174 
SIR G. H. DARWIN ON THE 
From the equation 
a 3 . 1 5 F' a 
13 + AT7T/Z + '° j - °> 
& 7 
we find, after some reductions, 
1 + 3 ^, 
a 
3 r 2 , v d 1 F' 
1 3x °®7 
a 
On substituting 
reduces to 
for F", G' their values in terms of X, I find that this 
(I — V 3 ) 2 r 2 
\ 1;3 (1 +xy a 2 ’ 
expression 
an essentially positive quantity. 
On substitution in A I find 
The factors outside [ ] are essentially positive and do not affect the sign of A, and 
2 
it is clear that A can only be positive if § G—F— 2 is positive. But A must be positive 
cl 
^2 
for secular stability; hence stability can only be secured by f G—F— 2 being positive, 
cl 
and it is not necessarily so secured. But if this function is positive, so also is {r, r}, 
and if this last is positive the system is unstable. Hence stability is always 
impossible. As a fact, in all the solutions given above {r, r} is positive, and we 
should have to move the spheres much further apart to make it negative, and 
therefore on this ground alone the system is always unstable. But A is sometimes 
positive and sometimes negative and vanishes for a certain value of X. As the 
vanishing of A puzzled me a good deal, I propose to examine the matter further. 
Before doing so, however, I will show that the instability of the system may be 
concluded from other considerations. 
It was proved in § 1 that two spheres, unconnected by a pipe, are in limiting- 
stability when their distance apart is given by 
^=A(i + w )( i + x)V3 = f |. 
This is the condition that {r, r } should vanish. 
When X is zero the two spheres in limiting stability are infinitely far apart, and 
when X is unity they are as near as possible, and r = 1738a. 
Now the table of solutions in the case where the two are connected by a pipe shows 
that they are furthest apart when X is unity, and that then r = l‘ 666 a. 
The removal of the constraint of one degree of freedom may destroy stability, but 
