1908-9.] On Energy Accelerations and Partition of Energy. 373 
1 
It is necessary that AB should he greater than 
2M 
0V \ 2 
ds) J 
in order 
that energy equilibrium may be possible, if the hypotheses are justifiable. 
Let us consider the question of stability. From the equations which we 
have obtained for the general case 
S^ 2+ ' 2) 
1_ 
M 2 
dr) 
1 _ 
M 
T 
,df 
rdr 
+ 
1 
M 
dr 2 
1 
_ _^2 + r2 ] 
2 M 2 
0 2 Y" 
0S 2 
avy 
ds) 
so that in this case 
l a 2 
3 
(d Y \2 
J 
" 2 M 2 
_ V 0 s / _ 
A ; 2 AB__A[- 2 ]_2[ 2 ] 
M 2 M M a u ] M u J 
' 02 Y' 
ds 2 
Let [ A q 2 ] = T 0 + > 7 , where T 0 is the value of [Jg 2 ] corresponding to energy 
equilibrium, and is a small variation in it. 
Then 
"0 2 V 
IP 
, 2 f A , 
v + — < — + 
M I a 
= 0. 
must be positive. This condition is necessarily 
For stability — + 
a L . 
satisfied if we suppose that A and a are of the same sign. 
(19) The case where the external force follows the law of proportion 
of mass to weight . — In this case the equations of energy equilibrium are 
scarcely simplified, but we have the following relations, if we suppose 
that = P*=k:— 
m 1 m 2 
m , 
_ \ dx^ / _ 
- Vic 
70<jy 2 ‘ 
i_ \03? 2 / _j 
= (wq - m 2 ) 
r (d J Y 
\dx) . 
0 Y\ 2 ' 
ds) 
= k 2 {^m l + m 2 ) < m 1 
2 A : 2 
m-pricy 
m 1 + m 
T\ 
0 <£A 2 
_V 0 ;r 1 / _ 
0<E>A 2 
+ m 
dx. 
- m ( 
0 $ x 0^> 2 
_ dx Y dx 2 
d% 
_dx 1 dx 2 
0 $ ] 0 $ 2 _ 
U 0 .r 1 dx 2 _ 
0 2 3» 1 
Jdxf _ 
_dx 2 _ 
and 
"a*v" 
dx 2 _ 
~0 2 Y~ 
k(m l + m 2 ) 
02 ^ 
_dxf _ 
lcmgn 2 r 0 2 $ 1 
_ ds 2 _ m l + m 2 \_dxf_ 
( 34 ). 
