OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
317 
If we add together the first two of these, and avail ourselves of the property that 
^ is homogeneous of degree — I, we easily prove that the stationary value of E is 
E = 
+ a 
da 
Since the potential of the ellipsoid must satisfy Poisson’s equation 
41so 
d^ 
' I ' 1 " 
ada hdb cdc ubc 
, dT' , dSir 
da- 
dc 
dc 
By means of these and two out of the three equations (34), we may eliminate the 
diflerentials of and writing p for the density find 
u>~ 
(o + TV + -w) — 6 
\a 
iTrp 
.( 35 )- 
I do not happen to have seen this form for the angular velocity of Jacobi’s ellipsoid 
in any book. 
It is easy also to show that the stationary value of E may he written 
P — - 9 - 1/3 _ 
+ + 1 + _ 4 
^ - 2 
W + 
abc 
/I . 1 
We may now express the potential, say V, of tlie system entirely in terms of 'V 
and a “ , for 
da 
V = f J/ 
^ a da ^ V 3J/ d I da , 3if 
— ^1/ 
— 4 m 
■ , d'^ (x^ 1/ ^ 
+ “IT (a* + P + 
We thus verify that V is constant over the surface of the ellipsoid. 
Let g denote the value of gravity at the surface. Then if dn be an element of the 
outward normal, g — — 
dV 
dn 
Since 
dx px dy py 
dn ~ cd ’ ~ b^ ’ 
& 
dn 
pz 
