150 Proceedings of the Royal Society of Edinburgh. [Sess. 
XVI. — A Law of Force giving Stability to the Rutherford Atom. 
By J. Marshall, M.A., B.Sc., University College, Nottingham. 
(Read June 21, 1920. Revised MS. received July 13, 1920.) 
Introduction. 
There is a large accumulation of evidence in support of Sir Ernest 
Rutherford’s conception of atomic structure. According to this conception 
an uncharged atom consists of certain groupings of negative electrons 
rotating about a positively charged nucleus. Dr Bohr’s theory of line 
spectra, which is based on this structure, frankly discards dynamical con- 
ditions of stability. There is no doubt, however, that this theory has been 
very successful in accounting for the numerical relationships between the 
frequencies of spectral lines. 
In this paper it is proposed to obtain a law of force which will preserve 
the stability of a group of electrons rotating in the same plane about a 
positive nucleus in the simple cases in which the Bohr theory is applicable. 
In order to account for atomic behaviour, we shall assume a law of force 
which for distances comparable with the radius of the atom is at variance 
with the inverse square law, but which for distances great in comparison 
with atomic dimensions differs by a negligible quantity from this law. 
Analysis. 
Let the positive charge on the nucleus be se and the number of 
electrons in the ring rotating round the nucleus as centre be p. 
In equilibrium the cylindrical co-ordinates of the pth electron will be 
( r , , 0) referred to the positive nucleus as centre. In the disturbed state 
the co-ordinates of the pth electron will be ( r + r v , 0 P + <p 9 , x 9 ). 
Let the force on the pth electron due to the positive nucleus be 
,se 2 k n 
where is a summation with respect to n, where n can have any 
value <t2. The forces acting on the pth electron can therefore be derived 
from the potential function 
p-i 
^ {(r + r t ) 2 + (r + r v f + {x v - x t f -2 (r + r t )(r + r P ) cos (0 P + cf> v - 6 t -</>#)}• 
2 ' 
se 2 A 
(w-l){(r + r 3 ,)2 + V} (w “ 1,/2 ‘ 
