1919-20.] Stability of the Rutherford Atom. 
151 
Hence for displacements perpendicular to the plane of the orbit we have 
the equation of motion 
1. 
mx n - 
cV 
dxJ 
We shall put 
0 P — 0 t = 2ir - 2zJa, where a = 7 r/p. 
Neglecting quantities of the second order, we have on expansion of the 
right side of equation 1 : 
1-1. mx „ = ^ g^-3 s e in3 - * t ) - 2 
If we put x 2 = fix-L , x 3 = /3x 2 , x 4 = fix 3 , x p = ^x p _ x , and cc = f3x v , 
we have 
/3 P = 1 = cos 2/br + i sin 2/for 
f3 = cos 2/ra + i sin 2ka, where k — 0, 1, 2, . . . p-1. 
For a vibration which involves e iqt as a time factor, we have 
1 ‘2. o 2 = y Se — "V cosec 3 ta + N' w — -5 cosec 3 zhx cos 2Atfa, 
L ^ mr tl+1 ^Smr 3 
since x t — ^x v and 
p - 1 
^ sin 2 kta cosec 3 ta — 0. 
t — i 
Displacements perpendicular to the plane of the orbit will be stable if 
1*3. y* SG ^ n > "y, — — cosec 3 ta(l - cos 2 kta). 
mr n+i Smr 3 ' 
For radial displacements we have the equation 
2. mr p - mru> 2 — 2mroo</> 2) - mr P w 2 
dr p 
On expanding the right side of equation 2 and neglecting terms of the 
second order, we have 
2*1. mr p — mra) 2 - 2 mrw<j> p — m?' p w 2 
S6 2 k vl 
= y — cosec ta - y' 
2-J ^. r 2 r n 
e 2 
+ — 2 cos ta cosec 2 ta((fi p - </>d 
e 2 
- y,-^-o(3 cosec hx - cosec 3 ^a)r^ 
- y (cosec ta + cosec 3 ta)r t 
+ 2' 
'8r 
’Se 2 k„nr, 
For steady motion we have 
2 - 2 . 
mrw 
__ ' se 2 k n _ 
4r 
cosec £a. 
