1919-20.] Stability of the Rutherford Atom. 155 
We shall now apply our results to particular cases. If we simplify the 
1 b n ~ 2 
law of force to the form — ; — , it is evident that the constant b will 
rp<-* ty*™ 
give a measure of the closest approach of the electron to the positive 
nucleus, when there are no other forces acting on the electron. 
Let 
K = — 
One necessary condition that this law must satisfy is, that for distances 
large relatively to the radius of an atom the law of the inverse square of 
the distance must hold. This condition is evidently satisfied, since b is a 
small quantity and n is <(;3. 
Special Cases. 
(1) p = s = 1, t — 0. 
The condition for stability is 
1 >K> 
n+ 1 
and is satisfied w<t3. 
(2) p = s = 2, £ = 1, a = ^ ,k = 0 and 1. 
H — 0 in each case, and G is positive. Stability is obtainable if 
•87>K> 
2-87 
n -\- 1 
If n<t3, F is positive, and F and G are each greater than (H/J) 2 
(3) p = s = 3, £ = 2, a = ^,Jc = 0, 1, and 2. 
In each case F and G are positive and greater than (H/J) 2 if 
2-76 
•8>K> 
n+ 1 ’ 
i.e. if n<^3. 
(4) p = s = 4i, t= 3, a = - ; ,Jc = 0, 1, 2, and 3. 
4 
In each case F and G are positive and greater than (H/J) 2 if 
2'81 
•65>K> 
n + 1 ’ 
i.e. if ^<f4. 
a = ~z. , Jc = 0, 1 , 2, 3, and 4. 
5 
(5) p = s = 5, t = 4>. 
