154 Proceedings of the Boyal Society of Edinburgh. [Sess. 
corresponding to the 2p degrees of freedom of the rotating group in 
the plane. 
Let 
(f = ^- 3 (B + 2C{ cos 2Jcta) 
e z 
= 2J[5 cosec ta - cosec 3 ta + cos 2kta (cosec ta + cosec 3 £a)] 
-«[3 - 2'( w + 1)K W _J 
where n > 2, and 3 - 1\n + 1 )K W _ 2 = 
G = mr 3 ( D _^ D ^ cog 2 
e A 
^ = ^2 — ^ (cot ta + A tan ta) COS ta 
sm^ ta 
H = sin 2kta 
e z 
= cos ta cosec 2 ta sin '2kta 
W? 2 - 
, Imr 3 
V n e * ■ 
The equation 4*7 now takes the form 
4*9. (F - q' 2 )(G - q'*) - (H - J q'f - 0. 
By consideration of signs of the left side of this equation it is easy to show 
that q' will have four real values if F and G are positive and each is 
greater than (H/J) 2 . 
For stability we must also satisfy the inequalities obtained in equation 
1*3 and 2*2, viz. : 
g ( 4s(l - 2'K n _ 2 ) > % cosec 3 ta sin 2 Ida. 
( 4s(l — S'K n _ 2 ) >3 cosec ta. 
This latter condition is necessary in order that co 2 given by the equation 
- w 2 = 4s(l - ^'K n _ 2 ) - 5 cosec ta 
e 2 
may be real. 
J 2 = 4s(l - S / K n _ 2 ) — S cosec ta. 
Particular Cases. 
In the previous work we have developed the conditions of stability 
on the assumption that the force between the positive nucleus and the 
1 k 
negative electron is represented by a function of the form 
where n> 2. 
