360 Proceedings of the Royal Society of Edinburgh. [Sess. 
It is also necessary to determine the mean values of the differential 
coefficients of V with respect to x v x 2 , y v y 2 , z v z 2 , in terms of the mean 
values of the coefficients with respect to x, y, z, r, 0, <fi. 
Now 
r 2 = ( x i ~ x z ) 2 + (Vi ~ .Vo ) 2 + (h - z 2 ) 2 > 
sin 0 cos 
dx 
Xi Xn - r\ • , /) -I y n r\ ^1 <3r 
sin 6 sin <f> = Vi — -1A , cos 6 = — = 
???., 
and 
dr x-. — x 9 . n , rdO n , 
— — , — = — = sin V cos <p , — = cos 6 cos </> , 
dx } nij 4 - m 2 dx 1 r dx l 
r sin $d<f> 
dx 1 
= - sin y> , 
Let us differentiate with respect to x 1 and x 2 , and we easily obtain, 
upon taking mean values, the relations 
A*1 
H 
A t l/ >c 2 
03q x 2 “ 
_ V 
dx , 
m-, 
(% + ^ 2 ) 5 
2 
avy 
0j;/ 
(ra x + ra 2 ) s 
0Y\2' 
dx) 
0$ 1 0<& 2 
_ dx Y 0^ 2 _i 
m x m 2 
70 v y 
AW 
+ 1 
^ 3 
_ 1 
3 
0vy- 
W . 
0Y\2- 
0r / 
W)' 
+ ± 
+ ir 
(S’: 
n 
0vy 
03/ 
+ i 
Y-T 
\dr) 
dr 
7VY 
\drJ 
(12). 
K + %) 5 
If the second differential coefficients be written down and mean values 
be taken, we have 
A 1 ! 
fx 2 
0 2 $ x 
??q 2 
p 2 vi 
”0 2 V” 
+i 
02 V 
1 
d 2 f 
_ 0^ x 2 _ 
(ra x + ??i 2 ) 2 
_0X 2 _ 
_0r 2 _ 
_ 0S 2 _ 
3 
\_dr 2 \ 
m 9 2 
'0 2 V” 
+l 
”0 2 Y” 
+§ 
■0 2 V” 
1 
~d 2 f ~ 1 
r 
L0^ 2 J 
( m i + ™») 2 
dx 2 _ 
L 0r 2 J 
_ 0S 2 _ 
3 
_dr 2 
(13). 
A 1 • 
Also, since - — -V — () 
ox 2 ox 1 
0 = 
m yn 9 
"0‘ 2 V" 
-02y- 
dr 2 
(m x + m 9 ) 2 [_0x 2 
From (12) we obtain the relation 
m 9 /xp 
+ 
~d 2 /~ 
dr 2 
-02y- 
0s s 
fd® A 2 
/0<L>\ 2 
/ \ 
r0<j>, 
03> 1 
_\0aJ 1 ) _ 
— m-yx. 2 
_\W _ 
_0£ 1 
0X 2 _ 
= 0 
(14). 
(15) 
when the masses are unequal ; and using the equations (11) we have 
0V\2" 
dx) 
m, + m 9 „ 
1 
m 1 — m 2 
d<S> l 
ni-i -H m 9 9 
>2 
_ \ 02’ 1 / J ?/q - m 2 
/0vy 
_ ... m 2 W 
r wi 
+ »iW 
\ 0s / _ 
m 1 2 — m 2 2 
L\0^ 1 / J 
wij 2 — m 2 2 
03y 2 
dx.-) 
0$, 
0£ o 
( 16 ), 
when m 1 4= m 2 . Also 
/0Y\2- 
\ 0r / 
df\ 2 ' 
dr, 
+ h 
Ivy 
' 08 / 
