156 Proceedings of the Royal Society of Edinburgh. [Sess. 
In each case F and G are positive and greater than (H/J) 2 if 
2-86 
•51>K> 
n+ 1 ’ 
i.e. if tk£5. 
(6) p = s = 6, t = 5, a = ^,k = 0, 1, 2, 3, 4, and 5. 
In each case F and G are positive and greater than (H/J) 2 if 
301 
•29>K> 
n+ \ ’ 
i.e. if <£10. 
(7) p = s = 7,t = 6, a = ~,7c = 0, 1, 2, 3, 4, 5, and 6. 
In each case F and G are positive and greater than (H/J) 2 if 
3*13 
•10>K> 
n+ 1 
i.e. if <£32. 
(8) p=s = 8, t = 7, a = J , k=0, 1, 2, 3, 4, 5, 6, and 7. 
O 
It is not necessary to discuss the conditions of stability in full, as the 
condition for displacements perpendicular to the plane of the orbit is not 
satisfied, since 
4s(l - K)>2 cosec 3 ta sin 2 Ida 
gives for k — 3 or 5, 
1 — K> 1*08. 
Since G, H, and J are independent of n for a fixed value of K, and F 
increases as n increases, it follows that if stability is obtainable for any 
value of n it is obtainable for all larger values of n. 
Conclusions. 
The preceding analysis shows that if a law of force between a positive 
1 f b n ~ 2 \ 
nucleus and a negative electron be of the form fed 1— ^zr 2 ) , an n can be 
found which will preserve the stability of a group of electrons, not 
exceeding seven in number, rotating in a circular orbit round a positive 
nucleus. Since b is smaller than the radius of an atom, for distances large 
in comparison with the radius of an atom this law of force will differ from 
the inverse square law by a negligible quantity. Stability for the simple 
cases of the Rutherford atom will therefore be established. 
The question of the stability of a series of rings of electrons rotating 
