1919-20.] Stability of the Rutherford Atom. 157 
about a positive nucleus introduces considerable complexity into the 
rigorous analysis required. If we make the tentative assumption that, so 
far as the effect of an inner set of rings on the outer ring is concerned, we 
can replace the inner set by an equivalent charge at the centre of the atom, 
the conditions for stability of an outer ring of p electrons rotating in a 
circular orbit will be determined by the equations 
/ gK\ 1,-1 
Api\- — J > ^ cosec 3 ta sin 2 kta , 
' P ' #=i 
and the reality of the roots of the equation 
(F- 2 ' 2 )(G-^) = (H-Jg) 2 , 
where 
p — i 
F = 2 J[5 cosec ta - cosec 3 ta + cos tka (cosec ta + cosec 3 
t = i 
and G and H have the same values as before, 
J 2 = 4jp^l - — ^ - 3 cosec ta. 
. . sK . 
In these conditions — replaces K; and since s>p, the new value of K will 
be less than the previous value of K. This can be obtained since r is 
greater for the outer set than for any one of the values of r for the inner 
set, the value of n being kept the same; or we may increase n as well 
as r, and the increased value of n will still maintain the stability of the 
inner set. 
The displacements of the electrons in the outer ring, perpendicular to 
the plane of the orbit, will be unstable, however, when p exceeds seven. 
This would seem to indicate that the atom could be built up of a series 
of rings of seven electrons, and we should expect a periodicity in the 
chemical properties of the atoms corresponding to MendeleefFs classifica- 
tion, which was stated by Newlands in 1864 in the form, u the eighth 
element starting from a given element is a kind of repetition of the first.” 
*«)] -p 
3 — («+ 1) : 
sK 
p J 
(Issued separately October 12, 1920.) 
