FIGURE AND STABILITY OF A LIQUID SATELLITE. 
237 
§ 24. Figures of Equilibrium of Two Masses in Limiting Stability. 
In this case both masses are liquid. We saw in § 2 that when one of the masses 
is infinitely small, stability only exists when the two are infinitely far apart. When 
X is less than 0'5 we may obtain fair results from the approximate investigation of 
§11), but for greater values of X it is necessary to employ the laborious method 
adopted in determining Roche’s ellipsoid. 
When X = 0’5 I obtain the following approximate results for the two figures :— 
r = 2-574 
a = 0-62, A = 0-81, 
b = 0-66, B = 0-87, 
c = 0-81, (7=0-95. 
It is probable that the value of r derived in this way is too small. 
For X = 0-4, I found r = 2*59, but did not calculate the axes. 
The only other case in which the problem has been solved is for equal masses, when 
X = 1. The two ellipsoids are exactly alike, and I find limiting stability occurs for 
the following values :— 
y =36°18', fc = sin 59° 33', r = 2-638, a = 0723, 6= 0771, c = 0-897 ; 
and 
This is illustrated in fig. 5. 
r—2c = 0"844. 
xb= 771 x 6=771 
§ 25. Unstable Figures of Equilibrium of Two Masses. 
When X = 0 the figure of minimum radius vector, when the larger body is rigid, 
is also that of minimum angular momentum, but for larger values of X there is an 
ellipsoid considerably nearer to the larger body than that which possesses limiting 
