552 Prof. J. W. Nicholson on the High-frequency 



The point can satisfy these three conditions simultaneously 

 for any triangle, for they are in accord with Oeva's theorem 



BL . CM . AN = LC . MA . NB 



relating to concurrent lines through the angular points of a 

 triangle. 



Thus other than circular orbits may exist for three par- 

 ticles moving with specified angular momenta about a nucleus, 

 and the condition of angular momentum alone does not 

 greatly restrict the orbits. We can express the radii to the 

 electrons in terms of (a, b, c) as variables at any instant, 

 for 



BL LC a 



and therefore 



c 3 "" b z £ 3 + c 3 ' 



(BL, LQ) = (ac\ a£ 3 )/(6 3 + c 3 ) 



(CM, MA) = (6a 3 , 6c 3 )/(a 3 +c 3 ) 



(AN, NB) = (cZ> 3 , ca 3 )/(a 3 + 6 3 ), 



AL = (c 2 + BL 2 - 2c BL cos B)* 



and we can easily derive 



OA=r 1 = aX+ h ° +c3 {(b + c)(b> + c°)-bca*}i . (1) 



with two cyclic expressions for r 2 and r 3 , so that the atom is 

 at any instant completely expressed in terms of the distances 

 between the electrons. In obtaining these formulae, we have 

 supposed, on the one hand, that the nucleus is at rest, on 

 account of its great inertia, but on the other hand, only that 

 the angular momenta of the electrons are constant, — not 

 necessarily equal, as in Bohr's theory. 



Consider now the force tending to increase OA. The 

 attraction of the nucleus on the electron A is Ne 2 /OA 2 , and 

 the resolved part of the electronic repulsions is 



/cos a, cos a 2 \ 



which, after considerable reduction, becomes 



e 2 OA(a 3 + b 3 + c 3 )/bV (2) 



If rj = OA, the steady orbit of the first electron A is given, 



