238 
SIR G. H. DARWIN ON THE 
stability. I have only determined the ellipsoid of minimum radius vector in two 
cases, viz., when X = 0 - 8 and 1*0. 
When X = 0*8 I find minimum radius vector to be r = 2’36, whereas limiting 
stability occurs for r =2*50. When X = 0'8, r = 2*36, the ellipsoids are determined 
by the following data :— 
whence 
y = 54° 20', k = sin 71° 51', 
a = 0-619, 
b = 0-675, 
c = 1-063, 
r = 46° 10', K = sin 64° 20'; 
A = 0-705, 
B = 0-774, 
C = 1-018. 
When X = 1 , the minimum radius vector occurs when y is about 54° and is then 
equal to 2 - 343, whereas limiting stability occurs when r = 2-514. I have not 
computed the axes, since it suffices to learn that there is an ellipsoidal solution when 
the two masses are considerably nearer than is consistent with stability. 
As y increases, the ellipsoids get longer and longer, and it is interesting to inquire 
whether they increase in length with such rapidity that, notwithstanding the increase 
of v, the interval between the two vertices continues to decrease, or whether the 
increase of r annuls the simultaneous increase of c. 
The following table of values, computed with fair but not extreme accuracy, affords 
the answer to this question. 
7- 
r. 
X = 1. 
c. 
r - 2c. 
Differences. 
44° 
2-429 
0-962 
0-506 
46° 
2-396 
0-983 
0-430 
-76 
o 
oo 
2-370 
1-007 
0-356 
-74 
50° 
2-354 
1-034 
0-286 
-70 
52° 
2-345 
1-064 
0-217 
-69 
54° 
2-343 
1-097 
0-149 
-68 
56° 
2-350 
1-134 
0-082 
-67 
58° 
2-367 
1-176 
0-015. 
-67 
The differences of r—2c hardly diminish at all, and it is clear that the next entry 
would be negative, or in other words the two figures would overlap. 
These results are obtained on the supposition that our approximation is adequate, 
but the small terms £, e, r/, which are really infinite series, show signs of bad con¬ 
vergence as y increases. I think it probable that when we get to these extreme 
