Spectra of the Elements, and Structure of the Atom. 553 

 after the usual manner of central orbits, by 



S^=i?( N -w(« 3+ * 3+ 4 • • .< 8 > 



where u 1 = l/r 1) mh 1 is the angular momentum of the electron, 

 and there are similar equations for the other two orbits. 



These equations cannot be accurately solved in general 

 terms unless the triangle ABC is at every instant equi- 

 lateral, — this does not include only circular motion. We shall 

 return to these equations later. Without solving them at 

 all, we may show more simply that Bohr's lithium atom is 

 not a possible configuration. 



In that atom, the two inner electrons rotate several times 

 faster than the outer one, so that positions must arise 

 frequently in which the three electrons are in a straight line, 

 and the mutual repulsions between the electrons then act 

 along this line. The only other forces being the attractions 

 of the nucleus, which are radial, we see that when the 

 electrons are in a line, this line must pass through the 

 nucleus. Otherwise the conservation of angular momentum 

 for each electron is destroyed by electrostatic forces, which 

 Bohr recognizes as causing accelerations. 



When the electrons are in this configuration, let the 

 instantaneous values of the radii of their orbits be (r 1? r 2 , r 3 ), 

 shown on the figure, where (A, B, C) are the electrons, and 



COB A 



is the nucleus. On Bohr's theory, r 2 and r 3 will be nearly 

 equal, and i\ much larger than either (r r = 3r 2 approximately). 

 The sides of the triangle are 



a=r 2 + r 3 , b = r 1 + r 3 , c = r 1 -r 2 , 

 and a = b — c, as the triangle is instantaneously a straight line. 

 Since (r 1? r 2 , r 3 ) are determined already in terms of (a, b, c) 

 by the condition of angular momentum, we can in this case 

 find an identical relation between the radii. Thus since 

 a = b + c, 





« ; ' + /> :, + c 3 



and therefore ar x br 2 



a 2 a 2 +b 2 



YW 



(5) 



