1908-9.] On Energy Accelerations and Partition of Energy. 365 
If Maxwell’s Law holds good, we have 
3 d 2 r/1 2X1 1 
2 dt 2 ^ 2 ^ _ M 2 
df_ y 
dr) 
2M 
'0 2 Y 
IP] 
i 
M 
2M 
fe 2 ] 
~d?f 
dr 2 
E 2 ] 
~ d£ 
3 
-4- 
/0V\ 2 
_rdr_ 
2M 2 
i\ds) J 
(23). 
Let [J^ 2 ] = P + ^ j 
where T 2 corresponds to energy equilibrium, and is a small variation in 
the rotational kinetic energy. Then 
V 
2 d f 
3M 
+ — + 
0.s 2 dr 2 7- dr 
For stability it is necessary that the average value of 
3 ajY + xy + 2 # 
0s 2 dr 2 r dr 
should be positive. 
Now, since 
~0 2 Y _ 
??q?7? 2 
0 2 W 
Las 2 ] 
(m l + ?a 2 ) 2 
_0X 2 _ 
.0s 2 . 
must be positive; for our assumptions require that 
-02y- 
0a: 2 
should 
be positive. 
The question of stability will be considered in particular examples. 
In the general case, suppose that we put 
JM = T 1 + ^ 
ib 2 ] = T 2 + % 
and 
fc 4 ] = fe 4 ]„ + f > 
where [g 4 ] 0 is the value of [q 4 ] corresponding to energy equilibrium, and 
t, = m 0 
' tWk 2 J„ 
with the corresponding rotation. 
We have the equations 
and 
d 2 iy.2 
dt 22 J 
1 
M 2 
i i 
^ 1 
to 
1 1 
+ 
(f~ 
_r 2 _ 
- 3 
r *9 9~ 1 
r*q z 
1 
M 
Ao0 2 V1 
dr 2 _ 
2 
M 
> d f 1 
A rdr_ 
d\ 2 “ 
dgE. 
1 
“M 2 
~/0Y\ 2 
\0S / 
- 
X 4 
2*2 
+ 3 
f 2 q 2 
7 ‘2 . 
1 
M 
r 2 0 2 V~ 
i 
+ M 
2 d f 
q 2 
rdr 
Now 
[>¥] = [r 2 ] [<f] 
= 4(T 1 + 1)l )(T 2 + % ) 
= + ‘Oi T s + nr i » 
neglecting terms of the second order. 
