FIGURE AND STABILITY OF A LIQUID SATELLITE. 
173 
last term inside the bracket is also negative. Hence if S J is negative, f\^j 1S 
positive, and vice versd. But if S (^j is negative, r is greater than r 0 and the two 
masses are too far apart to admit of junction, and vice versd. 
Therefore if for a given solution for detached masses / ( F j is positive, the masses 
are too far apart to admit of junction by a weightless pipe, and if it is negative they 
are too near. 
When in the general case we form a function / (a/r), such that when the ellipticities 
of the two masses are annulled, it reduces to the above function, its sign will afford 
the criterion as to whether the masses are too far or too near to admit of junction by 
a thin neck of liquid. I return to this subject below in § 13. 
The solution of the problem when the two masses are constrainedly spheres is so 
curious that it seems worth while to consider its stability. This may be done by the 
method of § 2. 
The system depends on two parameters r and X, and the stability will depend on 
three functions, which are defined as follows :—- 
\r. 
r) 
3 2 V x 2 3 2 I_oj 2 (dl\ 2 
ar 3 + ar 3 I \dr) ’ 
, a 2 F ! 2 a 2 i w 2 a/a/ 
“ 3r3X + 2<u a rax I dr 3X’ 
{X,X} 
a 2 F 12 ^_w 2 /a /\ 2 
ax 2 + 2 W ax 2 / \ax/' 
These functions correspond to a n , a 12 , a 22 of (2) in § 2, and we see that for secular 
stability {r, ?•} and {X, X} must be negative, and 
A = {r, r} {X, X}-[{r, X}] 3 
must be positive. 
Without giving the details of the several differentiations, I may state that it we 
write 
F- 
a 3 - 
F b + f 1 9 
so that H is essentially positive, we find 
lArAl 
H 
{r, r} 
H 
r 2 
&G-FL■ 
5 ^ r.2 ’ 
cL 
i r > k } 
H 
+ l + 3^ 
a 
