FIGURE AND STABILITY OF A LIQUID SATELLITE. 
183 
On substituting for q s its value, I find 
1 + 
) = ^3 
7 (1 + k 2 ) + 2D F 5{11 + 10k 2 +11k 4 +4(1 + k 2 ) D } F 
14k 3 
7 
168k 
Whence 
5 [©»+©»*)- 
]+k 2 £ 2 5(11 + 1 Ok 2 + 11 k 4 ) F 
2k 2 r 2 168k 4 
1 F 20 (1 + k 2 ) F 
7 k 2 r 2 168k 4 r 4 *" 
,4 * * ' 
Substituting these values in (6) we have for the potential of M/\ at the surface of 
the ellipsoid, as far as concerning terms involving x, y , z, 
F 
k) ' Jc 2 F' r 3 
1 + 
3 (3 + k 2 ) F 5 (5 + 2 k 2 +k 4 ) F ' 
14k 2 r 2 56k 4 r 4 
V 2 F 
2 F • r 3 
, £ 
F ‘ r 3 
1 + 3(1 +3k 2 ) F 5(5k 4 +2k 2 +1) F 
14 k 2 r 2 
56 k 4 
\ ! 3(1+k 2 )F ! 5 (3 + 2k 2 + 3k 4 ) k 
56k 4 r 
7 k 3 i* + 
( 10 ). 
If the system be rendered statical by the imposition of a rotation potential, we 
must add to the above such a potential, and that of the ellipsoid itself. 
The expression for the internal potential of an ellipsoid v 0 is given in (65) of 
“ Harmonics ” ; it is 
3d/ J <2to (^o) _ x 2 Qf (v 0 ) _ y 2 ad (+i) « a <Qi (v 0 ) 1 
2k (v 0 ) F Pd („ 0 ) F f d ( Vo ) F 13, („ 0 ) J * 
I will now introduce an abridged notation which was used in some of my previous 
papers, as follows 
Pfi w Q* 1 W = Ad, pd (v 0 ) ad w = fi w ©i (v 0 ) = 
Then, since 
Pd {y o) = /y/(V- ^), IPd (^o) = x/K-l), |3i (u>) = u>, 
we may, on omitting the term independent of x, y, z, write this potential in the form 
3 M 
2k 
x 
Ad + 
y 
F(V-1/k 2 ) 1 F(v»- 1) 
ad+ 
F ^ 2 
SCi 
( 11 ) 
The rotation with angular velocity takes place about an axis parallel to x through 
the centre of inertia of the system, which consists of two masses M and M/X distant r 
from one another. Hence the rotation potential is 
w 
r+ 
3 M\co 2 F(y 2 + z 2 \ 9 (FFr z] a> 2 r 2 
2k I 3/1/\ F / 3 (1 + X) M' k\ 2 (1+ \) 2 • 
( 12 ). 
