153 
1919-20.] Stability of the Kutherford Atom. 
In order to solve the equations 4*1 and 4 2, we use the artifice 
= ’) t'p-l — j • • ■ T \ = /^2>* 
. *. (3 P = 1 = cos 2kir + i sin 2/for, 
i.e. /3 — cos 2ka 4- i sin 2/ca. k = 0, 1, 2, . . . p-\. 
A similar set of equations is used for the (p’s, viz. : 
4*1) = } 4 > v-i = > • • • = PPv 
If the time factor enters in the form e iqt in each variable, we have 
4*4. (B — q 2 + 2C t(^) r p = (2 <dq - 2A t p*)r</)j>. 
4-5. (D - q 2 - 2D t ^)r(p v = ( - 2 <aiq + 2A f /3*)?> 
From equations 4'4 and 4'5 we obtain an equation determining q. 
4-6. (B — q 2 + 2C,0*)(D - q 2 - 2D t p) = - (2o aq - %A t p) 2 . 
This equation involves imaginary coefficients of q. But 
2A ffi* = 2{ A # cos 2kta + iA t sin 2kta). 
But 
^ cos fa 
sill 2 fa 
COS 2/ffa = 0 
And it can be shown that 
2A f /3* = aA*sin 2kta, 
2C t /3* = %C t cos2kta, 
2D^= SD; cos 2 kta. 
The equation for the frequency q becomes 
4-7. (B + 2C t cos 2k ta - q 2 )(D - 2D* cos 2 kta - q 2 ) - (2 a*q - 2A* sin 2kta) 2 = 0. 
This equation is a quartic, and will in general give 4 values of q. 
k has the p values 0, 1, 2, 3, . . . p—\. 
If p — k is written for k in 47, the values of q obtained merely differ 
in sign. 
If p is odd we obtain 1 - equations of the form 47. Also for the 
value k = 0 , 2A* sin 2 kta = 0 , so that we have a quadratic in q 2 . 
There are therefore 
+ 2 = 2p values of the frequency corre- 
sponding to 2 p degrees of freedom. When p is even we obtain ^ — 1 
equations of the form 47 giving 2p — 4 frequencies. 
When k — 0, as before we get another two values for q ; and when 
r p 
k = A , 2 A* sin 2kta = 2 A* sin pta = 'EA t sin tir — 0, so that we obtain another 
two values for q. The total number of values for q is 2p — 4 + 2 + 2 = 2p, 
