372 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The two last equations of energy equilibrium are 
,-.2 
<f 
1 
2M 2 
1 
M 
0Y\ 2 " 
0s 
0 2 V 
06' 2 
1 
+ M 2 
1 
M 2 
dr) 
1 
M 
0V\ 2 ' 
ds) 
_1 
M 
. 2 0 2 V~ 
r A — - 
0«S 2 _ 
2 df 
rdr 
M 
dr ' 1 
2 
M 
. df 
rdr 
In this case they become 
lr 4 . A 2 AB 2Ar on 
J M 2 M aM^ ^ 
a * 
1 
2M 2 
From the first equation, 
= -E 2 l 
M u J 
q 2 ^ A \ 2 
a M, 
p 2 v n 
1 
r/0V\ 2 i 
_ 06 2 _ 
M 2 
A 0s/ J 
©; 
A^ 
rtM 
fa 2 ]- 
2 M 2 
w 
AB 
M ’ 
which shows that AB must be positive. 
These two equations can be easily solved for [^ 2 ] and [g 4 ]. We obtain 
b 2 ] = 
0V\ 2 ' 
ds) 
+ A 2 - ABM 
A -fa a 
~0 2 V' 
06‘ 2 
or 
fa 2 ]= 
a 
a 
M 
A 2 - ABM + 3M/A \ £\ 
m , 
d^\ 2 ~ 
dx l 
_ ^2 
m c 
M 
Hence 
^ Clfl^ Tlh^ 
m 1 + m 2 
dr<P l 
_0aq 2 _i 
0<F d% 
dxj p; 2 _ 
also 
q 2 _ A 
a M 
q 2 A x2 
a~ M 
3 
1 1 
1 cc 
CO l<^> 
1 1 
i 
PP 
i 
~d 2 Y~ 
L 06 2 J 
M 
r A , 0 2 a~i 
ds 2 _ 
AB 1 
M 2M 2 
w 
(32), 
(33); 
it is much simpler to use these expressions than the corresponding ones for 
[ q 2 ] and [g 4 ] . 
3 r/0v\ 2 ' 
W\ 2 ’ 
ds) _ 
+ A 2 - ABM = 
2 LV 06' ) 
+ A 2 - M 2 
0Y\ 2 
ds) J 
+ A 2 - 
A 
i - A Y" 
a M/ 
My 2 \ 2 
0V\ 2 " 
ds) 
a ) 
and this is a positive quantity, if we suppose that A and a are of the 
same sign. 
