152 Proceedings of the Royal Society of Edinburgh. [Sess. 
Using this condition which determines w, we obtain 
2*3. mr p - '2mrw<ji p 
2 ^2 ( _ ^ 
cos ta cosec 2 ta<f) t - r p - (5 cosec ta - cosec 3 t a) 
Sr' z l 8?’ 3 
- 2 ' ^ + ? ,n+i— W ) ~ 2 £p( cosec ta + cosec3 ta ) r * 
p-i 
since 2 cos ^ a cosec 2 ta = 0. 
t=i 
For tangential displacements we have 
3. 
mr<f> p + 2 mr p oi = — 
dV 
{r + r P )dO P 
On expanding the right-hand side of equation 3, we have 
3-1. 
?nr(f> p + 2ma)f p 
2 3 e 2 cos ta COS ta 
8r'* «ir\2 i n ^ Q r 3 g [ n 2 f a * 
- COS ta / ^ f a + l tan f a \ 
^^4r 2 sinna V 2 ’ 
Z e 2 COS ta , , , , , , , \ i 6 2 cos £a 
( cot ^ i tan ta ^t - 2 472 • 
4r 2 sin 2 
Since 2, = 0, equation 3T becomes 
"sin 2 £a 
32. 
mr<f> p -I- 2 mu)f P 
= - ^2 4$ ( cot ta + i tan *0 
+ 2,775 Srr ( oot tn + a tan *“)& 
1 4r 2 sin 2 ta 
2 6 2 COS ^ 
8?* 3 sin 2 ta * 
The equations 23 and 3*2, which determine the stability of the con- 
figuration in the plane of the orbit, are of the form 
4*1. 
4-2. 
where 
r p -2ro)cf) P = -'%rA t cf) t -Br p -'ZC t r t pHl, 2, 3 . . . p. 
v<b p -\- 2<jir p = — T)v<b p + -l- p — 1, 2, 3 . . . p , 
A # = 
8 mr 3 
8mr 
cos ta cosec 2 ta 
Ul fi2 /k / o / v , (n + 1 )se 2 & ? 
B= V — * 0> cosec ta - cosec 3 ta) - y v ' 
8m, r 3 i 
t= l 
mi 
,71 + 1 
C t — - — - (cosec ta + cosec 3 ta) 
8 mr 
e 2 cos ta 
imr 2, sin 2 ta 
(cot ta + J tan ta) 
D = 2D*. 
4*3. 
