366 Proceedings of the Royal Society of Edinburgh. [Sess. 
Therefore, substituting for [p], [ q 2 ], and [r 2 ] in the equations, we obtain 
at once the equations 
V 2 ~ 
- 12Tj 
r- 
^2 
- 12T f 
m - 2^1 
n M 
■02y _ 
dr 2 
_ 4^2 
M 
df 
rdr 
+ 12T : 
%+ lL>T 2 
11 
P?2 
M 
£w 
0S 2 
+ M^ 2 
df_ 1 
It is next necessary to evaluate 
We know that 
[g 2 ] = 2 [r 2 # 2 ] . 
Now 
r 2 # 2 = {r 2 0 2 } o + e 2 , suppose, 
where e 2 is small, and {r 2 d 2 } 0 denotes a value of r 2 0' 2 corresponding to 
energy equilibrium ; so that, if we take mean values, 
7 /2 = W* 
Squaring both sides of the above relation, we have 
r 4 0 4 _ } 0 + 2e 2 {r 2 # 2 }, 
and when we take mean values this becomes 
[r 4 # 4 ] = [r 4 # 4 ] 0 + 2 ? 7 2 T 2 
or 
for we have proved that 
so that 
whence 
We thus have 
' 1_ L- 12T 
t[2 4 ] = tfe 4 ]o + 2 % T 2. 
[»"b 4 ] = lb 4 ] ; 
b 4 ] = b 4 lo + Tr>72 T 2 > 
£=-VVb . 
r, = ilT 
71 3 x 2 
r- 
r- 
*• = -¥T 2 
% 
1 
V 2 + 
^2-12T 2 
r/ 2 + 12T 2 
1 
_ 0 2 V' 
0 r 2 
M 
(if 
1 
*?i 
%2 
M 
0 2 y" 
0S 2 
+ %2 
M 
rdr 
df 
rdr 
The energy equilibrium is stable, if the equation 
L 2 - 12 I 2 
1 
^2 
1 2T 0 
~r 
^2 
2_ 
M 
W"| x 
3 r 2 J ’ 
dr 2 
, i> 2 -¥'b 
6 T> 
3 L 2 
/y*Z 
12 T, 
£ 
M 
rdr 
1 
+ 12T n 
1" 
0 
-j 
0 2 V 
+ 1 
df 
ry* 2 
1 
y>2 
M 
_ 0S 2 _ 
M 
_rdr_ 
0 . (24), 
treated as a quadratic in p 2 , lias positive roots. 
