370 Proceedings of the Royal Society of Edinburgh. [Sess. 
These are two of the equations of energy equilibrium in this case ; the 
third is an awkward equation involving [g 4 ], namely, 
• o 
r 
T 
T 
= [ 2 2 ]-! 
j 
_ x m i +m l2 I 
m x m 2 ) 
m Y m 2 
Pi 
r iwi 
_ \H 
0$j 
03>O 
m 1 
_\dx 1 J _ 
m 2 
_0aq 
dx 2 _ 
/ 
(30). 
The condition that Maxwell’s Law should hold good becomes in this case 
A 2 [r 2 ] - 3/q^ 2 
0 < J> 1 0<J> 2 " 
. ?)X l CX. 2 _ 
SV 
0-Tj 2 
3A 
Pi 
\ox 1 / J 
+ /X 2 
0$! 0$ 2 
_0aq dx 2 _ 
which may also be put in the form 
0 2 $ 
_ _ 
— 3ytq 
0^y 
Lva^ 
3fXr, 
'0$! 0^> 2 
dx l dx 2 _ 
) 
A + 
0 2 3> 
0^ x 2 J J 
u 
o 
• (31). 
Now, it has been shown that if Maxwell’s Law holds good, it is necessary 
that 
i I 
^ rS 
2 _ 
Splp2 
0<h 1 0<3> 2 
_dx x dx 2 _ 
d\f 2 df 
__dr 2 r dr _ 
should be positive. In this case, therefore, 
A[r 2 ] 
0<L 0$ 2 ~ 
dx x dx 9 _ 
_ *^P\p2 
A 
must be positive. 
If A is negative, it is clearly necessary that 
_ 3/x^ 0 ^ 0$ 2 
A L 0.c x dx 2 
or that ya ly a 2 
■a#! a$ 2 
L dx x dx 1 
should be positive, provided that Maxwell’s Law 
holds good. The condition is that 
l _dx 1 dx 2 _j 
should be negative in the 
case of two oppositely electrified equal masses, when A is negative. 
The question of stability will now be considered. From the general 
results, 
| 2 [ife^)]=^pV] + § 
0V\ 2 ' 
06 */ 
• - [V / 2 + r 2 ] 
M U J 
06* 2 
where we proceed as in the case of the law of the inverse square. 
Let [i(q 2 + r 2 )] = T 1 -he, where T 1 is the value of [o(<7 2 + 7 ‘ 2 )] determined 
from the equation of energy equilibrium, and e is a small variation in it. 
Then 
2e f 
"0 2 V~ 
) 
