FIGURE AND STABILITY OF A LIQUID SATELLITE. 
169 
Thus the variable part of the lost energy may be written 
l(^) 2 
~3 2 F 
1 2 (FI 2, 
/0A 2 \1 
+ 8e8f 
ra 2 F j 2 
(FI 
2 0/0 A1 
Lae 2 4 
2 W \0e 2 /( 
v 0e/ /_ 
[dedf 
\0e3/ 
/ 0e 0/A 
+ i (S /) 2 
VV, 1 2 /0 2 / 
-2 + 2 W 
W 
2 fdl 
a/ 2 i\dj: 
\ 2 
This is a quadratic function of the departures of the ellipticities from their 
equilibrium values, and the form is obvious which the result would have if there 
were any number of ellipticities. 
Since the condition for secular stability is that the energy shall be a minimum, 
the lost energy must be a maximum, and therefore this quadratic function of 
Se, 8f &c., must always be negative in order that the system may possess secular 
stability.* 
If F is a quadratic function of n variables, x x , x 2 , x 3 , &c., so that 
F = « u £Ci 2 + 2a 12 . x x x 2 + 2a 13 x 1 x 3 + ... 
+ a 22 x 2 + 2a 23 x 2 x 3 +... 
+ a^x 3 2 +..., 
it is known that the condition that F shall always be negative for all values of the 
variables is that the series of functions 
t*12j 
tt 13j 
shall be alternatively negative and positive. 
Since we might equally well begin with any one of the variables, it follows that 
a m a 2 ‘>-. ■ • • must all be negative; also a 12 — a n a 22 , ci 13 —a n a 33 , a 23 — a 22 a 33 ... must all 
be negative if F is always to be negative. 
Now, suppose that F is the function of lost energy for a system with n+ 1 degrees 
of freedom, but that a constraint destroys one of the degrees. If the system has 
secular stability, the n determinants must have their appropriate signs, and when the 
constraint is removed, the new additional determinant must have its proper sign in 
order to secure secular stability. It follows that stability can never be restored by 
the removal of a constraint if the system was unstable when the constraint existed ; 
but stability may be destroyed by the removal of a constraint. 
* This result is also given, but with less detail, in my paper on “ Maclaurin’s Spheroid,” in ‘ Trans. 
Amer. Math. Soc.,’ vol. 4, No. 2, pp. 113-133 (1903). 
VOL. CCVI.-A. Z 
ct n , 
Ct\2 
<^125 
C^22 
«13 
Oj 22 , 
C*23 
tt 23) 
^33 
