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

 Again, d 2 u/dp 2 must be positive for a minimum, and we find 



by + ay du + vf(fa\* + $fffiu =0 



(3/> 2 ~dp"dit dp "du*{dpj "du dp 2 

 and as dit/dp = 0, 



dp 2 'dp 2 / du 



~ 9 + 5u 



after some reduction. For a minimum of u, this should be 

 positive, and therefore the first value of u gives no minimum. 

 The second value gives r ± negative and is inadmissible. 



Finally, therefore, there is no minimum of u as p ranges 

 from unity, its smallest value, to infinity. Nor is there a 

 maximum between these limits, for p is less than unity with 

 the first value of u, in contradiction to the hypothesis that r 2 

 and r z were the two radii for the inner electrons. Thus the 

 divergencies from circular orbits must increase continuously 

 as the outer electron is nearer and nearer to the atom ; and 

 it is never possible to regard the two electrons nearer the 

 nucleus as forming a ring. 



In fact, in Bohr's atom, p is about equal to 3, the ratio of 

 the two radii of the circles to which the orbits most nearly 

 approach. Thus u is given by the cubic 



-9u + 3(3-2u)(l + u) + (l-uXl + n 2 ) = 



or 



u 3 + 5u 2 + 7u-10 = 



whose real root is about u = 0'85. Thus the radii of the orbits 

 of Bohr's inner electrons, in the permanent state of the atom, 

 should be in the ratio (1*85)/(0*15) or 12 to 1 when the 

 electrons are in a straight line. 



It is easy to verify this mode of reasoning in a variety of 

 ways. Thus if u=l, signifying r s ==0, r 3 = 2a, where one 

 inner electron is in the nucleus, then (7) gives p 2 = 4 or 0, so 

 that r x = () or 2a. Thus there is only one electron in the 

 atom, or there are two, equidistant from the nucleus. 



A similar process can be applied to the model atoms of 

 beryllium and boron treated by Bohr, and leave little doubt 

 that coplanar rings of electrons are as impossible on his 

 theory as on the ordinary theory. In fact, in order to retain 

 them at all, we must depart from the ordinary dynamics even 

 more completely, and admit ihe possibility of an uncom- 

 pensated force of electrostatic type producing no corresponding 



