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