1903-9.] On Energy Accelerations and Partition of Energy. 357 
We obtain 
J? \(r 2 d 2 + r 2 sin 2 0&) 
+ 
1 
M 
2 
M 2 
r*(P—( 
'0Y \ 
riO 
fo) 
) -f r 2 sin 2 Of 
0Y 
r sin 0dcf>\r sin 0dcf)/_ 
av _y 
avy 
v sin ' \rdO ) 
- [r 2 sin 4 0f~\ — [r 2 # 4 ] + ^ 
+ 3 
2 df 
rdr 
;.2 
—{f-Q 2 + r 2 sin 2 62 2 ) 
- 2[r 2 0 2 (fi 2 sin 2 #] 
or 
rj ^ 
i 
r ,a 2 v~ 
1 
r/0Y\ 2 i 
(fr l 
Upb 
M 
L 2 a?J 
+ M 2 
LU)J 
+ 3 
~ 9 
L r J 
- 2[r 2 0 2 </>' 2 sin 2 0] - [r 2 sin 4 6^</> 4 ] - [r 2 # 4 ] + 
M 
q- 
or 
2 
M 
02y- 
0£ 2 
+ 
M 2 
+ 3 
^2^.2 
r 
^2 
1 
+ M 
< Z 
df 
rdr _ 
2 d f 
rdr 
Similarly we obtain 
' d 2 
df 
(¥ 2 ) 
M* 
70 vy~ 
\0r7_ 
+ 
T 
2 
-3 
^2^2 
2 
M 
0 2 Y~ 
0r 2 
2 
M 
df 
rdr 
In accordance with our hypotheses we may suppose that 
x 
•2 
0£ 2 . 
0W “ 
0r 2 
2 0 2 Y 
r — 
0S 2 . 
= \x 2 ~\ 
= m 
wi 
. 0X 2 J 
■0W" 
0 r 2 _ 
-0W- 
_ 0S 2 . 
, etc. 
(8) We liave thus obtained three equations which determine the mean 
energy acceleration components corresponding to translation, vibration, 
and rotation ; namely, 
d 2 
a 
1 
/0VY 2 
1 Tf 2 l 
~0 2 V 
(m 1 + m 2 ) 2 
_\dx) _ 
m l + ra 2 
_0X 2 _ 
2 
M 2 
1 
M 2 
iypyi 
+ 
cf 
-3 
r r 2 i 
% 2 
\0r / 
_r 2 _ 
_r 2 
[©1 
- 
“2' 
+ 3 
i i 
-s l s- 
to| to 
to 
1 1 
2 
M 
, 9 0 2 Y" 
i 
dr 2 
2 
M 
T 
d£ 
rdr 
-k 2 l 
M Li J 
0S 2 
+ 
1 
M 
o df' 
r-y 
rdr 
(9)- 
The conditions for a stationary distribution require that the mean 
acceleration of energy shall vanish, just as in statics the conditions of 
equilibrium require that the accelerations of the bodies of the system shall 
vanish. 
