358 Proceedings of the Royal Society of Edinburgh. [Sess. 
The equations representing these conditions will be called equations 
of energy equilibrium, as in the Arch. Neerlandaises to which reference 
has been made. The equations of energy equilibrium are therefore 
/0V\ 2 ' 
Vc§ 
= ( TOj + Wl 2 )[flS 2 ] 
02V 
dx 2 
/.2 
-rA l 
*•2 
1 
M 2 
1 
M 
djy 
dr) 
' 2 0 2 Y' 
3 a? 
1 
" . <2 0 2 V 
2 " 
M 
. 0r 2 _ 
M [j 
1 
~ (d\\ 2 ~ 
1 " 
M 2 
_\ 0S / _ 
M _ 
df 
rdr _ 
2 d f 
rdr 
( 10 ). 
j 
f)2\7- ;/2y 32y 
(9) In the case of a Newtonian field — 9 -f = 0, and therefore 
our assumptions would require that 
dx 2 dy 2 dz 2 
~0 2 Y~ 
0 2 V 
0 2 Y 
. 0a£_ 
M 2 - 
_ dz 2 _ 
0. 
The first of the equations of energy equilibrium shows that these assumptions 
ray- 
are impossible in this case, and likewise in the case where 
Adx 2 . 
is negative; 
so that we must exclude this latter case and that of a Newtonian external 
field of force. 
(10) It is next necessary to obtain expressions for 
We know that 
v = /VE(lj> V\> O + Vv %)+/W • 
Therefore, 
0Y _ d% 
dx ^ l dx ^ 2 dx 2 
0Y 0$. 0<J> 9 
0Y 0$, 0<3> 9 
— Zfip, P'2 - 
CZ 0Z 1 CZ 2 
~ 0 2 Y~ 
r/ 0 V\ 2 i 
_ 0 a) 2 _ 
A 08/ - 
, etc. 
0V __ df~ 
0 ?" dr ni 
1 ( ( 8$, 0<I> 9 \ . a , 
, , r, % ; os<A 
, / 0$. 0<D 2 \ 03> x 03>,\ , 
+ ( m 2 /x 1 — - ??? 1 /x 2 — ■ 4 j sin 6 1 sm 9 + ( m 2 ju, 1 — ± - ??q/x 0 — ^ ] cos 6 
dy l 
S;/„/ 
0^ 1 
02 f 
0V _ 
rdO rn 
1 { / 0$, 0<3> 2 \ , , 
, + m, H ‘"ay ^ 
, / 04>, 04>,\ a ■ , ( S'**, 0*>,\ ■ O 
+ ( - 1 - ) cos 0 sin </> - ( m.yy-— U - ) sin 6 1 ] 
0Y 
1 
r sin Odcj) m l + ra. 
0$, 0<$ 2 \ • , 
+ 
1 
mj + m 2 
0^1 0$ 9 , , 
m o/*,— J - m.Uo — * cos 
‘ s 2/i W 
