362 Proceedings of the Royal Society of Edinburgh. [Sess. 
Maxwell’s Law of Equipartition. 
(12) Maxwell’s Law of Partition of Kinetic Energy requires that, if the 
kinetic energy of the system be expressed as a sum of squares, the mean 
values of these squares taken over a large number of systems distributed 
according to a certain permanent or stationary law shall be equal. 
In the case of this system, the kinetic energy is equal to 
Mm, + m v )(x 2 + y 2 + z 2 ) + /.;-2 + r 2 Q2 + r 2 s j n 2 5)^2 \ _ 
“ 1 *m l + m 2 
Therefore, since 
[r‘ 2 & 2 ] = [r 2 sin 2 Of 2 ] = pq 2 ] , 
it is necessary that 
(m 1 + m 2 )[x 2 ] = 7n ' m<2 [r 2 ] = J [q 2 ] 
m 1 + m 2 J m 2 + m 2 
or 
(m l + m 2 ) 2 [a? 2 ] = m 1 m 9 [r 2 ] = pn^mpq 2 ] , 
if Maxwell’s Law of Partition holds good. In this case, the equations of 
energy equilibrium show that 
M 2 
dp 9r 
dr, 
M 
<r 
i df 
rdr 
+ Vi ,( fh 
M l m 1 
03^ 
y 2 
0<lq 0$ 
02^ / J m 2 0flq 02;. 
1 
M 
_/q 
rfr 2 _ 
TOj _ 
/ 2 1 . o\0 2 ^] 
(s +r) &M 
0 
or 
M 2 
or 
M 2 
or 
1_ 
M 
i 
r>w 
O 1 
1 
2 
df 
_\drj 
M 
__ rdr_ 
( m i m f)" r,;,‘21 1 3/q j I 1 1 
Wl 
/q 
0<K 0<L> 
\02? 1 ) J 
m 2 
_02?j 02? 2 _ 
_ J_ ( TO ! + m 2 ) 2 U21 
M m 1 m 2 L J 
dff 
dr 2 
Vi 0% + y ^ 2 ) 2 m.2-| 
?rq L ' J 
m 1 m 2 
~ 0 2 <1> 1 ' 
_0aq 2 _ 
= 0 
MjA 2 ] 3/q j /q 
dr ) _ M I TOj 
0^1 
02 ?-, 
_ /*2 
■0$ T 0<K~ 
m 2 L02? 1 02? 
( »i, + «» 2 ) 2 r :t 2i J 2 
m 1 m 2 
M 
PL 
rdr 
+ 
M 
dpr 
dr 2 
+ 3qi 
■0 2 ^ 
m 1 _02? 1 2 _ 
vy 
.dr) 
1 
~ 0 2 <P 1 
_ 02' 1 2 _ 
+ 3q^ 
/q 
fh 
m, 
0iy 9_ 
02 ?, 
s<3> A 2 i , r 0^ 1 0^2 
0q)j +fti 
or 
vy 
dr) 
3 /q/q 
jU ., 0$ 1 0$ 
m 2 L02?i 02?, 
^ 0$ 
L02?2 02? 
03q 0$ 2 "| ( f0 2 $ ] 
02? x 2 _ 
0<Eq X 9_ 
0.r ( 
2 
9 -J 
2 _ 
M 
df 
rdr 
r-(Pf 
dr 2 
+ 
M 
+ 
3/q 
m, 
~d 2 p 
_02?y 
»- 
0 
_02?j 02? 2 
= i /q 
+ /x Q 
0<lq 0$., 
_02?j 02? 9 _ 
W + 2 PL 
dr 2 rdr 
( 22 ). 
