1908-9.] On Energy Accelerations and Partition of Energy. 355 
Also we have 
— (r 2 6 2 ) = - 2 rrO 2 4- 2 r 2 9 2 sin 0 cos 0<j >- 2 - 2 AL , 
dV ’ MdO 
upon substituting for the accelerations from Lagrange’s equations, as before. 
Then 
~(r 2 f? 2 ) - - 2 r 2 9 2 - 2rr0 2 - 4rr00 - 20‘I ~ 
dV ’ M dO M dt\dO 
+ 2 sin 9 cos 64> 2 —(r 2 9) + r 2 $(2 cos 2 $4> 2 9 + 2 sin 204>cfi ) . 
dr 
Substitute for r, etc., from Lagrange’s equations, as before, 
= 6 ™ 2 - 2 ^ 4 - *<«* e + + 2,^1 ^ + i i(5) 2 
8.,a0Y 1 4r . ^ /j, 2 0Y , 0 .ajo ■ a n Q/jl d/dY\ 
4 -r0— — - — sin 6 cos 9d > 2 — - 12rrtf<£ 2 sin 6 cos 0 - 20— — ( — T i 
r dO M M ^ r0(9 Md/V067 
- 4 cos 6^- + 2r 2 sin 2 6 cos 2 . 
M 
Upon taking mean values, we have 
‘ dt 2 
3[r 2 <9 2 ] — [?- 2 A] — [(cos 2 0 + 2 U 2 <9 2 (/> 2 ] + 
1Y1. 
1 /0Y\ 2 ' 
r 2 V dOJ . 
and 
sin 2 6<j> 2 ) 
1 
M 
1 |<yU v 
mL W- 
0 2' 
0 2 Y’ 
0A 
4- [r 2 sin 2 0 cos 2 0<p] + 
J. 
M 
dr 
+ M 2 
1 i - +M^i(r 2 sin 2 6l) ' 2 
dt 
r 2 sin 2 6 1 c<£ 
— [r 2 sin 2 f9</> 4 ] — [r 2 cos 2 $A</> 2 ] — [r 2 </> 2 sin 2 0] 4- 
M 
r</> 2 sin 2 0 
0V' 
dr 
The three equations determining the mean accelerations of the energy 
components corresponding to r, <f>, 0 are therefore 
W' 
dr 
" fJ2 
_ /l;- 
dt 2 
l 
M 2 
— ^ J 4- [r 2 (# 2 + sin 2 #</> 2 ) 2 ] — 3[r 2 ($ 2 4- sin 2 0</> 2 )] 
1_ 
M 
' ,o0 2 Y" 
r Z 
(J 2 
a /j 
dt 2 
(|r 2 sin 2 Ocfi 2 ) 
1_ 
M 
r 2 sin 2 6(j > 2 — . 
1 0 
0Y 
r sin 6 d(ft\r sin 6d<f>J _ 
dr 2 
1 
2_ 
M 
d 
df 
+ 
M 2 
rdr 
0 V \2 
(3) 
+ 3[r 2 </> 2 sin 2 0] + 3[r 2 A cos 2 #<£ 2 ] - [?’ 2 sin 2 
r sin <90<^> 
2 d f 
J 
rdr 
(U 
d 2 
dt 2 
„(ir 2 6» 2 ) 
= 3[r 2 A] — [r 2 A] — [(cos 2 0 4- 2)r 2 A</> 2 ] 4- 
1 
1 
+ [r 2 sin 2 6 cos 2 0<£ 4 ] 
M 
M 2 
orn 0 ( 0 V \ 1 
0V\ 2 ' 
,2 (9 2 " Ll 4 
rdOYrdOJ 2M 
T 
, df~ 
rdr 
(5). 
