198-200]
Pear-shaped Configurations
217
Write f, rj, f for — X , ^~r , so that P is a function of r) } and £
Qj "HA« O'* f A. C 1 4“ A
and introduce a differential operator D defined by
D _(l 1 \o i (1 l\jP ( 1 1 ] d 8
V« 2 a 2 + A./ 3| 2 + \b 2 b 2 + A/ dr) 2 \c 2 c 2 + A/ 3£ 2 '
Then I have shewn* that the gravitational potential of a uniform mass
of density p bounded by the surface
5 + |i + ^- 1 + «P.- 0 (199'5>
V—rtpabcV (1996)
Jo A
where 4 , = P-(i/) PP+A(i/) ! P>P-^ 3 , (i/)*P*P + ...
- ie [№ - (i/) 2>P> + Hi ffVP'- T 's(i/)*P‘P' + • • ■]
+ T fc*[P«P>- A/P*P* + jfe/’P'P*- ...] + (199-7).
Here F gives the potential either at an internal point or at the boundary
of the mass, A as before being given by
A 2 = (a 2 + A) (b 2 + A) (c 2 + A).
200. For the surface (199‘5) to be a possible boundary for a mass of
rotating liquid, it is necessary that
V + l© 2 ^ + y 2 )
should be constant over this surface.
If the term eP 0 represents a third zonal harmonic distortion, we may
assume
P = f (a£ 2 + fir ) 2 + ry £ 2 -f K )
For small displacements from the ellipsoidal configuration we may neglect
e 2 , and obtain from equation (199‘7)
~Uj + B + h~ 1 )(^I + ^5 + ^)z + ,£ l (200 ’ 2) '
so that we may write
f 00 (b
Jo A = X+ + a%Z * + a *) (200 3).
Thus the distortion introduces additional terms of degrees 3 and 1 into
the potential.
* Problems of Cosmogony and Stellar Dynamics, chap. iv. or Phil. Trans. 217 A (1916), p. 7.
J 14