hence if H be equal to h\2ir, H will diminish to one-half 

 of H in about 180,000 revolutions, provided ft 2 can be 

 neglected, and the orbit is so nearly circular that we may 

 put p/r equal to unity. So rapid a change of the angular 

 momentum can hardly be regarded as consistent with sta- 

 tionary motion, and therefore the hypothesis of the expanding 

 electron must for this reason alone be considered as incom- 

 patible with Bohr's theory. 



The proof given here is extremely general ; it assumes 

 nothing whatever concerning the mass of the electron — its 

 variation with the speed, or a possible secular change due to 

 expansion or any other change of structure of the electron — 

 or the force acting on it, beyond the fact that the latter must 

 be central. Hence we cannot account for invariability of 



