FICtUEE of EQITTLTBEIUM OF A ROTATING MASS OF LIQUID. 
25:1 
PART I. 
Analytical Investigation. 
§ 1 . Method of Procedure. 
The pear-shaped figure is a deformation ol the critical Jacobian ellipsoid, and to 
the first order of small cpiantities it is expressed hy the third zonal harmonic vfitli 
respect to the longest axis of the ellipsoid. In the higher approximation a niunher 
of other harmonic terms will arise, and the coeflicients of these new terms will be ot 
the second order of small cpiantities. The mass of an harmonic inecpiality vanishes 
only to the first order, and it can no longer be assumed that the centre of inertia of 
the pear coincides with the centre of the ellipsoid. 
In order to define tlie pear, I descrilie an ellipsoid similar to and concentric with 
the original critical Jacobian; this new ellipsoid is taken to be sufficiently large to 
enclose the whole of the pear. It is clearly itself a critical Jacobian, and I adopt it 
as the ellipsoid of reference, and call it J. I call the region between J and the 
pear R. The pear may then be defined hy density + p throughout .7, and density — p 
throughout R. 
O 
If k is the parameter which defines J, its axes are expressed in the notation of 
“ Harmonics ” liy Zvy, h [vq — l)% h I vq — 
1 -f/3 
1 - 
\h 
or in the notation of the “ Pear- 
shaped Figure ” by /c/sin 7 cos ; 8 /sin /3, k cos y/sin ^ 8 , where sin ^ = k sin y. 
Now let Sd denote any surface harmonic, so that aSV' is the same thing as 
[^A'’ (p) 01’ Pj (p)] X [(•Ti' {(fi) or Cj (<^)]. The third zonal harmonic deformation aaoII 
then be eS^ or CP 3 (p) C 3 {4>), where e is of the first order of small cpiantities. On 
account of the symmetry of the figure, the new terms cannot involve the sine 
functions ^ or S, and moreover, the rank s must necessarily be even. 
Suppose that the new terms are expressed by 'S/fSf for all values of ^ from 1 to 
infinity, and with s ecpial to 0 , 2 , 4 . . . / or ^ — 1 . Then all the 7’s are of order e", 
excepting which is zero. 
We have seen in “ Harmonics,” § 11 , that ifpo denotes the perpendicular from the 
centre of the ellipsoid Vq on to the tangent plane at p, (f), the equation to a harmonic 
deformation of the ellipsoid is 
L = 2.,S' 
Since this equation may he written in the form 
77- 
VfC — 
+ 
LA (vd - 1 ) ^ Idv, 
1 - ^ 
+ 70 o — 1 + 
