208 The Configurations of Rotating Liquid Masses [ch. vm 
and more generally put 
as can also be seen from the circumstance that V 2 Vi must be equal to — 4nryp. 
We may also note the formulae 
all of which are easily verified by algebraic transformations. 
190. The necessary and sufficient condition that the standard ellipsoid 
(189T) shall be a figure of equilibrium for a homogeneous mass of density p 
freely rotating with angular velocity m is that 
shall be constant over the boundary, V { being given by equation (189‘7). 
Now consider the function 
where 6 is a constant, as yet undetermined. Operating with V 2 , we find that 
this function will be a spherical harmonic if 
With this value for 6 , expression (190‘2) becomes harmonic. The necessary 
and sufficient condition that the standard ellipsoid (189T) shall be a figure of 
equilibrium is that this function shall have a constant value over the boundary. 
Since the function is harmonic, this is equivalent to the condition that the 
function shall have a constant value throughout the interior of the ellipsoid. 
We must accordingly have 
(189-6), 
this equation assumes the form 
Vi = — Trypabc (J A x 2 + J B y 2 + J c z 2 — J) 
It is easily verified that 
(189-7). 
(189-8), 
.(189-91), 
.(189-92), 
.(189-93), 
Vi + £<u 2 (<x 2 + y 2 ) 
(190-1) 
Vi + !<o 2 {a? + y 2 ) + dvypabc 
\a 
'x 2 y 2 z 2 
v‘ + b' + ? 
(190-2), 
a condition which can be satisfied by assigning to 6 the value 
(190-3). 
— Trypabc (J A x 2 + J B y 2 + J c z 2 — J) + ^o) 2 (x 2 + y 2 ) 
= cons.,