364 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Also 
1 
2M 
so that we have also 
\< 
to 
II 
|1= 
h— » 
1 to 
0$, 
03>o 
_\ds J 
L\0^/ J 
m 2 
_0<r 1 
0^ 2 _ 
m 
rpvi 
f 1 1/^2 
0$ 7 df 
LV0.V J 
m 2 
_0£t‘ 1 dx 2 _ 
Other inequalities which of necessity hold good are 
/*1 < /q 
and 
df 
dx 1 J 
+ [X . 2 
l 1 1 
'd*f 
_df_ 
"0^> 1 0<t> 2 " 
l _0,r 1 dx 2 _i 
> 0 . 
>0 
Stability in the General Case. 
(14) In discussing the stability of the energy equilibrium let us first 
consider the translational mean energy. 
Let 
[i* 2 ] = T„ + « . 
where T 0 is a third of the mean translational kinetic energy determined by 
the equation of energy equilibrium, and e is a small variation which 
may be due to initial disturbance. We thus obtain, by substituting in 
equations (9), 
e = — 
0 2 V 
m l + m 2 L0.r- 
For stability the variations in e must be periodic, and this condition is 
W 
satisfied if 
dx 2 _ 
is positive ; and this is precisely what has been assumed, 
in order that our hypotheses may allow energy equilibrium to be possible. 
The term involving [g 4 ] in the equations of energy equilibrium makes 
it very difficult, at any rate, to deal with the question of stability in the 
general case, as far as the mean vibrational and rotational kinetic energies 
are concerned. 
Let us consider, however, the question of stability, when we suppose 
Maxwell’s Law to hold good. 
From the equations which give the average energy accelerations it is 
clear that 
~i(g 2 +r 2 ) 
M 2 
0Y\ 2 /0V\ 2 ' 
dr) + \ds) 
1 
7 
1 
M 2 
\dr) _ 
M 
a 
M 
2 d f ' 
rdr 
' 9 0 2 V , .,0 2 V" 
a 'aF + r a? 
+ 
2M 2 
1 
M 
1_ 
M 
df 
rdr 
dr 2 
