368 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The Law of the Inverse Square. 
(16) The case will now be considered in which the law of force between 
the particles is that of the inverse square of the distance. 
Let f(r) = — — where X is a constant. 
/y* ' 
Then 
dj \ 
dr 
d?[_ 
~dr 2 
2A 
so that 
and 
r t7/ i 
= A 
:i ~ 
ryyi 
= A 2 
_rdr_ 
_r 3 _ 
LW; _ 
_r 4 _ 
ri 2 / 
= - 2A 
1 
_dr 2 _ 
— i 
CO 
1 
The equations of energy equilibrium become 
[U = 
/^i 
7— Y 2- 
_\0(T 1 / _ 
+ /x 2 
rd&, 
i 
_0^! 
dx 2 _ 
(?/?! + m 2 ) 
0 2 $ 1 
_0iC 1 2 _ 
\ 
r 2 r 
r L 
T 
r 2 
X 2 
M 2 
r 4 
2Ar 
- m [2 ] 
+ 2 hM 
> (27) 
III n 
r vq p^ 1 I /q j Ah 
Lory J M ] wq 
0<£p 2 ' 
_\0£C 1 / _ 
/y 
??2 n 
0^ 0$ 2 
0aq 0cc 2 _, 
= ^T// 2 ! 
7/0 
0-<t> 1 
i_0aq 2 _, 
“ / x i ) Aq 
M I m. 
V 0xy _ 
Ah 
//io 
'0^ 0<L, 
_0a?! 0rq_ 
/ 
Important simplifications occur, however, when we consider Maxwells 
Law of Equipartition. 
The condition, in the general case, that the Law holds good is 
# x2_ 
dr 
'Vi/q 
03^ 0<L 
yaq 0^ 2 _ 
y 2 <j>^ 
l _0Z 1 2 _ J 
- I *[©' 
+ /U 
0^3 0^2 
_0.r 1 0x 2 _ 
t_zy 2 df 
dr 2 r dr 
Now, in this case 
a 2 ^ 
d 2 f 2 #• 
—, + - — 0, 
a/ ,ij r ar 
and 
Lda;p 
simply 
cannot vanish ; so that the condition for equipartition becomes 
dfV 
dr J 
3/X]^X 2 
0 < J > 1 0^> 2 
_dx 1 dx 2 _ 
